Question

m∠ p = 50°, the m∠ q = 65°, and the length of pq is 4 cm.

Explanation:

To solve this, we use the Angle - Angle - Side (AAS) congruence criterion for triangles. AAS states that if two angles and a non - included side of one triangle are equal to the corresponding two angles and the non - included side of another triangle, then the triangles are congruent.

Step 1: Analyze the given angle measures

We know that the sum of the interior angles of a triangle is \(180^{\circ}\). For a triangle with angles \(m\angle P = 50^{\circ}\) and \(m\angle Q=65^{\circ}\), the third angle \(m\angle R=180-(50 + 65)=65^{\circ}\) (using the angle - sum property of a triangle: \(m\angle P+m\angle Q + m\angle R=180^{\circ}\)).

Step 2: Compare with the triangles in the options

We need to find a triangle where two angles match \(50^{\circ}\) and \(65^{\circ}\) and the length of the non - included side (the side \(PQ = 4\) cm in the reference triangle) is also \(4\) cm.

• For the first option (top - left triangle):
• Check the angles. If we calculate the third angle (using \(180-(50 + 65)=65^{\circ}\)), and the side \(PQ = 4\) cm. The angles \(50^{\circ}\), \(65^{\circ}\) and the side length \(4\) cm (non - included side) match the given triangle's angles and side length. So, by AAS congruence, this triangle is congruent to the reference triangle.
• For the other options:
• In the top - right triangle, the angle measures do not match (the third angle is not \(65^{\circ}\) as per the given angle labels).
• In the bottom - left triangle, the right - angle (if present) does not match the angle measures of the reference triangle.
• In the bottom - right triangle, the angle of \(45^{\circ}\) does not match the required angle measures of the reference triangle.

Answer:

The first option (the top - left triangle with the circle labeled 4) is the correct one.