Question

use the diagram to the right to find the measures of the angles listed below. 8. ∠ptq 9. ∠qtr 10. ∠pts none of the angles in △qpt are congruent. the measure of ∠ptq is 45 (type an integer or a decimal. do not include the degree symbol in your answer.) 9. in order to find m∠qtr, begin by finding a sum or difference of angle measures that is equal to m∠qtr. which of the following expressions a. m∠rtp - m∠ptq b. m∠rtp + m∠tpr c. m∠rtp - m∠tpr d. m∠rtp + m∠ptq

Explanation:

Step1: Analyze right - angled isosceles triangle

In right - angled triangle $\triangle QPT$, since $PQ = PT$, $\angle PTQ=45^{\circ}$ (in a right - angled isosceles triangle, the non - right angles are equal and their sum is $90^{\circ}$, so each is $45^{\circ}$).

Step2: Determine the relationship for $\angle QTR$

We know that $\angle RTP$ and $\angle QTR$ and $\angle PTQ$ are related. $\angle QTR=\angle RTP - \angle PTQ$.

Step3: Find $\angle RTP$

In $\triangle RTS$, it is an equilateral triangle (sides are equal), so $\angle RTS = 60^{\circ}$, and $\angle RTP=180^{\circ}-\angle RTS=120^{\circ}$ (linear pair).

Step4: Calculate $\angle QTR$

$\angle QTR=\angle RTP-\angle PTQ = 120 - 45=75$.

Step5: Calculate $\angle PTS$

$\angle PTS=\angle RTP+\angle RTS=120 + 60=180$.

Answer:

1. 45
2. A. $m\angle RTP - m\angle PTQ$, and $m\angle QTR = 75$
3. 180