Question

use the diagram to the right to find the measures of the angles listed below. 8. ∠ptq 9. ∠qtr 10. ∠pts 8. to find m∠ptq, begin by examining △qpt. in △qpt, which of the following angles are congruent? select all that apply. a. ∠pqt b. ∠ptq c. ∠qpt d. none of the angles in △qpt are congruent.

Explanation:

Step1: Analyze triangle QPT

In right - triangle QPT, we know that \(\angle QPT = 90^{\circ}\). Let's assume triangle QPT is isosceles right - triangle (from the congruence markings on the sides). In an isosceles right - triangle, \(\angle PQT=\angle PTQ\). Since the sum of angles in a triangle is \(180^{\circ}\), and \(\angle QPT = 90^{\circ}\), we have \(\angle PQT+\angle PTQ=180^{\circ}-\angle QPT\).

Step2: Calculate \(\angle PTQ\)

Let \(x = \angle PQT=\angle PTQ\). Then \(2x=180^{\circ}- 90^{\circ}=90^{\circ}\), so \(x = 45^{\circ}\), thus \(m\angle PTQ = 45^{\circ}\).

Step3: Analyze \(\angle QTR\)

We know that \(\angle PTR = 60^{\circ}\) (given in the diagram). Since \(\angle QTR=\angle PTR-\angle PTQ\), substituting the values of \(\angle PTR = 60^{\circ}\) and \(\angle PTQ = 45^{\circ}\), we get \(\angle QTR=60^{\circ}-45^{\circ}=15^{\circ}\).

Step4: Analyze \(\angle PTS\)

\(\angle PTS = 180^{\circ}-\angle PTR\) (linear - pair of angles). Since \(\angle PTR = 60^{\circ}\), then \(\angle PTS=180^{\circ}-60^{\circ}=120^{\circ}\).

Answer:

1. \(m\angle PTQ = 45^{\circ}\)
2. \(m\angle QTR = 15^{\circ}\)
3. \(m\angle PTS = 120^{\circ}\)