Question

1. which two triangles must be congruent? triangles a, b, c, d with angle and side labels a. triangle a and triangle b b. triangle b and triangle c c. triangle a and triangle d d. triangle a and triangle c

Explanation:

Step1: Find angles of Triangle A

In Triangle A, angles are \(50^\circ\), \(75^\circ\), so the third angle is \(180 - 50 - 75 = 55^\circ\). Side length is 12 (between \(75^\circ\) and \(50^\circ\)? Wait, no, let's check sides and angles. Wait, Triangle A has angles \(50^\circ\), \(75^\circ\), and side 12. Triangle C has angles \(55^\circ\), \(75^\circ\), and side 12. Wait, no, let's recalculate Triangle A's angles: sum of angles in triangle is \(180^\circ\). So Triangle A: \(50^\circ + 75^\circ + x = 180^\circ\) → \(x = 55^\circ\). So angles of A: \(50^\circ\), \(75^\circ\), \(55^\circ\). Triangle C: angles \(55^\circ\), \(75^\circ\), so third angle is \(180 - 55 - 75 = 50^\circ\). So Triangle A and Triangle C have angles \(50^\circ\), \(75^\circ\), \(55^\circ\) (in some order) and a side of length 12. Let's check the side: in Triangle A, the side 12 is between which angles? Wait, Triangle A: angle \(50^\circ\), angle \(75^\circ\), side 12 (opposite? No, let's see the diagram. Triangle A: one side 12, angles \(50^\circ\) and \(75^\circ\). Triangle C: side 12, angles \(55^\circ\) and \(75^\circ\). Wait, no, wait Triangle A's angles: \(50^\circ\), \(75^\circ\), so third angle \(55^\circ\). Triangle C: angles \(55^\circ\), \(75^\circ\), third angle \(50^\circ\). So both triangles have angles \(50^\circ\), \(75^\circ\), \(55^\circ\) and a side of length 12. So by ASA (Angle-Side-Angle) or AAS (Angle-Angle-Side), they should be congruent. Let's check other options:

Triangle B: angles \(65^\circ\), \(55^\circ\), so third angle \(60^\circ\) (180 - 65 - 55 = 60). So angles different from A and C. Triangle D: angles \(50^\circ\), \(70^\circ\), third angle \(60^\circ\), so different. So Triangle A and C have same angles and a side of 12, so congruent.

Step2: Verify other options

• Option A: Triangle A (angles 50,75,55) and B (65,55,60) → different angles, not congruent.
• Option B: Triangle B (65,55,60) and C (50,75,55) → different angles, not congruent.
• Option C: Triangle A (50,75,55) and D (50,70,60) → different angles, not congruent.
• Option D: Triangle A and C: same angles (50,75,55) and a side of 12, so congruent by AAS or ASA.

Answer:

D. Triangle A and Triangle C