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Ideas Math Geometry Answers Ch 5 Congruent Triangles in a simple and easy to understand language. Boost your math skills and have a deeper understanding of concepts taking help from the Congruent Triangles BIM Book Geometry Solution Key. Prepare using the Big Ideas Math Book Geometry chapter 5 Congruent Triangles Answer Key and get a good hold of the entire concepts. Clarify all your doubts taking help of the BIM Book Geometry Ch 5 Congruent Triangles Solutions provided. Simply tap on
the BIM Geometry Chapter 5 Congruent Triangles Answers and prepare the corresponding topic in no time. Big Ideas Math Geometry Congruent Triangles Solution Key covers questions from Lessons 5.1 to 5.8, Practice Tests, Assessment Tests, Chapter Tests, etc. Attempt the exam with confidence and score better grades in exams. Find the coordinates of the midpoint M of the segment with the given endpoints. Then find the distance between the two points. Question 1. Question 2. Question 3. Solve the equation. Question 4. Question 5. Question 6. Question 7. Question 8. Question 9. Question 10. Monitoring Progress Classify each statement as a definition, a postulate, or a theorem. Explain your reasoning. Question 1. Question 2. Question 3. Question 4. Exploration
1 Writing a Conjecture Work with a partner. a. Use dynamic geometry software to draw any triangle and label it ∆ABC. b. Find the measures of the interior angles of the
triangle. Hence, from the above, The measures of the interior angles are: α = 62.1°, β = 64.1°, and γ = 53.8° c. Find the sum of the interior angle measures. d. Repeat parts (a)-(c) with several other triangles. Then write a conjecture about
the sum of the measures of the interior angles of a triangle. Hence, from the above, We can conclude that the conjecture about the sum of the measures of the interior angles of a triangle is: The sum of the internal angle measures of a triangle is always: 180° CONSTRUCTING VIABLE ARGUMENTS Exploration 2 Writing a Conjecture Work With a partner. a. Use dynamic geometry software to draw any
triangle and label it ∆ABC. Hence, from the above, We can conclude that the vertices of the triangle are: A, B, and C b. Draw an exterior angle at any vertex and find its measure. Hence, From the above, We can conclude that The external angle measures of the triangle are: α = 310.7°, β = 299.3°, and γ = 290° c. Find the measures of the two nonadjacent interior angles of the triangle. Hence, from the above, The angle measures of two non-adjacent sides are: α = 70°, β = 60.7°, and γ = 49.3° d. Find the sum of the measures of the two nonadjacent interior angles. Compare this sum to the measure of the exterior angle. e. Repeat parts (a)-(d) with several other triangles. Then write a conjecture that compares the measure of an exterior angle
with the sum of the measures of the two nonadjacent interior angles. Hence, from the above, We can conclude that The external angle measure of a vertex for a given triangle = 360° – (Internal angle measure of a vertex that we are finding the external angle measure) The sum of the internal angle measures of the triangle is: 180° Communicate Your Answer Question 3. Question 4. Lesson 5.1 Angles of TrianglesMonitoring Progress Question 1. Question 2. Question 3. Answer: The given figure is: We know that, The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles From the given triangle, The exterior angle is: (5x – 10)° The interior angles are: 40°, 3x°, ∠1 So, (5x – 10)° = 40° + 3x° 5x° – 3x° = 40° + 10° 2x° = 50° x = 50° ÷ 2 x = 25° So, The interior angles are 40°, 3 (25)°, ∠1 = 40°, 75°, ∠1 We know that, The sum of the interior angles of a triangle is: 180° So, 40° + 75° + ∠1 = 180° 115° + ∠1 = 180° ∠1 = 180° – 115° ∠1 = 65° Hence, from the above, We can conclude that the value of ∠1 is: 65° Question 4. Answer: The given figure is: We know that, The sum of the interior angles in a triangle is: 180° From the given figure, The interior angles of the right-angled triangle are: 90°, 2x°, and (x – 6)° So, 90° + 2x° + (x – 6)° = 180° 84°+ 3x° = 180° 3x° = 180° – 84° 3x° = 96° x = 96° ÷ 3° x = 32° So, The measure of each acute angle is 90°, 2x°, (x – 6)° = 90°, 2(32)°, (32 – 6)° = 90°, 64°, 26° Hence, from the above, We can conclude that, The measure of each acute angle is 90°, 64°, and 26° Exercise 5.1 Angles of TrianglesVocabulary and Core Concept Check Question 1. Question 2. Monitoring Progress and Modeling with Mathematics In Exercises 3-6, classify the triangle by its sides and by measuring its angles. Question 3. Answer: Question 4. Answer: The given figure is: We know that, “|” represents the “Congruent” or “Equal” in geometry So, From the given figure, We can observe that all three sides of the given triangle are equal We know that, If a triangle has all the sides equal, then the triangle is called an “Equilateral triangle” Hence, from the above, We can conclude that the ΔLMN is an “Equilateral triangle” Question 5. Answer: Question 6. Answer: The given figure is: We know that, If any side is not equal to each other in the triangle, then the triangle is called a “Scalene triangle” The angle greater than 90° is called as “Obtuse angle” An angle less than 90° is called an “Acute angle” Hence, from the above, We can conclude that ΔABC is an “Acute scalene triangle” In Exercises 7-10, classify ∆ABC by its sides. Then determine whether it is a right triangle. Question 7. Question 8. Question 9. Question 10. In Exercises 11 – 14. find m∠1. Then classify the triangle by its angles Question 11. Answer: Question 12. Answer: The given figure is: We know that, The sum of interior angles in a triangle is: 180° So, From the above, The interior angles of the given triangle are: 40°, 30°, ∠1 Now, 40° + 30° + ∠1 = 180° 70 + ∠1 = 180° ∠1 = 180° – 70° ∠1 = 110° We know that, The angle greater than 90° is called an “Obtuse angle” Hence, from the above, We can conclude that the given triangle is an “Obtuse angled triangle” Question 13. Answer: Question 14. Answer: The given figure is: We know that, The sum of interior angles in a triangle is: 180° So, From the above, The interior angles of the given triangle are: 60°, 60°, ∠1 Now, 60° + 60° + ∠1 = 180° 120 + ∠1 = 180° ∠1 = 180° – 120° ∠1 = 60° We know that, An angle less than 90° is called an “Acute angle” The triangle that all the angles 60° is called an “Equilateral triangle” Hence, from the above, We can conclude that the given triangle is an “Equilateral triangle” In Exercises 15-18, find the measure of the exterior angle. Question 15. Answer: Question 16. Answer: The given figure is: We know that, An exterior angle is equal to the sum of the two non-adjacent interior angles in a triangle So, (2x – 2)° = x° + 45° 2x° – x° = 45° + 2° x = 47° Hence, The measure of the exterior angle is: (2x – 2)° = (2 (47) – 2)° = (94 – 2)° = 92° Hence, from the above, We can conclude that the measure of the exterior angle is: 92° Question 17. Answer: Question 18. Answer: The given figure is: We know that, An exterior angle is equal to the sum of the two non-adjacent interior angles in a triangle So, (7x – 16)° = (x + 8)° + 4x° 7x° – 5x° = 16° + 8° 2x = 24° x = 24° ÷ 2 x = 12° Hence, The measure of the exterior angle is: (7x – 16)° = (7 (12) – 16)° = (84 – 16)° = 68° Hence, from the above, We can conclude that the measure of the exterior angle is: 68° In Exercises 19-22, find the measure of each acute angle. Question 19. Answer: Question 20. Answer: The given figure is: From the given figure, We can observe that one angle is 90° and the 2 sides are perpendicular So, We can say that the given triangle is a right-angled triangle We know that, The sum of interior angles of a triangle is: 180° So, x° + (3x + 2)° + 90° = 180° 4x° + 2° + 90° = 180° 4x° = 180° – 90° – 2° 4x° = 88° x = 88° ÷ 4° x = 22° So, The 2 acute angle measures are: x° and (3x + 2)° = 22° and (3(22) + 2)° = 22° and (66 + 2)° = 22° and 68° Hence, from the above, We can conclude that the 2 acute angle measures are: 22° and 68° Question 21. Answer: Question 22. Answer: The given figure is: From the given figure, We can observe that one angle is 90° and the 2 sides are perpendicular So, We can say that the given triangle is a right-angled triangle We know that, The sum of interior angles of a triangle is: 180° So, (19x – 1)° + (13x – 5)° + 90° = 180° 32x° – 6° + 90° = 180° 32x° = 180° – 90° – 6° 4x° = 84° x = 84° ÷ 4° x = 21° So, The 2 acute angle measures are: (19x – 1)° and (13x – 5)° = (19 (21) – 1)° and (13(21) – 5)° = 398° and (273 – 5)° = 398° and 268° Hence, from the above, We can conclude that the 2 acute angle measures are: 398° and 268° In Exercises 23-26. find the measure of each acute angle in the right triangle. Question
23. Question 24. Question 25. Question 26. ERROR
ANALYSIS Question 27. Answer: Question 28. Answer: We know that, The exterior angle of a triangle is equal to the sum of the non-adjacent interior angles of a triangle So, From the figure, The external angle is: ∠1 The interior angles are 80°, 50° So, ∠1 = 80° + 50° ∠1 = 130° Now, The interior angle measure of ∠1= 180° – (External angle measure of 130°) = 180° – 130° = 50° Hence, from the above, The internal angle measure of ∠1 is: 50° In Exercises 29-36, find the measure of the numbered angle. Question 29. Question 30. Question 31. Question 32. Question 33. Question 34. Question 35. Question 36. Question 37. Answer: Question 38. (B) 96°, 74°, 10° (C) 165°, 113°, 82° (D) 101°, 41°, 38° (E) 90°, 45°, 45° (F) 84°, 62°, 34° Question 39. Question 40. From the above figure, We can say that The sum of the interior angles of a given triangle is: 180° The sum of the exterior angles of a given triangle is: 360° The relation between the interior angles and the exterior angles is: The exterior angle measure = Sum of the two non-adjacent interior angles Question 41. Answer: Question 42. Answer: It is given that In ΔABC, the exterior angle is ∠ACD We have to prove that m∠A + m∠B = m∠ACD Proof: Hence, from the above, We can conclude that m∠A + m∠B = m∠ACD is proven Question 43. Question 44. Question 45. Question 46. Answer: The given figure is: From the figure, We can observe that all the length of the sides of the triangle are equal We know that, The triangle that has the length of all the sides equal is called an “Equilateral triangle” Hence, from the above, We can conclude that the given triangle is an “Equilateral triangle” b. Answer: The given figure is: From the figure, We can observe that the lengths of all the 3 sides are different We know that, The triangle that has all the different side lengths is called a “Scalene triangle” Hence, from the above, We can conclude that the given triangle is called a “Scalene triangle” c. Answer: The given figure is: From the figure, We can observe that the length of all the 3 sides are different and 1 angle is obtuse i.e., greater than 90° We know that, The triangle that has any angle obtuse is called an “Obtuse angled triangle” Hence, from the above, We can conclude that the given triangle is an “Obtuse angled scalene triangle” d. Answer: The given figure is: From the figure, We can observe that 1 angle is 90° and the 2 sides are perpendicular to each other We know that, The triangle that has an angle of 90° and the slope -1 is called a “Right-angled triangle” Hence, from the above, We can conclude that the given triangle is called a “Right-angled triangle” Question 47. Question 48. Explanation: MATHEMATICAL CONNECTIONS Question 49. Answer: Question 50. Answer: The given figure is: From the figure, We have to obtain the values of x and y Now, By using the alternate angles theorem, x = 118° Now, By using the exterior angle theorem, x = y + 22° y = x – 22° y = 118° – 22° y = 96° Hence, from the above, We can conclude that the values of x and y are: 118° and 96° respectively Question 51. Answer: Question 52. Answer: The given figure is: From the above figure, We have to find the values of x and y Now, By using the sum of interior angle measures, x° + 64° + 90° = 180° x° + 154° = 180° x° = 180° – 154° x° = 26° Now, By using the exterior angle theorem, y° = x° + 64° y° = 26° + 64° y° = 90° Hence, from the above, We can conclude that the values of x and y are: 26° and 90° respectively Question 53. Answer: Maintaining Mathematical Proficiency Use the diagram to find the measure of the segment or angle. Question 54. Question 55. Question 56. Question 57. 