Question

the measure of ∠5 is 110°. find the remaining angle measures.
look for pairs of vertical angles and supplementary angles to find the missing measures.
m∠6 = 70°
m∠7 =?°
m∠8 =?°

Explanation:

Step1: Find \( m\angle7 \) (vertical angles with \( \angle5 \))

Vertical angles are equal. \( \angle5 \) and \( \angle7 \) are vertical angles, so \( m\angle7 = m\angle5 = 110^\circ \).

Step2: Find \( m\angle8 \) (supplementary with \( \angle7 \))

Supplementary angles sum to \( 180^\circ \). Let \( m\angle8 = x \), then \( x + 110^\circ = 180^\circ \), so \( x = 180^\circ - 110^\circ = 70^\circ \). Also, \( \angle6 \) and \( \angle8 \) are vertical angles, so \( m\angle8 = m\angle6 = 70^\circ \) (consistent). Wait, correction: Wait, \( \angle5 \) and \( \angle6 \): Wait, original \( m\angle6 = 70^\circ \) (given). Then \( \angle5 \) and \( \angle6 \): Wait, maybe linear pair. Wait, \( \angle5 \) and \( \angle6 \): Wait, the diagram: \( \angle5 \) and \( \angle6 \) are adjacent? Wait, no, the first part: \( m\angle6 = 70^\circ \) (given). Then \( \angle5 \) and \( \angle7 \) are vertical, so \( m\angle7 = 110^\circ \) (since \( m\angle5 = 110^\circ \)). Then \( \angle7 \) and \( \angle8 \) are supplementary, so \( m\angle8 = 180 - 110 = 70^\circ \). Wait, but \( \angle6 \) is 70, so \( \angle8 \) is vertical to \( \angle6 \), so \( m\angle8 = 70^\circ \), and \( \angle7 \) is vertical to \( \angle5 \), so \( m\angle7 = 110^\circ \).

Wait, let's re-express:

Given \( m\angle5 = 110^\circ \).

• \( \angle5 \) and \( \angle7 \) are vertical angles: \( m\angle7 = m\angle5 = 110^\circ \).
• \( \angle6 \) and \( \angle8 \) are vertical angles: \( m\angle8 = m\angle6 = 70^\circ \).
• Also, \( \angle5 \) and \( \angle6 \) are supplementary? Wait, \( 110 + 70 = 180 \), so they are linear pair (supplementary), which makes sense. Then \( \angle7 \) and \( \angle8 \) are also linear pair: \( 110 + 70 = 180 \), correct.

So:

Step1: \( m\angle7 \) (vertical to \( \angle5 \))

\( m\angle7 = m\angle5 = 110^\circ \)

Step2: \( m\angle8 \) (vertical to \( \angle6 \) or supplementary to \( \angle7 \))

\( m\angle8 = m\angle6 = 70^\circ \) (or \( 180 - 110 = 70 \))

Answer:

\( m\angle7 = \boldsymbol{110^\circ} \), \( m\angle8 = \boldsymbol{70^\circ} \)