Question

in the figure, ( mangle3 = 75^{circ} ) and ( mangle10 = 105^{circ} ). find the measure of ( angle5 ).

Explanation:

Step1: Find the measure of ∠7

Since ∠3 and ∠7 are vertical angles, vertical angles are equal.
So, \(m\angle7 = m\angle3=75^{\circ}\)

Step2: Check the relationship between ∠7 and ∠10

We know that \(m\angle7 + m\angle10=75^{\circ}+ 105^{\circ}=180^{\circ}\). When the sum of two angles is \(180^{\circ}\), they are supplementary. If a transversal intersects two lines such that a pair of same - side interior angles are supplementary, then the two lines are parallel. So, the line with angles ∠7 and ∠10 (assuming the transversal) makes the two lines parallel.

Step3: Use the property of parallel lines and corresponding angles

Since the lines are parallel (from Step 2), ∠5 and ∠7 are corresponding angles. Corresponding angles formed by parallel lines and a transversal are equal.

Answer:

\(75^{\circ}\)