Question

use the diagram to the right to find the measures of the angles listed below. 8. ∠ptq 9. ∠qtr 10. ∠pts d. m∠rtp + m∠ptq the measure of m∠rtp is 60. (type an integer or a decimal. do not include the degree symbol in your answer.) the measure of ∠qtr is 15. (type an integer or a decimal. do not include the degree symbol in your answer.) 10. to find the measure of ∠pts, add the measures of ∠rtp and ∠rts. what is the measure of ∠rts? (type an integer or a decimal. do not include the degree symbol in your answer.)

Explanation:

Step1: Analyze right - angled triangle

In the right - angled triangle with right - angle at the relevant vertex, assume some angle - relationships. Since the given $\angle RTP = 60$. And we know that in a right - angled triangle formed by the lines related to point $T$, $\angle PTQ$ and $\angle RTP$ are complementary (as they form a right - angle). So $\angle PTQ=90 - 60=30$.

Step2: Given value of $\angle QTR$

It is given that $\angle QTR = 15$.

Step3: Find $\angle RTS$

In triangle $RTS$, since the triangle is isosceles (from the congruence marks), and the sum of angles in a triangle is 180. Let the base - angles be equal. If we consider the angle at $R$ in $\triangle RTS$, and using the fact that the vertical - angle to the given 60 - degree angle is also 60 degrees. The sum of angles in $\triangle RTS$ gives us the base - angle $\angle RTS=(180 - 60)\div2 = 60$.

Step4: Calculate $\angle PTS$

$\angle PTS=\angle RTP+\angle RTS$. Substituting the values of $\angle RTP = 60$ and $\angle RTS = 60$, we get $\angle PTS=60 + 60=120$.

Answer:

1. 30
2. 15
3. 120