5.2 Congruent PolygonsExploration 1 Describing Rigid Motions Work with a partner: of the four transformations you studied in Chapter 4, which are rigid motions? Under a rigid motion. why is the image of a triangle always congruent to the original triangle? Explain your reasoning. Answer: Rigid motion occurs in geometry when an object moves but maintains its shape and size, which is unlike non-rigid motions, such as dilations, in which the object’s size changes. All rigid motion starts with the original object, called the pre-image, and results in the transformed object, called the image. There are 4 types of rigid motion. They are: a. Translation b. Rotation c. Reflection d. Glide reflection We know that, Rotation only occurs in terms of 90° or 180° Now, The given transformations are: So, From the above figure, The first figure and the second figure are different The second figure and the third figure are the same in shape The first figure and the fourth figure are the same in shape So, We can say that the first and the fourth figures are rigid motions W can say that the second and the third figures are rigid motions In the second and the third figures, The “Rotation” takes place i.e., the second figure is rotated 180° keeping the original shape In the first and the fourth figures, The “Reflection” takes place i.e., the first figure is reflected keeping the original shape Now, The image of the triangle is always congruent to the original triangle because of the “Translation” i.e., the original triangle and the image of the triangle have the same sides and the same angles but not in the same position. Exploration 2 Finding a Composition of Rigid Motions Work with a partner. Describe a composition of rigid motions that maps ∆ABC to ∆DEF. Use dynamic
geometry software to verify your answer. a. ∆ABC ≅ ∆DEF Answer: b. ∆ABC ≅ ∆DEF Answer: c. ∆ABC ≅ ∆DEF Answer: d. ∆ABC ≅ ∆DEF Answer: Communicate Your Answer Question 3. Question 4. Lesson 5.2 Congruent PolygonsMonitoring Progress In the diagram, ABGH ≅ CDEF. Question 1. Question 2. Answer: Explanation: Question 3. Answer: Use the diagram. Question 4. Answer: Explanation: Question 5. Answer: Exercise 5.2 Congruent PolygonsQuestion 1. Question 2. Is ∆ABC ≅ ∆RST? Answer: Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4. identify all pairs of congruent corresponding parts. Then write another congruence statement for the polygons. Question 3. Answer: Question 4. Answer: Corresponding angles: ∠G = ∠Q, ∠H = ∠R, ∠K = ∠T, ∠J = ∠T Correspoding sides: ST = JK, TQ = KG, QR = GH, RS = HJ So, GHJK ≅ QRST In Exercises 5-8, ∆XYZ ≅ ∆MNL. Copy and complete the statement. Question 5. Question 6. Question 7. Question 8. In Exercises 9 and 10. find the values of x and y. Question 9. Answer: Question 10. Answer: x = 32°, y = 51 Explanation: In Exercises 11 and 12. show that the polygons are congruent. Explain your reasoning. Question 11. Answer: Question 12. Answer: ∠X = ∠Z, ∠XWY = ∠ZWY, ∠XYW = ∠ZWY WX = YZ, XY = WZ All corresponding parts of polygons are congruent So, △ XWZ ≅ △ ZWY In Exercises 13 and 14, find m∠1. Question 13. Answer: Question 14. Answer: ∠1 = 180 – (80 + 45) = 180 – 125 ∠1 = 55° Question 15. Given \(\overline{A B}\) || \(\overline{D C}\), \(\overline{A B}\) ≅ \(\overline{D C}\) is the midpoint of \(\overline{A C}\) and \(\overline{B D}\) Prove ∆AEB ≅ ∆CED Answer: Question 16. Answer: Proved Explanation: ERROR
ANALYSIS Question 17. Answer: Question 18. Answer: ∠N = ∠S, ∠M = ∠R, MN ≠ RS The corresponding sides are not congruent. So, ∆MNP is not similar to ∆RSP. Question 19. Question 20. The interior angles of these triangles are equal because one pair of angles is equal because they are cross angles and the other pairs are equal because the angles are on the transversal. When the angles are equal, then the sides of these triangles are also equal that means these triangles are congruent. Question 21. Question 22. a. Explain how you know that \(\overline{B E}\) ≅ \(\overline{D E}\) and ∠ABE ≅∠CDE. Answer: ABEF ≅ CDEF So by using the property of corresponding parts of congruent triangles \(\overline{B E}\) ≅ \(\overline{D E}\) are corresponding parts of congruent triangles ∠ABE ≅∠CDE are congruent b. Explain how you know that ∠GBE ≅ ∠GDE. c. Explain how you know that ∠GEB ≅ ∠GED. d. Do you have enough
information to prove that ∠BEG ≅ ∠DEG? Explain. MATHEMATICAL CONNECTIONS Question 23. Question 24. x = 5, y = 2 Explanation: Question 25. Maintaining Mathematical Proficiency What can you conclude from the diagram? Question 26. Answer: ∠Z = ∠W Question 27. Answer: Question 28. Answer: JK = KM ∠J = ∠M Question 29. Answer: 5.3 Proving Triangle Congruence by SASExploration 1 Drawing Triangles Work with a partner. Use dynamic geometry software. b. Locate the point where one ray of the angle intersects the smaller
circle and label this point B. Locate the point where the other ray of the angle intersects the larger circle and label this point C. Then draw ∆ABC. c. Find BC, m∠B, and m∠C. d. Repeat parts (a)-(c) several times. redrawing the angle indifferent positions. Keep track of your results by copying and completing the table below. What can you conclude? Communicate Your Answer Question 2. Question 3. Explanation: Consider the sides AB, AC and m∠A. From the table, it is clear that the lengths of the sides are constant. It is observed that the coordinates of B and C, and the angle are formed are constant. By SAS congruence theorem, the given set of triangles is constant. Hence it can be concluded that the given set of triangles is congruent. Lesson 5.3 Proving Triangle Congruence by SASMonitoring Progress In the diagram, ABCD is a square with four congruent sides and four right Question 1. Prove that ∆SVR ≅ ∆UVR. Answer: Question 2. Answer: Question 3. Answer: Exercise 5.3 Proving Triangle Congruence by SASvocabulary and core concept check Question 1. Question 2. Monitoring progress and Modeling with Mathematics In Exercises 3-8, name the included an1e between the pair of sides given. Question 3. Question 4. Question 5. Question 6. Question 7. Question 8. In Exercises 9-14, decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Theorem (Theorem 5.5). Explain. Question 9. Answer: Question 10. Answer: Yes, two pairs of sides and the included angles are congruent. Question 11. Answer: Question 12. Answer: Yes, two pairs of sides and the included angles are congruent. Question 13. Answer: Question 14. Answer: No, one of the congruent angles is not the included angle. In Exercises 15 – 18, write a proof. Question 15. Answer: Question 16. Answer: Since AB || CD ∠BAC = ∠ACD (Alternative Interior Angle) ∠DAC = ∠BCA (Alternative Interior Angle) AC = AC (Reflexive property of congruence) ∆ABC ≅ ∆CDA by Angle Side Angle Congruence Theorem. Question
