Question

1. write pertinent congruent corresponding relationships to support your reason for triangle congruency

____≅__, __≅__, __≅____
which postulate or theorem shows
△abc ≅ △def? (lesson 5-3)
images of triangles abc and def

1. complete the table to answer the question.

select the pairs of triangles that must be
congruent to each other. (lesson 5-5)
a. image of triangle pair a b. image of triangle pair b
c. image of triangle pair c d. image of triangle pair d
table with rows a, b, c, d and columns congruent? (yes, no), reason, if yes

Explanation:

Step1: Analyze Triangle A

In triangle pair A, we have right triangles with one leg marked equal (the horizontal leg) and the hypotenuse marked equal (the slanted side with one mark). By the Hypotenuse - Leg (HL) congruence theorem for right triangles, if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. So triangle pair A is congruent.

Step2: Analyze Triangle B

In triangle pair B, we have right triangles with one leg (horizontal) marked equal and the other leg (vertical) marked equal (two marks). By the Side - Angle - Side (SAS) congruence postulate (since the right angle is included between the two legs), if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. So triangle pair B is congruent.

Step3: Analyze Triangle C

In triangle pair C, we have right triangles with one leg (slanted side with one mark) and an acute angle marked equal. By the Angle - Angle - Side (AAS) congruence theorem (since we have a right angle, one acute angle, and a side), if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the triangles are congruent. So triangle pair C is congruent.

Step4: Analyze Triangle D

In triangle pair D, we have right triangles with an acute angle marked equal and no other sides or angles marked as equal (except the right angle). We do not have enough information to prove congruence (we need at least one side to use AAS or ASA, or two sides for SAS/HL). So triangle pair D is not congruent.

For the table:

TRIANGLE PAIR	YES	NO	REASON, IF YES
B	$\boldsymbol{\checkmark}$		SAS (Side - Angle - Side)
C	$\boldsymbol{\checkmark}$		AAS (Angle - Angle - Side)
D		$\boldsymbol{\checkmark}$	Not enough info

For the first part (triangle \( \triangle ABC\cong\triangle DEF\)):
Looking at the triangles \( \triangle ABC\) and \( \triangle DEF\), we can see that \( AB\cong DE\) (one mark on the slanted side), \( AC\cong DF\) (two marks on the vertical side), and \( \angle A\cong\angle D\) (the included angle between the two sides). By the SAS (Side - Angle - Side) postulate, \( \triangle ABC\cong\triangle DEF\). The congruent corresponding relationships are \( AB\cong DE\), \( \angle A\cong\angle D\), \( AC\cong DF\).

Answer:

For question 9:

Congruent relationships: \( AB\cong DE\), \( \angle A\cong\angle D\), \( AC\cong DF\); Postulate/Theorem: SAS (Side - Angle - Side)

For question 10 (table):

TRIANGLE PAIR	YES	NO	REASON, IF YES
B	$\checkmark$		SAS
C	$\checkmark$		AAS
D		$\checkmark$	-

(For the multiple - choice - like part of question 10, the pairs that must be congruent are A, B, C)