Question

select the correct answer.
which triangles in the diagram are congruent?
triangle 1
triangle 2
triangle 3
triangle 4
a. triangle 1 and triangle 2
b. triangle 1, triangle 2, and triangle 3
c. triangle 1 and triangle 3
d. triangle 1, triangle 3, and triangle 4

Explanation:

Step1: Recall congruence (SAS rule)

Two triangles are congruent if two sides and the included angle are equal (SAS: Side-Angle-Side).

Step2: Analyze Triangle 1

Triangle 1 has: one side with 2 marks, one side with 1 mark, included angle $70^\circ$.

Step3: Analyze Triangle 2

Triangle 2 has: two sides with 1 mark, base with 2 marks; no $70^\circ$ included angle between the marked sides.

Step4: Analyze Triangle 3

Triangle 3 has: two sides with 1 mark, included angle $70^\circ$ between them. This matches the side-angle-side combination of Triangle 1 (the side with 2 marks in Triangle 1 is the base, corresponding to the unmarked base in Triangle 3, while the two 1-mark sides and included $70^\circ$ angle are congruent).

Step5: Analyze Triangle 4

Triangle 4 has: side with 2 marks, side with 1 mark, but the $70^\circ$ angle is not included between these two marked sides, so it does not fit SAS.

Step6: Match congruent triangles

Only Triangle 1 and Triangle 3 satisfy the SAS congruence condition.

Answer:

C. triangle 1 and triangle 3