Question

question 9 of 10
the triangles shown below must be congruent.

a. true

b. false

Explanation:

Step1: Analyze Triangle Angles

First, recall that the sum of angles in a triangle is \(180^\circ\). For the first triangle (right - angled, one angle \(60^\circ\), another \(30^\circ\)), the right angle is \(90^\circ\), \(60^\circ + 30^\circ+90^\circ = 180^\circ\). For the second triangle (right - angled, one angle \(60^\circ\), another \(30^\circ\)), \(60^\circ+30^\circ + 90^\circ=180^\circ\). So both triangles are \(30^\circ - 60^\circ - 90^\circ\) triangles.

Step2: Analyze Congruence Conditions

We know the length of the side adjacent to the \(30^\circ\) angle (the side of length 5) in both triangles. In a \(30^\circ - 60^\circ - 90^\circ\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\) (opposite \(30^\circ\), \(60^\circ\), \(90^\circ\) respectively). The side of length 5 is adjacent to the \(30^\circ\) angle (in the first triangle, the right angle, \(30^\circ\) angle, and the side of length 5: the side of length 5 is adjacent to \(30^\circ\); in the second triangle, the right angle, \(60^\circ\) angle, and the side of length 5: the side of length 5 is also adjacent to \(30^\circ\) when we consider the angle measures). Using the Angle - Side - Angle (ASA) congruence criterion: we have two angles (\(30^\circ\) and \(90^\circ\)) and the included side (length 5) equal in both triangles. So the triangles are congruent.

Answer:

A. True