17. Answer: Question 18. Answer: \(\overline{P T}\) ≅ \(\overline{R T}\) \(\overline{Q T}\) ≅ \(\overline{S T}\) In a parallelogram, diagonals bisect at 90 degrees. ∠PTQ = ∠STR (Vertical Angles Congruence Theorem) ∆PQT ≅ ∆RST (SAS Congruence Theorem) In Exercises 19-22, use the given information to name two triangles that are congruent. Explain your reasoning. Question 19. Answer: Question 20. Answer: BC ≅ AD, AB ≅ CD as four sides are congruent BD ≅ BD by the reflexive property of congruence. ∠A = ∠C, ∠DBA = ∠CBD All the corresponding sides and angles are congruent So, △BAD ≅ △BCD Question 21. Answer: Question 22. Answer: ∠M = ∠L = 90° MK = LN NK = NK by the reflexive property of congruence So, △ MKN ≅ △LKN CONSTRUCTION Question 23. Answer: Question 24. Answer: Construct side DE which is congruent to AC. Construct ∠D with vertex D and side DE so that it is congruent to ∠A. Construct DF that is congruent to AB. Draw △DFE. By SAS congruence theorem △ABC ≅ △DFE Question 25. Answer: Question 26. Answer: One included congruent angle is needed to prove that ∆ABC ≅ ∆DBC. Question 27. Answer: Question 28. SSS, SAS, ASA and AAS
are congruence theorems. AAA does not prove two triangles congruent. For example, these two triangles have the same angles but are not similar. Question 29. Answer: Question 30. Answer: My friend is wrong. Because to copy the ∠C, you need to construct the vertex C first. But by constructing \(\overline{A B}\) and \(\overline{A C}\), you will get vertex A. Question 31. Maintaining Mathematical Proficiency Classify the triangle by its sides and by measuring its angles. Question 32. Answer: Two sides are equal and one angle is the right angle. So, the triangle is right angles isosceles triangle Question 33. Answer: Question 34. Answer: Three sides of the triangle are equal. So, it is an equilateral triangle. Question 35. Answer: 5.4 Equilateral and Isosceles TrianglesExploration 1 Writing a Conjecture about Isosceles Triangles Work with a partner: Use dynamic geometry software. a. Construct a circle with a radius of 3 units centered at the origin. b. Construct ∆ABC so that B and C are on the circle and A is at the origin. c. Recall that a triangle is isosceles if it has at least two congruent sides. Explain why ∆ABC is an isosceles triangle. d. What do you observe about the angles of ∆ABC? e. Repeat parts (a)-(d) with several other isosceles triangles using circles of different radii. Keep track of your observations by copying and completing the table below. Then write a conjecture about the angle measures of an isosceles triangle. f. Write the converse of the
conjecture you wrote in part (e). Is the converse true? Communicate Your Answer Question 2. Question 3. The above figure shows the first conjecture for an isosceles triangle. The above figure, shows the second conjecture for an isosceles triangle. Lesson 5.4 Equilateral and Isosceles TrianglesMonitoring Progress Copy and complete the statement. Question 1. Answer: Question 2. Answer: Question 3. Answer: Question 4. Answer: Question 5. Answer: Exercise 5.4 Equilateral and Isosceles TrianglesVocabulary and Core Concept Check Question
1. Question 2. Explanation: Monitoring Progress and Modeling with Mathematics In Exercises 3-6. copy and complete the statement. State which theorem you used. Question 3. Question 4. Question 5. Question 6. In Exercises 7-10. find the value of x. Question 7. Answer: Question 8. Answer: x = 16 Explanation: Question 9. Answer: Question 10. Answer: x = 20° Explanation: Question 11. Answer: Question 12. Explanation: In Exercises 13-16, find the values of x and y. Question 13. Answer: Question 14. Answer: Question 15. Answer: Question 16. Answer: x = 7.416, y = 5.25 Explanation: CONSTRUCTION Question 17. Question 18. Question 19. Answer: Question 20. a. Explain why ∆ABC is isosceles. b. Explain ∠BAE ≅ ∠BCE. c. Show that ∆ABE and ∆CBE arc congruent. d. Find the measure of ∠BAE. Question 21. a. Explain how you know that an triangle made out of equilateral triangles is equilateral. b. Find the areas of the first four triangles in the pattern. c. Describe any patterns in the areas. Predict the area of the seventh triangle in the pattern. Explain your reasoning. Answer: Question 22. Explanation: In Exercises 23 and 24, find the perimeter of the triangle. Question 23. Answer: Question 24. Answer: Perimeter = 39 Explanation: MODELING WITH MATHEMATICS Question 25. Question 26. Explanation: Question 27. Question 28. Question 29. Question 30. Question 31. Question 32. Question 33. Question 34. a. Name two angles congruent to ∠WUX. Explain your reasoning. b. Find the distance between points U and V. Answer: a. ∠WUX ≅ ∠XVY ∠WUX ≅ ∠UXV b. The distance between thee points U and V is 8 m. Question 35. a. Find SL. Explain your reasoning. b. Explain how to find the distance between the boat and the shoreline. Answer: Question 36. Question 37. Question 38. a. Explain why ∆ABE ≅ ∆DCE. b. Name the isosceles triangles in the purse. c. Name three angles that are congruent to ∠EAD. Answer: a. AB ≅ CD, AE ≅ ED and ∠A ≅ ∠D So, ∆ABE ≅ ∆DCE by SAS theorem. b. ∠EAD ≅ ∠AED ≅ ∠EDA Question 39. Question 40. Explanation: When we connect the points T, V, U we can see that we have not obtained an isosceles triangle. Question 41. Given ∆ABC is equilateral ∠CAD ≅ ∠ABE ≅ ∠BCF Prove ∆DEF is equilateral Answer: Maintaining Mathematical Proficiency Use the given property to complete the statement. Question 42. Question 43. Question 44. 5.1 to 5.4 QuizFind the measure of the exterior angle. Question 1. Answer: x° = 110° Explanation: Question 2. Answer: Explanation: Question 3. Answer: Explanation: Identify all pairs of congruent corresponding parts. Then write another congruence statement for the polygons. Question 4. Answer: Question 5. Answer: Decide whether enough information is given to prove that the triangles are congruent using the SAS Congruence Theorem (Thm 5.5). If so, write a proof. If not, explain why. Question 6. Answer: Question 7. Answer: Question 8. Answer: Copy and complete the statement. State which theorem you used. Question 9. Answer: Question 10. Answer: Question 11. Answer: Question 12. Answer: Find the values of x and y. Question 13. Answer: Explanation: Question 14. Answer: Explanation: Question 15. Answer: Explanation: Question
