Question

use the diagram to the right to find the measures of the angles listed below. 8. \\( \angle p t q \\) 9. \\( \angle q t r \\) 10. \\( \angle p t s \\) 8. to find \\( m \angle p t q \\), begin by examining \\( \triangle q p t \\). in \\( \triangle q p t \\), which of the following angles are congruent? select all that apply. \\( \square \\) a. \\( \angle q p t \\) \\( \square \\) b. \\( \angle p q t \\) \\( \square \\) c. \\( \angle p t q \\) \\( \square \\) d. none of the angles in \\( \triangle q p t \\) are congruent.

Explanation:

Step1: Analyze triangle \( \triangle QPT \)

In \( \triangle QPT \), we know that \( \angle QPT = 90^{\circ} \). Also, since two sides are marked as equal (the two short lines on the sides of the triangle), \( \triangle QPT \) is a right - isosceles triangle. In a right - isosceles triangle, the two non - right angles are congruent. Let \( \angle PQT=\angle QTP = x \). Using the angle - sum property of a triangle (\( \angle QPT+\angle PQT+\angle QTP = 180^{\circ} \)), and substituting \( \angle QPT = 90^{\circ} \), we get \( 90^{\circ}+x + x=180^{\circ} \), or \( 2x=90^{\circ} \), \( x = 45^{\circ} \). So \( \angle PQT\) and \( \angle QTP\) (i.e., \( \angle PTQ\)) are congruent.

Step2: Analyze \( \angle QTR \)

We know that \( \angle QTR+\angle RTS = \angle QTS \). Also, since \( \triangle RTS\) is an equilateral triangle (all sides are marked equal), \( \angle RTS = 60^{\circ} \). And \( \angle QTS=\angle QTP+\angle PTS\). But from step 1, \( \angle QTP = 45^{\circ} \). Wait, no. Let's use another approach. We know that \( \angle QTR = 180^{\circ}-\angle QTP-\angle RTS\) (since \( \angle QTP+\angle QTR+\angle RTS = 180^{\circ}\) as they form a linear - pair at point \( T\)). Substituting \( \angle QTP = 45^{\circ}\) and \( \angle RTS = 60^{\circ}\), we get \( \angle QTR=180^{\circ}-45^{\circ}-60^{\circ}=75^{\circ}\).

Step3: Analyze \( \angle PTS \)

\( \angle PTS=\angle QTR\) (vertically opposite angles are equal). So \( \angle PTS = 75^{\circ}\)

Answer:

1. \( \angle PQT\) and \( \angle PTQ\) are congruent (B and C are incorrect. Wait, no. Wait, in \( \triangle QPT\), \( \angle PQT\) and \( \angle PTQ\) are congruent. So for question 8 (the multiple - choice part), the answer is B and C. But let's check again. In \( \triangle QPT\), \( \angle QPT = 90^{\circ}\), and the two legs (the sides adjacent to the right angle) are equal. So by the base - angle theorem (in an isosceles triangle, the angles opposite to the equal sides are equal), \( \angle PQT=\angle PTQ\).
2. \( \angle QTR = 75^{\circ}\)
3. \( \angle PTS=75^{\circ}\)

For the multiple - choice part of question 8:
B. \( \angle PQT\)
C. \( \angle PTQ\)