Question

1. (\triangle rsu) es equilátero. ¿cuál es la medida de (angle tou)? (2pts.)

Explanation:

Step1: Find the measure of ∠TUQ

In right - triangle \( \triangle QUP\), we know that the sum of angles in a triangle is \(180^{\circ}\). Given \( \angle Q = 55^{\circ}\) and \( \angle P=90^{\circ}\).

$$ \angle TUQ=180^{\circ}-\angle Q - \angle P $$

$$ \angle TUQ = 180^{\circ}-55^{\circ}-90^{\circ}=35^{\circ} $$

Step2: Use the property of an equilateral triangle

Since \( \triangle RSU\) is equilateral, \( \angle RSU = 60^{\circ}\).

Step3: Find the measure of \( \angle TOU\)

Using the exterior - angle property of a triangle (\( \angle TOU\) is the exterior angle of \( \triangle TUQ\)). Also, \( \angle STU = 180^{\circ}-\angle RSU=180^{\circ}- 60^{\circ}=120^{\circ}\) (linear pair).
In \( \triangle TUQ\), \( \angle TOU=\angle TUQ+\angle STU\) (exterior - angle theorem: the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles)

$$ \angle TOU=35^{\circ}+120^{\circ} $$

$$ \angle TOU = 155^{\circ} $$

Answer:

\(155^{\circ}\)