Question

use the diagram to the right to find the measures of the angles listed below. 8. \\( \angle ptq \\) 9. \\( \angle qtr \\) 10. \\( \angle pts \\) the measure of \\( \angle ptq \\) is (type an integer or a decimal. do not include the degree symbol in your answer)

Explanation:

Step1: Analyze the triangle \( \triangle QPT \)

Since \( \triangle QPT \) is an isosceles right - triangle (two sides are equal and \( \angle QPT = 90^{\circ} \)).
In an isosceles right - triangle, the two non - right angles are equal. Let \( \angle PTQ=\angle PQT = x \).
Using the angle - sum property of a triangle (\( \angle QPT+\angle PTQ+\angle PQT = 180^{\circ} \)).
Substitute \( \angle QPT = 90^{\circ} \) into the formula: \( 90 + x+x=180 \).

Step2: Solve the equation for \( x \)

Simplify the equation \( 90 + 2x=180 \).
Subtract 90 from both sides: \( 2x=180 - 90=90 \).
Divide both sides by 2: \( x = 45 \).

Answer:

45