Question

use the image to answer the question. in quadrilateral rstu, diagonal (overline{rt}) makes a pair of alternate interior angles, (angle srt) and (angle utr). if (mangle srt=\frac{1}{2}x + 21) and (mangle utr = 2x-7\frac{1}{2}), what value of (x) makes rstu a trapezoid? round to the nearest tenth. (1 point) 28.5 24 19.0 11.4

Explanation:

Step1: Set the angles equal

Since alternate - interior angles are equal for parallel lines (and for $RSTU$ to be a trapezoid, $RS\parallel UT$ and the alternate - interior angles $\angle SRT$ and $\angle UTR$ are equal), we set up the equation $\frac{1}{2}x + 21=2x - 7.5$.

Step2: Rearrange the equation

Subtract $\frac{1}{2}x$ from both sides: $21 = 2x-\frac{1}{2}x - 7.5$. Combine like - terms: $21=\frac{4x - x}{2}-7.5$, so $21=\frac{3}{2}x - 7.5$.

Step3: Isolate the variable term

Add 7.5 to both sides of the equation: $21 + 7.5=\frac{3}{2}x$, which gives $28.5=\frac{3}{2}x$.

Step4: Solve for x

Multiply both sides by $\frac{2}{3}$ to get $x=\frac{28.5\times2}{3}$. Calculate $28.5\times2 = 57$, then $x=\frac{57}{3}=19.0$.

Answer:

19.0