Question

question 4 of 10
the triangles shown below must be congruent.

12

12

a. true

b. false

Explanation:

Step1: Recall Congruence Criteria

For right - angled triangles, we can use criteria like ASA (Angle - Side - Angle), SAS (Side - Angle - Side), SSS (Side - Side - Side), etc. Here, both triangles are right - angled (one angle is \(90^{\circ}\)), both have a \(45^{\circ}\) angle, and the side adjacent to the \(45^{\circ}\) angle (the leg with length 12) is equal.

Step2: Apply ASA Congruence

In triangle 1: Angles are \(90^{\circ}\), \(45^{\circ}\), and \(45^{\circ}\) (since the sum of angles in a triangle is \(180^{\circ}\)), and the side between the \(90^{\circ}\) and \(45^{\circ}\) angle is 12. In triangle 2: Angles are \(90^{\circ}\), \(45^{\circ}\), and \(45^{\circ}\), and the side between the \(90^{\circ}\) and \(45^{\circ}\) angle is 12. By the ASA (Angle - Side - Angle) congruence criterion, if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent. So these two right - angled triangles with a \(45^{\circ}\) angle, a right angle, and the included side of length 12 are congruent.

Answer:

A. True