Question

in the figure, m∠7 = 100°. find the measure of ∠11.

Explanation:

Step1: Identify Parallel Lines and Transversal

Lines \( t \) and \( u \) are parallel, and the transversal is the line intersecting \( t \) and \( u \) (same as the one intersecting \( s \)). \( \angle 7 \) and \( \angle 11 \) are corresponding angles? Wait, no, let's check the positions. Wait, actually, lines \( s \), \( t \), \( u \) are parallel? Wait, looking at the figure, lines \( s \), \( t \), \( u \) are all parallel (since they have the same direction arrows), and the transversal is the line \( m \) (the slanted line). Wait, \( \angle 7 \) is at the intersection of transversal and \( t \), \( \angle 11 \) is at intersection of transversal and \( u \). Since \( t \parallel u \), and the transversal cuts them, \( \angle 7 \) and \( \angle 11 \) are corresponding angles? Wait, no, wait: actually, \( \angle 7 \) and \( \angle 11 \) – let's see the vertical angles or corresponding. Wait, another way: \( \angle 7 \) and \( \angle 5 \) are vertical? No, \( \angle 7 \) and \( \angle 5 \) are adjacent? Wait, maybe \( \angle 7 \) and \( \angle 11 \) are corresponding angles because lines \( t \) and \( u \) are parallel, so corresponding angles are equal? Wait, no, wait: actually, the transversal is the same, so if \( t \parallel u \), then \( \angle 7 \) and \( \angle 11 \) – wait, maybe \( \angle 7 \) and \( \angle 11 \) are equal? Wait, no, wait, let's check the other way. Wait, \( \angle 7 \) and \( \angle 6 \) are supplementary? Wait, no, the problem is to find \( \angle 11 \). Wait, actually, lines \( t \) and \( u \) are parallel, and the transversal is the line that makes \( \angle 7 \) and \( \angle 11 \) corresponding angles. Wait, no, maybe \( \angle 7 \) and \( \angle 11 \) are equal because of parallel lines and corresponding angles. Wait, but let's think again. Wait, the lines \( s \), \( t \), \( u \) are all parallel (since they have the same direction), and the transversal is the slanted line (let's call it \( n \)). So \( \angle 7 \) is at \( t \cap n \), \( \angle 11 \) is at \( u \cap n \). So since \( t \parallel u \), corresponding angles are equal. Wait, but \( \angle 7 \) and \( \angle 11 \) – are they corresponding? Let's see the positions: \( \angle 7 \) is below the transversal, on the right side of line \( t \); \( \angle 11 \) is below the transversal, on the right side of line \( u \). So yes, corresponding angles. Therefore, \( m\angle 11 = m\angle 7 \)? Wait, no, wait, that can't be. Wait, maybe I made a mistake. Wait, \( \angle 7 \) and \( \angle 5 \) are vertical? No, \( \angle 7 \) and \( \angle 5 \) are adjacent? Wait, \( \angle 7 \) and \( \angle 8 \) are supplementary? Wait, no, let's look at the figure again. The figure has three parallel lines (s, t, u) and a transversal (m) and another transversal? Wait, no, the transversal is the slanted line (let's say line \( l \)) that intersects s, t, u. So at line t, the angles are 5,6,7,8; at line u, angles are 9,10,11,12. So \( \angle 7 \) and \( \angle 11 \): are they corresponding? Let's see, \( \angle 7 \) is in the lower right of the intersection with t, \( \angle 11 \) is in the lower right of the intersection with u. Since t and u are parallel, corresponding angles are equal. Wait, but \( m\angle 7 = 100^\circ \), so \( m\angle 11 = 100^\circ \)? Wait, no, maybe they are alternate interior angles? Wait, no, alternate interior would be between the lines. Wait, maybe I messed up. Wait, another approach: \( \angle 7 \) and \( \angle 5 \) are vertical? No, \( \angle 7 \) and \( \angle 5 \) are adjacent? Wait, \( \angle 7 \) and \( \angle 6 \) are supplementary? Wait, no, the key is that lines t and u are parallel, so the transversal creates corresponding angles. So \( \angle 7 \) and \( \angle 11 \) are corresponding angles, so they are equal. Wait, but let's check with another angle. \( \angle 7 \) and \( \angle 3 \): are they corresponding? No, s and t are parallel, so \( \angle 2 \) and \( \angle 6 \) are corresponding, \( \angle 3 \) and \( \angle 7 \) are corresponding? Wait, maybe. So if s || t, then \( \angle 3 = \angle 7 = 100^\circ \). Then, since t || u, \( \angle 3 \) and \( \angle 11 \) are corresponding? Wait, no, \( \angle 3 \) is at s and transversal, \( \angle 11 \) is at u and transversal. Wait, maybe the transversal is the same, so \( \angle 7 \) and \( \angle 11 \) are equal because t || u. So the measure of \( \angle 11 \) is 100 degrees.

Step2: Confirm Corresponding Angles

Since lines \( t \) and \( u \) are parallel (indicated by the same direction arrows) and cut by a transversal, \( \angle 7 \) and \( \angle 11 \) are corresponding angles. By the Corresponding Angles Postulate, corresponding angles are equal when lines are parallel. Thus, \( m\angle 11 = m\angle 7 \).

Answer:

\( 100^\circ \)