Question

1. what is the criterion for proving the similarity of triangles by comparing two pairs of corresponding angles?a. asab. sasc. sssd. aa9. given △abc ≅ △def, which of the following can be proven using cpctc?a. △abc and △def have equal perimeters.b. the corresponding medians of △abc and △def are equal.c. corresponding angles in △abc and △def are equal.d. the areas of △abc and △def are identical.10. what ensures a triangle constructed using the sss method is unique?a. the presence of a right angleb. the inclusion of two equal anglesc. the flexibility of side lengths d. the fixed lengths of all three sides11. what is the common term for the longest side of a right triangle?a. altitudeb. medianc. legd. hypotenuse12. which of the following is not a postulate used to prove triangle congruence?a. sssb. asac. aad. sas13. what property distinguishes similar triangles from congruent triangles?a. similar triangles are identical in size and , while congruent triangles are not.b. similar triangles have equal side lengths, while congruent triangles do not.c. congruent triangles have proportional sides, while similar triangles do not.d. similar triangles have proportional sides and congruent angles, while congruent triangles have equal sides and angles.14. which postulate or theorem is typically used to establish triangle congruence before applying cpctc?a. midpoint theoremb. sas postulatec. cpctc itselfd. pythagorean theorem

Explanation:

• Question 8: The AA (Angle - Angle) criterion is used to prove the similarity of triangles by comparing two pairs of corresponding angles. ASA (Angle - Side - Angle) is for congruence, SAS (Side - Angle - Side) is for congruence, and SSS (Side - Side - Side) is for congruence.
• Question 9: CPCTC (Corresponding Parts of Congruent Triangles are Congruent) is used to prove that corresponding angles (or sides) of congruent triangles are equal. If \(\triangle ABC\cong\triangle DEF\), then corresponding angles in \(\triangle ABC\) and \(\triangle DEF\) are equal. Equal perimeters, equal medians, and identical areas are not directly proven by CPCTC.
• Question 10: The SSS (Side - Side - Side) method constructs a unique triangle when the fixed lengths of all three sides are given. A right - angle (for SSS) doesn't ensure uniqueness in the context of the SSS construction method itself. Two equal angles are related to angle - based criteria, and flexibility of side lengths would not give a unique triangle.
• Question 11: The hypotenuse is the common term for the longest side of a right - triangle. An altitude is a perpendicular segment from a vertex to the opposite side, a median is a segment from a vertex to the mid - point of the opposite side, and a leg is one of the non - hypotenuse sides of a right - triangle.
• Question 12: AA (Angle - Angle) is a similarity criterion, not a congruence criterion. SSS (Side - Side - Side), ASA (Angle - Side - Angle), and SAS (Side - Angle - Side) are postulates used to prove triangle congruence.
• Question 13: Similar triangles have proportional sides and congruent angles, while congruent triangles have equal sides and angles. Similar triangles are not identical in size (so their areas are not identical), congruent triangles have equal side lengths, and congruent triangles have equal (not just proportional) sides.
• Question 14: SAS (Side - Angle - Side) Postulate (or other congruence postulates like SSS, ASA) is typically used to establish triangle congruence before applying CPCTC. The Midpoint Theorem is about the mid - points of sides of a triangle, CPCTC is used after congruence is established, and the Pythagorean Theorem is about the relationship between the sides of a right - triangle.

Answer:

• 8. d. AA
• 9. c. Corresponding angles in \(\triangle ABC\) and \(\triangle DEF\) are equal.
• 10. d. The fixed lengths of all three sides
• 11. d. Hypotenuse
• 12. c. AA
• 13. d. Similar triangles have proportional sides and congruent angles, while congruent triangles have equal sides and angles.
• 14. b. SAS Postulate