use the diagram to the right to find the measures of the angles listed below. 8. ∠ptq 9. ∠qtr 10. ∠pts 8. to…

Explanation:

Step1: Analyze triangle QPT

In $\triangle QPT$, we see that $PQ = PT$ (marked by the equal - side symbols). In an isosceles triangle, the base - angles are congruent. The base - angles of $\triangle QPT$ are $\angle PQT$ and $\angle PTQ$.

Step2: Find $\angle PTQ$ in $\triangle QPT$

Since $\angle QPT = 90^{\circ}$ and the sum of angles in a triangle is $180^{\circ}$, and $\angle PQT=\angle PTQ$ in $\triangle QPT$. Let $x = \angle PQT=\angle PTQ$. Then $90 + x + x=180$. Combining like terms gives $90 + 2x = 180$. Subtracting 90 from both sides: $2x=90$, so $x = 45^{\circ}$. So, $m\angle PTQ = 45^{\circ}$.

Step3: Find $\angle QTR$

In $\triangle RPT$, we know that $\angle RTP$ and $\angle QTR$ are vertical angles. Also, in $\triangle RST$, we can find related angle - measures. But since $\triangle QPT$ and $\triangle RPT$ share side $PT$, and we know from the figure's symmetry and angle - relationships. $\angle QTR = 40^{\circ}$ (by observing the angle marked as $80^{\circ}$ in $\triangle RST$ and using the fact that the line $RT$ bisects the angle formed at $T$ related to the overall figure's symmetry and angle - sum properties).

Step4: Find $\angle PTS$

$\angle PTS=\angle PTQ+\angle QTR+\angle RTS$. We know $\angle PTQ = 45^{\circ}$, $\angle QTR = 40^{\circ}$, and from the isosceles $\triangle RST$ (since $RT = ST$), and the marked $80^{\circ}$ angle in $\triangle RST$, the base - angles of $\triangle RST$ are $50^{\circ}$ each. $\angle RTS = 80^{\circ}$, so $\angle PTS=45 + 40+80=165^{\circ}$.

Step5: Answer for congruent angles in $\triangle QPT$

In $\triangle QPT$, since $PQ = PT$, $\angle PQT$ and $\angle PTQ$ are congruent. So the correct options are A. $\angle PQT$ and B. $\angle PTQ$.

Answer:

8. $m\angle PTQ = 45^{\circ}$
9. $m\angle QTR = 40^{\circ}$
10. $m\angle PTS = 165^{\circ}$ For the multiple - choice part: A. $\angle PQT$, B. $\angle PTQ$