16. a. Classify triangles 1 – 4 by their angles. Answer: b. Classify triangles 4 – 6 by their sides. Answer: c. Is there enough information given to prove that ∆7 ≅ ∆8? If so, label the vertices Answer: 5.5 Proving Triangle Congruence by SSSExploration 1 Drawing Triangles Work with a partner. a. Construct circles with radii of 2 units and 3 units centered at the origin. Label the origin A. Then draw \(\overline{B C}\) of length 4 units. b. Move \(\overline{B C}\) so that B is on the smaller circle and C is on the larger circle. Then draw ∆ABC. c. Explain
why the side lengths of ∆ABC are 2, 3, and 4 units. d. Find m∠A, m∠B, and m∠C. e. Repeat parts (b)and (d) several times, moving \(\overline{B C}\) to different locations. Keep track of ‘our results by copying and completing the table below. What can you conclude? Communicate Your Answer Question 2. Question 3. Lesson 5.5 Proving Triangle Congruence by SSSMonitoring Progress Decide whether the congruence statement is true. Explain your reasoning. Question 1. Answer: Explanation: Question 2. Answer: Explanation: Question 3. Answer: Explanation: Determine whether the figure is stable. Explain your reasoning. Question 4. Answer: The figure is not stable. Because it doesn’t have a triangle. By the SSS Congruence Theorem, those triangles cannot change shape. Question 5. Answer: Question 6. Answer: Use the diagram. Question 7. Answer: Question 8. Answer: Exercise 5.5 Proving Triangle Congruence by SSSVocabulary and Core Concept Check Question 1. Question 2. Answer: The second triangle legs do not belong with the other three. Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, decide whether enough information is given to prove that the triangles are congruent using the SSS Congruence Theorem (Theorem 5.8). Explain. Question 3. Answer: Question 4. Answer: No, PQ ≅ QR, QS ≅ QS by reflexive property. We need one more congruence property to prove those triangles are congruent. In Exercises 5 and 6, decide whether enough information is given to prove that the triangles are congruent using the HL Congruence Theorem (Theorem 5.9). Explain. Question 5. Answer: Question 6. Answer: Yes Explanation: In Exercises 7-10. decide whether the congruence statement is true. Explain your reasoning. Question 7. Answer: Question 8. Answer: Yes Explanation: Question 9. Answer: Question 10. Answer: Yes Explanation: In Exercises 11 and 12, determine whether the figure is stable. Explain your reasoning. Question 11. Answer: Question 12. Answer: No, the figure has no diagonals. So the figure is not stable. In Exercises 13 and 14, redraw the triangles so they are side by side with corresponding parts in the same position. Then write a proof. Question 13. Answer: Question 14. Answer: EG ≅ GH as G is the midpoint of \(\overline{E H}\) \(\overline{F G}\) ≅ \(\overline{G I}\), ∠E ≅ ∠H ∆EFG ≅ ∆HIG by SAS congruence theorem. In Exercises 15 and 16. write a proof. Question 15. Answer: Question 16. Answer: XY ≅ YZ, WY ≅ VY, WX ≅ VZ ∆VWX ≅ ∆WVZ by SSS Congruence theorem CONSTRUCTION Question 17. Answer: Question 18. Answer: At first, construct a side that is congruent to QS. Draw an arc with the compass with Q as center and radius as QR. Draw another arc that intersects the first arc with S as center and radius as SR. Join the point to Q and S to form a circle that is congruent to ∆QRS. Question 19. Answer: Question 20. Answer: 4x + 4 = 6x 2x = 4 x = 2 Question 21. Question 22. Answer: Given that, HS ≅ FT FS ≅ FS by reflexive property of congruence ∠HFS ≅ ∠TFS So, ∆HFS ≅ ∆FST by SAS congruence theorem ST ≅ ST by reflexive property of congruence HS ≅ FT ∠FST ≅ ∠TSH So, ∆FST ≅ ∆STH by SAS congruence theorem Therefore, ∆HFS ≅ ∆FST ≅ ∆STH Question 23. a. What additional information do you need to use the HL Congruence Theorem (Theorem 5.9) to prove that ∆JKL ≅ ∆MKL? b. Suppose K is the midpoint of JM. Name a theorem you could use to prove that ∆JKL ≅ ∆MKL. Explain your reasoning. Answer: Question 24. a. What additional intormation do you need to use the SSS Congruence Theorem (Theorem 5.8) to prove that ∆ABC ≅ ∆CDE? b. What additional information do you need to use the HL Congruence Theorem (Theorem 5.9) to prove that ∆ABC ≅ ∆CDE? Answer: a. AB ≅ CD is required o use the SSS Congruence Theorem (Theorem 5.8) to prove that ∆ABC ≅ ∆CDE b. One angle should be the right angle in those triangles to use the HL Congruence Theorem (Theorem 5.9) to prove that ∆ABC ≅ ∆CDE In Exercises 25-28. use the given coordinates to determine whether ∆ABC ≅ ∆DEF. Question 25. Question 26. Explanation: Question 27. Question 28. Explanation: Question 29. Question 30. Answer: Question 31. Answer: Question 32. USING TOOLS Question 33. Answer: Question 34. Answer: Question 35. Answer: a. Show that ∆ABD ≅ ∆CBD. State which theorem or postulate you used and explain your reasoning. b. Are all four right triangles shown in the diagram Congruent? Explain your reasoning. Answer: Question 36. Answer: The possible values of x are 3, 6, 5. Explanation: Maintaining Mathematical proficiency Use the congruent triangles. Question 37. Question 38. Question 39. Question 40. 5.6 Proving Triangle Congruence by ASA and AASExploration 1 Determining Whether SSA Is Sufficient Work with a partner. b. Construct a circle with a radius of 2 units centered at the origin. Locate point D where the circle intersects \(\overline{A C}\). Draw \(\overline{B D}\). Answer: c. ∆ABC and ∆ABD have two congruent sides and a non included
congruent angle. d. Is ∆ABC ≅ ∆ABD? Explain your reasoning. e. Is SSA sufficient to determine whether two triangles are congruent? Explain your reasoning. Exploration 2 Determining Valid Congruence Theorems Work with a partner. Use dynamic geometry software to determine which of the following are valid triangle congruence theorems. For those that are not valid. write a counter example. Explain your
reasoning.
Answer: Communicate Your Answer Question 3.
