Question

consider the two triangles shown below.
(image of two triangles with angles and side lengths: first triangle has angles 84°, 43° and side length 7; second triangle has angles 61°, 41° and side length 8. note: the triangles are not drawn to scale.)
are the two triangles congruent?
choose 1 answer:
a yes
b no
c there is not enough information to say.

Explanation:

Step1: Calculate the third angle of the first triangle

The sum of angles in a triangle is \(180^\circ\). For the first triangle with angles \(84^\circ\) and \(43^\circ\), the third angle \(A_1\) is \(180 - 84 - 43 = 53^\circ\)? Wait, no, wait. Wait, let's recalculate. Wait, \(84 + 43 = 127\), so \(180 - 127 = 53\)? Wait, no, the second triangle: angles \(61^\circ\) and \(41^\circ\), so third angle \(A_2 = 180 - 61 - 41 = 78^\circ\)? Wait, no, that can't be. Wait, no, maybe I made a mistake. Wait, the first triangle: angles \(84^\circ\), \(43^\circ\), so third angle is \(180 - 84 - 43 = 53^\circ\)? Wait, no, \(84 + 43 = 127\), \(180 - 127 = 53\). The second triangle: angles \(61^\circ\), \(41^\circ\), so third angle is \(180 - 61 - 41 = 78^\circ\). Wait, but also, the sides: first triangle has a side of length 7, second has 8. For triangles to be congruent, corresponding angles and sides must be equal. Let's check the angles again. Wait, maybe I miscalculated the first triangle's angles. Wait, no, the first triangle: \(84^\circ\), \(43^\circ\), so third angle is \(180 - 84 - 43 = 53^\circ\). The second triangle: \(61^\circ\), \(41^\circ\), third angle \(180 - 61 - 41 = 78^\circ\). So the angle sets are different: first triangle angles \(84^\circ\), \(43^\circ\), \(53^\circ\); second triangle \(61^\circ\), \(41^\circ\), \(78^\circ\). Also, the sides: first has a side of length 7, second has 8. So the angles don't match, and the sides don't match (7 vs 8). Therefore, the triangles are not congruent.

Wait, maybe I made a mistake in angle calculation. Let's recheck the first triangle: \(84 + 43 = 127\), \(180 - 127 = 53\). Second triangle: \(61 + 41 = 102\), \(180 - 102 = 78\). So the angle measures are different. Also, the given sides: 7 and 8 are different. So the triangles can't be congruent.

Step2: Compare angles and sides

Congruent triangles must have all corresponding angles equal and corresponding sides equal. Here, the angle sets are different (first triangle: 84, 43, 53; second: 61, 41, 78) and the given sides (7 and 8) are different. So they are not congruent.

Answer:

B. No