Question: Find the measure of $\angle t$ given that $p \parallel q$. [asy] pair A,B,C,D,E,F,G,H; A =…

Find the measure of $\angle t$ given that $p \parallel q$.

[asy] pair A,B,C,D,E,F,G,H; A = (0,0); B = (1,0); C = (2,0); D = (3,0); E = (0,1); F = (1,1); G = (2,1); H = (3,1); draw(A—B—C—D—A); draw(E—F—G—H—E); draw(A—E); draw(B—F); draw(C—G); draw(D—H); label(“$p$”, (3,0.5), E); label(“$q$”, (3,1.5), E); label(“$x=150^\circ$”, (2.5,0.5), S); label(“$y=30^\circ$”, (2.5,1.5), S); label(“$t$”, (1.5,1), N); label(“$u$”, (1.5,1), E); label(“$v$”, (1.5,0.5), W); label(“$w$”, (1.5,0.5), S); [/asy]

Solution

To find the measure of angle \( \angle u \), given that lines \( p \) and \( q \) are parallel and using the information in the image: Since \( \angle x \) and \( \angle y \) are given as \( 150^\circ \) and \( 30^\circ \) respectively, and these angles are alternate interior angles when lines \( p \parallel q \), they are congruent to their corresponding angles. By the Alternate Interior Angles Theorem, \( \angle u = \angle x \). Thus, the measure of \( \angle u \) is \( 150^\circ \).