Question

use the diagram to the right to find the measures of the angles listed below.

1. ∠ptq
2. ∠qtr
3. ∠pts
4. 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.
the measure of ∠ptq is
(type an integer or a decimal. do not include the degree symbol in your answer.)

Explanation:

Step1: Analyze triangle QPT

In $\triangle QPT$, we need to find congruent - angles first. Since there are no markings indicating equal - sides in $\triangle QPT$, none of the angles in $\triangle QPT$ are congruent. So the answer to the multiple - choice part is D.

Step2: Use angle - sum property of a right - triangle

In right - triangle $QPT$ with $\angle QPT = 90^{\circ}$. Let's assume we know some other angle relationships from the diagram (not fully shown in the provided image, but if we assume a right - triangle property). In a right - triangle, if one angle is $90^{\circ}$, and let's say we know the measure of $\angle PQT$. Then, using the fact that the sum of the interior angles of a triangle is $180^{\circ}$, for $\triangle QPT$ with $\angle QPT = 90^{\circ}$, we have $\angle PQT+\angle PTQ + 90^{\circ}=180^{\circ}$. So, $\angle PTQ=180^{\circ}-90^{\circ}-\angle PQT$. Without knowing $\angle PQT$, if we assume it's $45^{\circ}$ (for the sake of example, since the diagram is incomplete), then $\angle PTQ = 45$.

Answer:

45 (Note: This answer is based on an assumed value for $\angle PQT$ as the diagram is incomplete. In a complete problem, the value of $\angle PQT$ should be given or derivable from the diagram.)