Question 4. Lesson 5.6 Proving Triangle Congruence by ASA and AASMonitoring Progress Question 1. Answer: WX ≅ YZ, XY ≅ WZ, and ∠1 ≅ ∠3 So, WXY ≅ WYZ by the AAS congruence theorem. Question 2. Answer: \(\overline{A C}\) ≅ \(\overline{D C}\), ∠A ≅ ∠D So, the given information is not enough to prove that ∆ABC ≅ ∆DEF. Question 3. Answer: ∠S ≅ ∠U, RS ≅ UV So, the given information is not enough to prove that ∆RST ≅ ∆VYT Lesson 5.6 Proving Triangle Congruence by ASA and AASVocabulary and Core Concept Check Question 1. Question 2. Monitoring Progress and Modeling with Mathematics In Exercises 3-6, decide whether enough information is given to prove that the triangles are congruent. If so, state the theorem you would use. Question 3. Answer: Question 4. Answer: No, two angles are not sufficient to determine congruence. Question 5. Answer: Question 6. Answer: No, one side and one included angle is not sufficient to determine congruence. In Exercises 7 and 8, state the third congruence statement that is needed to prove that ∆FGH ≅ ∆LMN the given theorem. Question 7. Given \(\overline{G H}\) ≅ \(\overline{M N}\), ∠G ≅ ∠M, _______ = ________ Use the AAS Congruence Theorem (Thm. 5.11). Answer: Question 8. In Exercises 9 – 12. decide whether you can use the given information to prove that ∆ABC ≅ ∆DEF Explain your reasoning. Question 9. Question 10. Question 11. Question 12. CONSTRUCTION Question 13. Answer: Question 14. Answer: Construct a side that is similar to JK. With J as center and JL as radius, draw an arc. With K as center and KL as radius draw another arc that intersects the first arc. Label the intersection of arcs as L. Connect LK and LJ. ERROR ANALYSIS Question 15. Answer: Question 16. Answer: △QRS ≅ △VWX by ASA congruence theorem. PROOF Question 17. Prove ∆NQM ≅ ∆MPL Answer: Question 18. Answer: AJ ≅ KC, ∠K ≅ ∠J, ∠A ≅ ∠C So, ∆ABK ≅ ∆CBJ by ASA congruence theorem. PROOF Question 19. Answer: Question 20. Answer: Given ∠NKM ≅∠LMK, ∠L ≅∠N KM ≅ KM ≅ by reflexive property of congruence ∆NMK ≅ ∆LKM by AAS congruence theorem. PROOF Question 21. Question 22. Question 23. Question 24. (A) \(\overline{K M}\) ≅ \(\overline{K J}\) (B) \(\overline{K H}\) ≅ \(\overline{N H}\) (C) ∠M ≅ ∠J (D) ∠LKJ ≅ ∠LNM Answer: Question 25. m∠ABC = (8x – 32)° m∠DBC = (4y – 24)° m∠BCA = (5x + 10)° m∠BCD = (3y + 2)° m∠CAB = (2x – 8)° m∠CDB = (y – 6)° Answer: Question 26. (A) \(\overline{B D}\) ≅ \(\overline{B D}\) (B) ∆STV ≅ ∆XVW (C) ∆TVS ≅ ∆VWU (D) ∆VST ≅ ∆VUW Answer: (C) ∆TVS ≅ ∆VWU by ASA congruence theorem. ∠T ≅ ∠V, TS ≅ VU, ∠S ≅ ∠U (D) ∆VST ≅ ∆VUW by ASA congruence theorem. ∠T ≅ ∠V, TS ≅ VU, ∠S ≅ ∠U Question 27. Question 28. Question 29. a. Prove that ∆ABD is Congruent to ∆CBD. Given ∠CBD ≅∠ABD DB ⊥ AC Prove ∆ABD ≅ ∆CBD b. Verify that ∆ACD is isosceles. c. Does moving away from the mirror have an effect on the amount of his or her reflection a person sees? Explain. Answer: Question 30. Answer: Consider ΔPTS and ΔQTR PT = RT ST = QT PS = QR ΔPTS ≅ ΔQTR consider ΔPTQ and ΔRTS RT = PT PQ = SR QT = ST ΔPTQ ≅ ΔRTS Question 31. Question 32. Question 33. a. List all combinations of three given statements that could provide enough information to prove that ∆TUV is congruent to ∆XYZ. b. You choose three statements at random. What is the probability that the statements you choose provide enough information to prove that the triangles are congruent? Answer: Maintaining Mathematical proficiency Find the coordinates of the midpoint of the line segment with the given endpoints. Question 34. Question 35. Question 36. Copy and angle using a compass and straightedge. Question 37. Answer: Question 38. Answer: Draw a segment. Label a point D on the segment. Draw an arc with center B, and label the intersection points A and C. Using the same radius, draw an arc with center D. Label the point of intersection of the arc with radius BC with center D. Label the intersection F. Draw DF. So, ∠B ≅ ∠FDE. 5.7 Using Congruent TrianglesExploration 1 Measuring the Width of a River Work with a partner: a. Study the figure. Then explain how the surveyor can find the width of the river. b. Write a proof to verify that the method you described in part (a) is valid. c. Exchange Proofs with your partner and discuss the reasoning used. Exploration 2 Measuring the Width of a River Work with a partner. It was reported that one of Napoleon’s offers estimated the width of a river as follows. The officer stood on the hank of the river and lowered the visor on his cap until the farthest thin visible was the edge of the bank on the other side. He then turned and rioted the point on his side that was in line with the tip of his visor and his eye. The officer then paced the distance to this point and concluded that distance was the width of the river. a. Study the figure. Then explain how the officer concluded that the width of the river is EG. b. Write a proof to verify that the conclusion the officer made is correct. c. Exchange proofs with your partner and discuss the reasoning used. Communicate Your Answer Question 3. Question 4. Lesson 5.7 Using Congruent TrianglesMonitoring Progress Question 1. Answer: If you can prove that △ABD ≅ △CBD, then ∠A ≅ ∠C. AB ≅ BC, AD ≅ CD BD ≅ BD by reflexive property of congruence So, △ABD ≅ △CBD by SSS congruence theorem. Question 2. Question 3. Answer: TU ≅ PQ ∠PTU ≅ ∠UQP PU ≅ PU by reflexive property of congruence ∆PTU ≅ ∆UQP by SAS congruence theorem. Question 4. Exercise 5.7 Using Congruent TrianglesVocabulary and core concept check Question 1. Question 2. Monitoring Progress and Modeling With Mathematics In Exercise 3-8, explain how to prove that the statement is true. Question 3. Answer: Question 4. Answer: QP ≅ PT, RP ≅ SP, QR ≅ ST All pairs of sides are congruent by SSS congruence theorem. △QPR ≅ △STP. Because corresponding parts of congruent triangles are congruent, ∠Q ≅∠T. Question 5. Answer: Question 6. Answer: \(\overline{A C}\) ≅ \(\overline{D B}\), \(\overline{A D}\) ≅ \(\overline{A D}\) by reflexive property of congruence ∠C ≅∠B △ACD ≅ △BDC by SAS congruence theorem. Question 7. Answer: Question 8. Answer: VW ≅ RT, ∠Q ≅ ∠S, ∠W ≅ ∠T △QVW ≅ △VRT by the AAS congruence theorem So, \(\overline{Q W}\) ≅ \(\overline{V T}\) In Exercises 9-12, write a plan to prove that ∠1 ≅∠2. Question 9. Answer: Question 10. Answer: ∠A ≅ ∠D, AB ≅ CD and ∠BEA ≅ ∠CED So, △ABE ≅ △EDC by AAS congruence theorem ∠ABE ≅ ∠DCE Use the congruent complements theorem to prove that ∠1 ≅ ∠2 Question 11. Answer: Question 12. Answer: AF ≅ CD, ∠AEF ≅ ∠CED, ∠FAE ≅ ∠ECD So, △AFE ≅ △CDE by ASA congruence thorem Then all parts of the triangles are congruent. So, ∠1 ≅ ∠2 In Exercises 13 and 14. write a proof to verify that the construction is valid. Question 13. Plan for proof ∆APQ ≅ ∆BPQ by the congruence Theorem (Theorem 5.8). Then show the ∆APM ≅ ∆BPM using the SAS Congruence Theorem (Theorem 5.5). Use corresponding parts of congruent triangles to show that ∠AMP and ∠BMP are right angles. Answer: Question 14. Plan for Proof Show that ∆APQ ≅ ∆BPQ by the SSS Congruence Theorem (Theorem 5.8) Use corresponding parts of congruent triangles to show that ∠QPA and ∠QPB are right angles. Answer: In Exercises 15 and 16, use the information given in the diagram to write a proof. Question 15. Answer: Question 16. Answer: Consider the side SU. Show that SU is congruent to itself SU ≅ SU Consider the sides RS, VU and SU. RS = VU SU = SU Consider the sides RU and VS. RU = RS + SU VS = VU + SU Use the transitive property of equality RU = VS Consider the sides RU and VS RU ≅ VS ∠URP ≅ ∠SVQ RU ≅ VS ∠PUR ≅ ∠QSV Consider the angles ∠URP and ∠URX Consider the angles ∠SVQ and ∠SVY The angles ∠URP and ∠URX form a linear pair. The angles ∠SVQ and ∠SVY form a linear pair. Consider the angles ∠URP and ∠URX Consider the angles ∠SVQ and ∠SVY The angles ∠URP and ∠URX are supplementary The angles ∠SVQ and ∠SVY are supplementary Consider the angles ∠URX and ∠SVY ∠URX ≅ ∠SVY Consider the triangles ΔURX and ΔSVY ∠URX ≅ ∠SVY RU ≅ VS ∠XUR ≅ ∠YSV ΔURX ≅ ΔSVY Consider the triangles ΔPRU and ΔQVS Consider the triangles ΔURX and ΔSVY ∠X ≅ ∠Y ∠P ≅ ∠Q PU ≅ SQ Consider the triangles ΔPUX and ΔQSY ∠X ≅ ∠Y ∠P ≅ ∠Q PU ≅ SQ Therefore ΔPUX ≅ ΔQSY Question 17. Answer: Question 18. a. Which triangle(s) have an area that is twice the area of the purple triangle? Question 19. Answer: Question 20. Answer: Question 21. Answer: Question 22. Question 23. Answer: Maintaining Mathematical Proficiency Find the perimeter of the polygon with the given vertices. Question 24. Explanation: Question 25. 5.8 Coordinate ProofsExploration 1 Writing a coordinate Proof Work with a partner. a. Use dynamic geometry software to draw \(\overline{A B}\) with endpoints A(0, 0) and B(6, 0). b. Draw the vertical line x = 3. c. Draw ∆ABC so that C lies on the line x = 3. d. Use your drawing to prove that ∆ABC is an isosceles triangle. Exploration 2 Writing a Coordinate proof Work with a partner. a. Use dynamic geometry software to draw \(\overline{A B}\) with endpoints A(0, 0) and B(6, 0). b. Draw the vertical line x = 3. c. Plot the point C(3, 3) and draw ∆ABC. Then use your drawing to prove that ∆ABC is an isosceles right triangle. d. Change the coordinates of C so that C lies below the x-axis and ∆ABC is an isosceles right triangle. e. Write a coordinate proof to show that
if C lies on the line x = 3 and ∆ABC is an isosceles right triangle. then C must be the point (3, 3) or the point found in part (d). Communicate Your Answer Question 3. Question 4. Lesson 5.8 Coordinate ProofsMonitoring Progress Question 1. Place the
base of the rectangle anywhere you want on the coordinate plane. Question 2. Answer: Explanation: Question 3. Answer: Question 4. Answer: Side lengths are OA = √(m – 0)² + (n – 0)² = √m² + n² OB = √(m – 0)² + (0 – 0)² = √m² = m AB = √(m – m)² + (n – 0)² = √n² = n By using the Pythagorean theorem, OA² = OB² + AB² m² + n² = m² + n² So, the trinagle is a right-angled triangle. Question 5. Given Coordinates of vertices of ∆NPO and ∆NMO Prove ∆NPO ≅ ∆NMO Answer: Exercise 5.8 Coordinate ProofsVocabulary and Core Concept Check Question 1. Question 2. Answer: Because the right triangle has the base and another leg on the same line in the coordinate plane. Maintaining Progress and Modeling with Mathematics In Exercises 3-6, place (he figure in a coordinate plane in a convenient way. Assign coordinates to each vertex. Explain the advantages of your placement. Question 3. Question 4. It is easy to find the lengths of horizontal and vertical segments and distances from the origin. Question 5. Question 6. In Exercises 7 and 8, write a plan for the proof. Question 7. Answer: Question 8. Answer: The coordinates of G are (3, 2) OG = √(3 – 0)² + (2 – 0)² = √9 + 4 = √13 OF = √(5 – 0)² + (0 – 0)² = 5 GF = √(5 – 3)² + (0 – 2)² = √2² + 2² = √8 GH = √(3 – 1)² + (2 – 4)² = √2² + 2² = √8 HJ = √(6 – 1)² + (4 – 4)² = √5² = 5 GJ = √(6 – 3)² + (4 – 2)² = √9 + 4 = √13 OG ≅ GJ, OF ≅ HJ, GF ≅ GH All the sides are congruent. So, ∆GHJ ≅ ∆GFO by SSS congruence theorem. In Exercises 9-12, place the figure in a coordinate plane and find the indicated length. Question 9. Question 10. The length of one of the legs = 58.31 Explanation: Question 11. Question 12. The length of the diagonal is n√2 Explanation: The length of diagonal = √(n – 0)² + (n – 0)² = √n² + n² = n√2 In Exercises 13 and 14, graph the triangle with the given vertices. Find the length and the slope of each side of the triangle. Then find the coordinates of the midpoint of each side. Is the triangle a right triangle? isosceles? Explain. Assume all variables are positive and in m ≠ n.) Question 13. Question 14. In Exercises 15 and 16, find the coordinates of any unlabeled vertices. Then find the indicated length(s). Question 15. Answer: Question 16. Answer: OT = √(2k – 0)² + (2k – 0)² = √(2k)² + (2k)² = √8k² = 2k√2 units PROOF Question 17. Answer: Question 18. Answer: G is the mid point of \(\overline{E A}\) So, G = (\(\frac { 2h + 0 }{ 2 } \), \(\frac { 0 + 2k }{ 2 } \)) = (h, k) H is the midpoint of \(\overline{D A}\) So, H = (\(\frac { -2h + 0 }{ 2 } \), \(\frac { 2k + 0 }{ 2 } \)) = (-h, k) \(\overline{D G}\) = √(h + 2h)² + (k – 0)² = √9h² + k² \(\overline{E H}\) = √(-h – 2h)² + (0 – k)² = √9h² + k² Question 19. Answer: Question 20. It can be seen that PS || QR and PQ || SR. If PQRS is a rectangle then PS ⊥ PQ, therefore find slopes of PS and PQ. Slope of PS = \(\frac { 1 – 2 }{ -2 – 0 } \) = \(\frac { 1 }{ 2 } \) Slope of PQ = \(\frac { -4 – 2 }{ 3 – 0 } \) = -2 It can be seen that slope of PS is negative reciprocal of slope of PQ. So, PS is perpendicular to PQ and PQRS is a rectangle. Question
