Question

consider the two triangles shown in the figure.
which statement must be true?
○ the two triangles are neither similar nor congruent.
○ the two triangles are similar but not congruent.
○ the two triangles are congruent but not similar.
○ the two triangles are both similar and congruent.

Explanation:

Step1: Find the third angle of the first triangle

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle of the first triangle be \(x\). Then \(x + 43^{\circ}+80^{\circ}=180^{\circ}\), so \(x = 180^{\circ}-(43^{\circ}+80^{\circ})=57^{\circ}\).

Step2: Analyze the second triangle

The second triangle has angles \(43^{\circ}\) and \(57^{\circ}\). Since the sum of angles in a triangle is \(180^{\circ}\), the third angle is \(180^{\circ}-(43^{\circ}+57^{\circ}) = 80^{\circ}\).

Step3: Check for similarity

Two triangles are similar if their corresponding angles are equal. Here, both triangles have angles \(43^{\circ},57^{\circ},80^{\circ}\), so they are similar.

Step4: Check for congruence

Two triangles are congruent if their corresponding sides and angles are equal. The first triangle has two sides (marked with one dash) and the second triangle has two sides (marked with one dash). Using the ASA (Angle - Side - Angle) congruence criterion (since two angles and the included side are equal), the two triangles are congruent.

Answer:

The two triangles are both similar and congruent.