Question

△abc ~ △srq. what is m∠c =? there are two triangles, one labeled abc with angle at a being 108°, angle at b being 34°, and the other labeled srq with angle at r being 34°, angle at q being 38°. work it out. not feeling ready yet? these can help: identify similar figures, solve proportions, lesson: similarity

Explanation:

Step1: Recall triangle angle sum

The sum of angles in a triangle is \(180^\circ\).

Step2: Calculate angle in similar triangle

For \(\triangle ABC\) (similar to \(\triangle SRQ\)), we know two angles: \(\angle A = 108^\circ\), \(\angle B = 34^\circ\). So \(\angle C=180^\circ - 108^\circ - 34^\circ\).
\(180 - 108 - 34 = 38^\circ\)? Wait, no, wait. Wait, similar triangles have corresponding angles equal. Wait, maybe I misread. Wait, the other triangle \(\triangle SRQ\) has angles \(34^\circ\) (at R), \(38^\circ\) (at Q). Wait, no, the first triangle \(\triangle ABC\) has angle at A: \(108^\circ\), angle at B: \(34^\circ\). So angle at C: \(180 - 108 - 34 = 38^\circ\)? Wait, but maybe the similarity is \(\triangle ABC \sim \triangle SRQ\), so corresponding angles. Wait, maybe I made a mistake. Wait, let's check again. The sum of angles in a triangle is \(180^\circ\). So for \(\triangle ABC\), angles are \(\angle A = 108^\circ\), \(\angle B = 34^\circ\), so \(\angle C = 180 - 108 - 34 = 38^\circ\)? Wait, but the other triangle has angle at Q: \(38^\circ\), angle at R: \(34^\circ\), so angle at S would be \(180 - 38 - 34 = 108^\circ\), which matches \(\angle A\). So yes, similarity: \(\angle A\) corresponds to \(\angle S\), \(\angle B\) corresponds to \(\angle R\), \(\angle C\) corresponds to \(\angle Q\). So \(\angle C = \angle Q = 38^\circ\)? Wait, no, wait the problem says \(\triangle ABC \sim \triangle SRQ\). So the order is A->S, B->R, C->Q. So \(\angle A = \angle S = 108^\circ\), \(\angle B = \angle R = 34^\circ\), \(\angle C = \angle Q = 38^\circ\). So calculating \(\angle C\): \(180 - 108 - 34 = 38\). Wait, but let's do the calculation again: \(180 - 108 = 72\), \(72 - 34 = 38\). So \(\angle C = 38^\circ\).

Wait, maybe I messed up the similarity notation. But regardless, using the angle sum property: sum of angles in a triangle is \(180^\circ\). So for \(\triangle ABC\), \(\angle A = 108^\circ\), \(\angle B = 34^\circ\), so \(\angle C = 180 - 108 - 34 = 38^\circ\).

Answer:

\(38\)