21. Question 22. Explanation: Question 23. Question 24. Question 25. Question 26. Answer: As the vertices are in the opposite sign, the diagonals of square TUVW are perpendicular to each other. Question 27. Maintaining Mathematical proficiency \(\vec{Y}\)W bisects ∠XYZ such that m∠XYW = (3x – 7)° and m∠WYZ = (2x + 1)°. Question 28. Explanation: Question 29. Congruent Triangles Chapter Review5.1 Angles of TrianglesQuestion 1. Answer: The trinagle have two congruent sides. So, the triangle is an isosceles triangle. Therefore, the measure of angles is less than 90 degrees. So these are obtuse angles. Find the measure of the exterior angle. Question 2. Answer: Explanation: Question
3. Answer: Explanation: Find the measure of each acute angle. Question 4. Answer: Explanation: Question 5. Answer: Explanation: 5.2 Congruent PolygonsQuestion 6. Answer: Corresponding sides: GH ≅ LM, KJ ≅ PN Corresponding angles: ∠G ≅ ∠H ≅ ∠L ≅ ∠M, ∠K ≅ ∠J ≅ ∠P ≅ ∠N Question 7. Answer: Explanation: 5.3 Proving Triangle Congruence by SASDecide whether enough information is given to prove that ∆WXZ ≅ ∆YZX using the SAS Congruence Theorem (Theorem 5.5). If so, write a proof. If not, explain why. Question 8. Answer: Yes Explanation: Question 9. Answer: Yes Explanation: 5.4 Equilateral and Isosceles TrianglesCopy and Complete the statement. Question 10. Answer: Question 11. Answer: Question 12. Question 13. Question 14. Answer: 5.5 Proving Triangle Congruence by SSSQuestion 15. Answer: LM ≅ NP, MP ≅ MP by reflexive property of congruence. We need one more side to be congruent to prove that ∆LMP ≅ ∆NPM using the SSS Congruence Theorem Question 16. Answer: XZ ≅ XZ by reflexive property of congruence. The hypotenuse leg of ∆WXZ and ∆YZX are congruent. So, ∆WXZ ≅ ∆YZX 5.6 Proving Triangle Congruence by ASA and AASQuestion 17. Answer: ∠E ≅ ∠H, ∠F ≅ ∠J, FG ≅ KJ So, ∆EFG ≅ ∆HJK by AAS Congruence Theorem. Question 18. Answer: ∠T ≅ ∠Q, TU ≅ QR The given information is not enough to prove that ∆TUS ≅ ∆QRS using AASCongruence Theorem. Decide whether enough information is given to prove that the triangles are congruent using the ASA Congruence Theorem (Thm. 5.10). If so, write a proof, If not, explain why. Question 19. Answer: ∠PNL ≅ ∠MNL, ∠PLN ≅ ∠MLN LN ≅ LN by reflexive property of congruence So, ∆LPN ≅ ∆LMN using ASA Congruence Theorem. Question 20. Answer: WZ ≅ XY XZ ≅ XZ by reflexive property of congruence theorem So, the given information is not enough to prove that ∆WXZ ≅ ∆YZX using ASA Congruence Theorem. 5.7 Using Congruent TrianglesQuestion 21. Answer: HJ ≅ ML KJ ≅ NM ∠J ≅ ∠M So, ∆HKJ ≅ ∆MNL using SAS congruence theorem. Therefore, ∠K ≅∠N Question 22. Answer: Given ∆QSV, ∆QVT, ∆PUS, ∆RTV SV ≅ QT, SV ≅ VT ∠1 ≅ ∠2 ∠QSV ≅ ∠QTV ∠QSV and ∠1 are vertically opposite angles. ∠QTV and ∠2 are vertically opposite angles. 5.8 Coordinate ProofsQuestion 23. Answer: Find the distance of all sides OP = √(h – 0)² + (k – 0)² = √h² + k² OQ = √(h -0)² + (k + j – 0 )² = √h² + (k + j)² PQ = √(h – h)² + (k – k – j)² = √0² + j² = j OR = √(0 – 0)² + (0 – j)² = √j² = j OQ = √(h – 0)² + (k + j – 0)² = √h² + (k + j)² QR = √(h – 0)² + (k + j – j)² = √h² + k² OP ≅ QR, OQ ≅ OQ, PQ ≅ OR So, ∆OPQ ≅ ∆QRO using SSS Congruence Theorem. Question 24. Question 25. From the image, the fourth vertex is (2k, k). Congruent Triangles TestWrite a Proof. Question 1. Answer: Given \(\overline{C A} \cong \overline{C B} \cong \overline{C D} \cong \overline{C E}\) As ∆ABC and ∆EDC are isosceles triangles. So, ∆ABC ≅ ∆EDC Question 2. Answer: \(\overline{J K}\|\overline{M L}, \overline{M J}\| \overline{K L}\) MK ≅ MK by reflexive property of congruence So, ∆MJK ≅ ∆KLM using SSS congruence theorem. Question 3. Answer: QR ≅ RS, ∠P ≅ ∠T ∠R ≅ ∠R by reflexive property of congruence So, ∆SRP ≅ ∆QRT using AAS Congruence Theorem Question
4. Answer: x = 12 Explanation: Question 5. Explanation: Question 6. ∠B ≅ ∠E, ∠C ≅ ∠F Then ∠A ≅ ∠D using third angles theorem. Write a plan through that ∠1 ≅∠2 Question 7. Answer: Question 8. Answer: Question 9. Answer: Yes Explanation: Question 10. Answer: The coordinates of T(3, 0), S(21, 0), R (12, 15), P(3, 30), Q(21, 30) PQ = √(21 – 3)² + (30 – 30)² = 18 PR = √(3 – 12)² + (30 – 15)² = √9² + 15² = √306 RQ = √(21 – 12)² + (30 – 15)² = √9² + 15² = √306 TR = √(12 – 3)² + (15 – 0)² = √9² + 15² = √306 TS = √(21 – 3)² + (0 – 0)² = √18² = 18 RS = √(21 – 12)² + (0 – 15)² = √9² + 15² = √306 PQ ≅ TS, PR ≅ TR, RQ ≅ RS So, △PQR ≅ △TRS using SSS congruence theorem Question 11. a. Classify the triangle shown by its sides. Answer: The triangle is an isosceles triangle. b. The measure of ∠3 is 40° What are the measures of ∠1 and ∠2? Explain your reasoning. Explanation: Congruent Triangles Cumulative AssessmentQuestion 1. Question 2. Answer: From step 4 the red line is parallel to m and passes through the pint P. So, point P is parallel to line m. Question 3. a. Write a composition of transformations that maps ∆JKL to ∆XYZ Answer: The coordinates of J(-3, 2), L (0, 2) K(-2, 4), X(1, -2), Y(2, -4), Z(4, -2) JL = √(0 + 3)² + (2 – 2)² = 3 XZ = √(4 – 1)² + (-2 + 2)² = 3 JK = √(-2 + 3)² + (4 – 2)² = √5 XY = √(2 – 1)² + (-4 + 2)² = √5 KL = √(-2 – 0)² + (4 – 2)² = √8 YZ = √(4 – 2)² + (-2 + 4)²= √8 JK ≅ XY, JL ≅ XZ, KL ≅ YZ b. Is the composition a congruence transformation? If so, identify all congruent corresponding parts. Question 4. (A) Q(1, 2, 3) (B) Q(4, 2) (C) Q(2, 3) (D) Q(-6, 7) Answer: (B) Q(4, 2) Explanation: RQ : RS = 2 : 3 The coordinates of R(-2, 5), S(8, 0) Let us validate answer with Q(2, 3) RQ = √(2 + 2)² + (3 – 5)² = √16 + 4 = √20 = 4.47 RS = √(8 + 2)² + (0 – 5)² = √125 = 11.18 If Q(4, 2) RQ = √(4 + 2)² + (2 – 5)² = √45 Now, RQ : RS = 2 : 3 So, Q(4, 2) Question 5. a. Prove ∆ABC ≅ ∆DEF using the given information. Answer: ∠B ≅ ∠E, BC ≅ EF So, ∆ABC ≅ ∆DEF b. Describe the composition of rigid motions that maps ∆ABC to ∆DEF Question 6. Question 7. (B) (C) (D) Answer:(D) Because by rotating the symbol, we get the same image. Question 8. Answer: The vertices of A(2, 5), B(4, 7), C(7, 4), D(5, 2) The slope of AB = \(\frac { 4 – 2 }{ 7 – 5 } \) = \(\frac { 2 }{ 2 } \) = 1 The slope of AD = \(\frac { 5 – 2 }{ 2 – 5 } \) = \(\frac { 3 }{ -3 } \) = -1 The slope of AB and AD are negative reciprocals So, AB is perpendicular to AD So, ABCD is a rectangle. Question 9. Answer: At step 1, we need to draw the base AB With A as centre and AB as radius draw an arc and with B as centre and AB as radius draw another arc intersecting the first arc at C. Name that point as C. Now formed triangle ABC is an equilateral triangle. |