Question

in the figure, m∠7 = 100°. find the measure of ∠2.

Explanation:

Step1: Identify angle relationships

∠7 and ∠5 are supplementary? No, wait, ∠7 and ∠8? Wait, no, looking at the transversal and parallel lines (assuming lines are parallel as per the diagram with multiple parallel lines cut by a transversal). ∠7 and ∠2: Let's see, ∠7 and ∠5? Wait, no, ∠7 and ∠2: actually, ∠7 and ∠2 are corresponding? Wait, no, let's check vertical angles or alternate exterior. Wait, ∠7 and ∠2: Wait, ∠7 and ∠5 are supplementary? No, ∠7 and ∠8 are supplementary? Wait, m∠7 = 100°, so ∠8 would be 80°, but ∠2 and ∠8: are they corresponding? Wait, lines s, t, u are parallel, cut by transversal m. So ∠2 and ∠8: are they corresponding? Wait, ∠2 and ∠5: no, ∠2 and ∠8: let's see, ∠7 and ∠2: actually, ∠7 and ∠2 are equal? No, wait, ∠7 and ∠2: wait, ∠7 and ∠5 are vertical? No, ∠5 and ∠7: ∠5 and ∠7 are adjacent supplementary? Wait, m∠7 = 100°, so ∠5 would be 80°? Wait, no, ∠5 and ∠7: they are adjacent, forming a linear pair? Wait, ∠5 + ∠7 = 180°, so ∠5 = 80°. Then ∠2 and ∠5: are they corresponding angles? If lines s and t are parallel, cut by transversal m, then ∠2 and ∠5 would be corresponding, so equal. Wait, no, ∠2 and ∠5: let's see the diagram. ∠2 is at the intersection of s and m, ∠5 is at t and m. So if s || t, then ∠2 and ∠5 are corresponding, so m∠2 = m∠5. But ∠5 and ∠7: linear pair, so m∠5 + m∠7 = 180°, so m∠5 = 180 - 100 = 80°. Then ∠2 = ∠5 = 80°, or is ∠2 equal to ∠7? Wait, no, maybe ∠2 and ∠7 are alternate exterior? Wait, no, let's re-examine. The transversal is the line with angles 1,2,3,4. The parallel lines are s, t, u. So ∠7 is at t and the transversal, ∠2 is at s and the transversal. So ∠2 and ∠7: are they alternate exterior? Wait, ∠2 is above line s, ∠7 is below line t. Wait, maybe ∠2 and ∠7 are equal? No, that can't be. Wait, maybe ∠2 and ∠8: ∠8 is vertical to ∠5? No, ∠8 and ∠5 are vertical? Wait, ∠5 and ∠8: no, ∠5 and ∠7 are adjacent, ∠5 and ∠6 are vertical, ∠7 and ∠8 are vertical. Wait, m∠7 = 100°, so m∠8 = 100°? No, ∠7 and ∠8 are adjacent, forming a linear pair? Wait, no, ∠7 and ∠8: are they vertical? No, ∠5 and ∠7: ∠5 is adjacent to ∠7, so ∠5 + ∠7 = 180°, so ∠5 = 80°, ∠7 = 100°. Then ∠2 and ∠5: corresponding angles, so m∠2 = 80°. So the measure of ∠2 is 80 degrees.

Step2: Calculate m∠2

Since ∠5 and ∠7 form a linear pair, \( m\angle5 + m\angle7 = 180^\circ \). Given \( m\angle7 = 100^\circ \), we solve for \( m\angle5 \):
\( m\angle5 = 180^\circ - 100^\circ = 80^\circ \).

Now, ∠2 and ∠5 are corresponding angles (since lines \( s \) and \( t \) are parallel, cut by transversal \( m \)). Corresponding angles are equal, so \( m\angle2 = m\angle5 = 80^\circ \).

Answer:

80