Find the measure of x.

Question

Line PU has points R and S between points P and U, lines QR and ST are parallel, line QR intersects line PU at point R, line ST intersects line PU at point S, the measure of angle PRQ is 135 degrees, and the measure of angle UST is 15 ( x plus 2 ) degrees.

x = −1
x = 7
x = 9
x = 13

x = −1
 x = 7
 x = 9
 x = 13

GPT-4o mini · Answer

To solve for $ x $, we can use the properties of parallel lines and angles formed when a transversal intersects them. Given: - Line $ QR $ intersects line $ PU $ at point $ R $, creating an angle $ PRQ $ of $ 135^\circ $. - Line $ ST $ intersects line $ PU $ at point $ S $, creating an angle $ UST $ which is equal to $ 15(x + 2) $ degrees. - Lines $ QR $ and $ ST $ are parallel. By the properties of parallel lines, we know corresponding angles are equal. Since angles $ PRQ $ and $ UST $ are on the same side of the transversal $ PU $, they are corresponding angles. This means: $$ \angle PRQ = \angle UST $$ Substituting the expressions we have: $$ 135^\circ = 15(x + 2) $$ Now, we'll solve for $ x $: 1. Divide both sides by $ 15 $: $$ \frac{135}{15} = x + 2 $$ $$ 9 = x + 2 $$ 2. Subtract $ 2 $ from both sides: $$ 9 - 2 = x $$ $$ x = 7 $$ Thus, the measure of $ x $ is $ \boxed{7} $.