5.2 congruent polygons (pp. 385-394) learning target: understand congruence in terms of rigid m 10. in the diagram, ghjk ≅ lmnp. identify all pairs of congruent corresponding parts. then write another congruence statement for the quadrilaterals. 11. find m∠v. 12. the figure shows the flag of the czech republic. write a congruence statement for two of the polygons. then show that those polygons are congruent.

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### Problem 11: Find $ m\angle V $
# Explanation:
## Step 1: Identify Congruent Triangles
The triangles $ \triangle TUV $ and $ \triangle QRS $ are right triangles (since $ \angle U $ and $ \angle R $ are right angles, $ 90^\circ $) and appear to be congruent (marked with congruent angles and right angles). So, corresponding angles are equal. In $ \triangle QRS $, we know $ \angle S = 74^\circ $, $ \angle R = 90^\circ $. The sum of angles in a triangle is $ 180^\circ $, so $ \angle Q = 180^\circ - 90^\circ - 74^\circ = 16^\circ $. But since $ \triangle TUV \cong \triangle QRS $, $ \angle V $ corresponds to $ \angle S $? Wait, no, let's check the markings. The marked angle in $ \triangle TUV $ is at $ T $, and in $ \triangle QRS $ at $ S $. Wait, actually, the right angle is $ \angle U $ and $ \angle R $, and the other marked angle (acute) is at $ T $ and $ S $? Wait, no, the triangle $ \triangle QRS $ has $ \angle R = 90^\circ $, $ \angle S = 74^\circ $, so $ \angle Q = 180 - 90 - 74 = 16^\circ $? Wait, no, maybe the triangles are congruent, so $ \angle V $ corresponds to $ \angle Q $? Wait, no, let's re-examine. The right angle is $ \angle U $ (in $ \triangle TUV $) and $ \angle R $ (in $ \triangle QRS $). The angle at $ T $ (in $ \triangle TUV $) is marked congruent to the angle at $ S $ (in $ \triangle QRS $)? Wait, the diagram shows $ \triangle TUV $ with right angle at $ U $, angle at $ T $ marked, and $ \triangle QRS $ with right angle at $ R $, angle at $ S $ marked $ 74^\circ $. So if the triangles are congruent, then $ \angle T \cong \angle S $, so $ \angle T = 74^\circ $. Then in $ \triangle TUV $, $ \angle U = 90^\circ $, $ \angle T = 74^\circ $, so $ \angle V = 180^\circ - 90^\circ - 74^\circ = 16^\circ $? Wait, no, that can't be. Wait, maybe the correspondence is $ \triangle TUV \cong \triangle SRQ $? Wait, maybe I mixed up the correspondence. Let's assume the triangles are congruent, so corresponding angles are equal. The right angles are $ \angle U $ and $ \angle R $, so $ \angle U \cong \angle R $. The angle at $ T $ (in $ \triangle TUV $) and angle at $ S $ (in $ \triangle QRS $) are congruent, so $ \angle T \cong \angle S $. Therefore, the third angles $ \angle V $ and $ \angle Q $ must be congruent. Wait, but in $ \triangle QRS $, $ \angle S = 74^\circ $, $ \angle R = 90^\circ $, so $ \angle Q = 180 - 90 - 74 = 16^\circ $. But that seems small. Wait, maybe the angle at $ S $ is $ 74^\circ $, so the angle at $ T $ is also $ 74^\circ $, so in $ \triangle TUV $, angles are $ 90^\circ $ (at $ U $), $ 74^\circ $ (at $ T $), so $ \angle V = 180 - 90 - 74 = 16^\circ $? Wait, that must be it. So $ m\angle V = 16^\circ $? Wait, no, wait, maybe I got the correspondence wrong. Let's do it properly.

In a right triangle, the two acute angles are complementary (sum to $ 90^\circ $). So in $ \triangle QRS $, right-angled at $ R $, $ \angle S = 74^\circ $, so the other acute angle $ \angle Q = 90^\circ - 74^\circ = 16^\circ $. Since $ \triangle TUV \cong \triangle QRS $ (by congruent right angles and congruent acute angles), $ \angle V $ corresponds to $ \angle Q $, so $ m\angle V = 16^\circ $? Wait, no, wait, maybe $ \angle V $ corresponds to $ \angle S $? No, that would be $ 74^\circ $, but that would make the sum exceed $ 180^\circ $. Wait, no, let's check the triangle angle sum. For $ \triangle TUV $, angles are $ \angle U = 90^\circ $, $ \angle T $, and $ \angle V $. If $ \triangle TUV \cong \triangle QRS $, then $ \angle T \cong \angle S = 74^\circ $, $ \angle U \cong \angle R = 90^\circ $, so $ \angle V = 180 - 90 - 74 = 16^\circ $. Yes, that makes sense. So $ m\angle V = 16^\circ $? Wait, no, wait, the problem is to find $ m\angle V $. Let's confirm:

Sum of angles in a triangle: $ \angle U + \angle T + \angle V = 180^\circ $. $ \angle U = 90^\circ $, $ \angle T = \angle S = 74^\circ $ (since triangles are congruent, corresponding angles), so $ 90 + 74 + \angle V = 180 $. Then $ \angle V = 180 - 90 - 74 = 16^\circ $. Wait, but that seems small. Wait, maybe the angle at $ S $ is $ 74^\circ $, so the angle at $ T $ is $ 74^\circ $, so $ \angle V = 180 - 90 - 74 = 16^\circ $. Yes, that's correct.

# Answer:
$ m\angle V = 16^\circ $

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Explanation:

Problem 11: Find \( m\angle V \)

Step 1: Identify Congruent Triangles

Answer: