Question

in this exercise, lines m and n are parallel. find the measure of each numbered angle. the figure is not to scale
m∠6 = 125°
m∠7 = 55°
m∠8 = □°

Explanation:

Step1: Identify vertical angles

∠6 and ∠8 are vertical angles, so they are equal. But also, ∠7 and ∠8 are supplementary to the 125° angle? Wait, no, ∠7 and the 125° angle are supplementary (they form a linear pair). Wait, but we need ∠8. Wait, ∠6 and ∠8: vertical angles are equal. Wait, ∠6 is given? Wait, no, the problem is to find m∠8. Wait, ∠6 and ∠8 are vertical angles? Wait, no, looking at the diagram, ∠6 and ∠8: let's see, the transversal intersects line n, so ∠6 and ∠8 are vertical angles? Wait, no, ∠7 and the 125° angle are supplementary, so m∠7 = 180° - 125° = 55°, which is given. Then ∠6 and ∠8: ∠6 and ∠7 are supplementary? Wait, no, ∠6 and ∠8: vertical angles. Wait, ∠6 and the angle adjacent to 125°: wait, the 125° angle and ∠7 are supplementary, so m∠7 = 55°, then ∠6 and ∠8: since ∠6 and ∠7 are adjacent? Wait, no, let's think again. ∠8 and the 125° angle: are they vertical angles? Wait, the 125° angle and ∠8: no, the 125° angle and ∠6? Wait, maybe I made a mistake. Wait, the transversal crosses line n, so the angle marked 125° and ∠8: are they vertical angles? Wait, no, the angle marked 125° and ∠6: maybe. Wait, no, let's use vertical angles. ∠8 and ∠6: vertical angles? Wait, no, ∠6 and ∠8: let's see, the two angles formed by the transversal and line n: ∠6, ∠7, ∠8. Wait, the 125° angle is adjacent to ∠7, so ∠7 + 125° = 180°, so ∠7 = 55°, which is given. Then ∠8 and ∠6: ∠6 and ∠8 are vertical angles? Wait, no, ∠6 and ∠8: if ∠6 is vertical to the 125° angle? Wait, no, the 125° angle and ∠6: are they vertical? Wait, maybe the 125° angle and ∠6 are vertical angles? Wait, no, the diagram: line n is horizontal, transversal is a line, so the angle above line n (125°) and ∠6 (below line n) – no, vertical angles are opposite each other. Wait, maybe ∠8 is equal to 125°? Wait, no, ∠7 is 55°, so ∠8 should be equal to ∠6? Wait, no, let's check the given: m∠6 is 125°? Wait, the problem says m∠6 = 125°, m∠7 = 55°, then m∠8: since ∠6 and ∠8 are vertical angles, they are equal. Wait, vertical angles are congruent. So if ∠6 is 125°, then ∠8 is also 125°? Wait, no, that can't be, because ∠7 and ∠8 should be supplementary? Wait, no, ∠7 and ∠8: are they adjacent? Wait, ∠7 and ∠8: if line n is horizontal, and the transversal is a line, then ∠7 and ∠8 are adjacent and form a linear pair? No, ∠7 and ∠8: let's see, ∠7 is between the transversal and line n, and ∠8 is opposite? Wait, maybe I messed up. Wait, the 125° angle and ∠7 are supplementary, so ∠7 = 55°, then ∠8 is vertical to the 125° angle, so ∠8 = 125°? Wait, no, vertical angles: the angle opposite to 125° is ∠6, so ∠6 = 125°, then ∠8 is vertical to ∠6? No, ∠6 and ∠8: maybe ∠6 and ∠8 are vertical angles, so they are equal. So if ∠6 is 125°, then ∠8 is 125°? Wait, but ∠7 is 55°, and ∠7 and ∠8 should be supplementary? 55 + 125 = 180, yes! So ∠7 + ∠8 = 180°, so ∠8 = 180° - ∠7 = 180° - 55° = 125°? Wait, no, 180 - 55 is 125? Wait, 55 + 125 is 180, yes. So ∠8 is 125°? Wait, but also, ∠6 and ∠8 are vertical angles, so they are equal. Since ∠6 is 125°, ∠8 is 125°. Alternatively, since ∠7 and ∠8 are supplementary, and ∠7 is 55°, then ∠8 = 180 - 55 = 125°.

Step1: Determine relationship of ∠7 and ∠8

∠7 and ∠8 form a linear pair, so they are supplementary (sum to 180°).
$$m\angle7 + m\angle8 = 180^\circ$$

Step2: Substitute m∠7 = 55°

$$55^\circ + m\angle8 = 180^\circ$$

Step3: Solve for m∠8

$$m\angle8 = 180^\circ - 55^\circ = 125^\circ$$

Answer:

125