Question

in the figure, m∠3 = 75° and m∠10 = 105°. find the measure of ∠15.

Explanation:

Step1: Identify Parallel Lines

First, check if lines \( w \) and \( x \) are parallel. Since \( m\angle3 = 75^\circ \) and \( m\angle10 = 105^\circ \), and \( \angle3 \) and \( \angle10 \) are same - side interior angles. The sum of same - side interior angles is \( 75^\circ+ 105^\circ=180^\circ \), so lines \( w \) and \( x \) are parallel (by the converse of the same - side interior angles theorem).

Step2: Identify Corresponding Angles

Now, look at \( \angle3 \) and \( \angle15 \). \( \angle3 \) and \( \angle15 \) are corresponding angles. When two parallel lines are cut by a transversal, corresponding angles are congruent.
Since \( m\angle3 = 75^\circ \), then \( m\angle15=m\angle3 = 75^\circ \).

Answer:

$75^\circ$