Question

the diagram shows a straight line with points r, t, v, s (r---t---v---s), a 90° angle at t between r-t and u-t, a triangle u-t-v with ∠tuv = 64°, and an angle ((x + 8)^circ) at v between u-v and v-s.

Explanation:

Step1: Find the angle at V in triangle UVT

In triangle \( UVT \), we know that \( \angle UTV = 90^\circ \) and \( \angle TUV = 64^\circ \). The sum of angles in a triangle is \( 180^\circ \), so the angle at \( V \) (let's call it \( \angle UVT \)) is \( 180^\circ - 90^\circ - 64^\circ = 26^\circ \).

Step2: Use linear pair to find x

Angles \( \angle UVT \) and \( (x + 8)^\circ \) form a linear pair, so their sum is \( 180^\circ \)? Wait, no, actually, looking at the diagram, \( \angle UVT \) and \( (x + 8)^\circ \) are supplementary? Wait, no, \( \angle UVT \) and \( (x + 8)^\circ \) should be supplementary? Wait, no, actually, \( \angle UVT \) is \( 26^\circ \), and \( \angle UVT + (x + 8)^\circ = 180^\circ \)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, \( \angle UTV \) is \( 90^\circ \), so \( \angle UVT \) is \( 26^\circ \), and \( \angle UVT \) and \( (x + 8)^\circ \) are adjacent angles on a straight line? Wait, no, \( T, V, S \) are on a straight line, and \( T, U, V \) form a triangle. Wait, actually, \( \angle UVT \) and \( (x + 8)^\circ \) are supplementary? Wait, no, \( \angle UVT \) is \( 26^\circ \), so \( 26^\circ + (x + 8)^\circ = 180^\circ \)? No, that would mean \( x + 34 = 180 \), \( x = 146 \), which doesn't make sense. Wait, maybe I messed up the angle in the triangle. Wait, \( \angle UTV \) is \( 90^\circ \), \( \angle TUV = 64^\circ \), so \( \angle UVT = 180 - 90 - 64 = 26^\circ \). Then, \( \angle UVT \) and \( (x + 8)^\circ \) are vertical angles? No, they are adjacent. Wait, maybe \( \angle UVT \) and \( (x + 8)^\circ \) are equal? Wait, no, the diagram shows that \( \angle UVT \) is \( 26^\circ \), and \( (x + 8)^\circ \) is adjacent to it. Wait, maybe the angle at \( V \) in the triangle is \( 26^\circ \), and \( (x + 8)^\circ \) is equal to that? Wait, no, that would be if they are vertical angles, but they are on a straight line. Wait, maybe I made a mistake. Wait, let's re-examine. The line \( RTS \) is straight, so \( \angle RTV = 90^\circ \), so \( \angle UTV = 90^\circ \). Then triangle \( UVT \) has angles \( 90^\circ \), \( 64^\circ \), so the third angle \( \angle UVT = 26^\circ \). Then, \( \angle UVT \) and \( (x + 8)^\circ \) are supplementary? No, \( \angle UVT + (x + 8)^\circ = 180^\circ \)? Wait, no, \( \angle UVT \) is \( 26^\circ \), so \( 26 + (x + 8) = 180 \)? That would be \( x + 34 = 180 \), \( x = 146 \), which seems too big. Wait, maybe the angle \( (x + 8)^\circ \) is equal to \( \angle UVT \)? Wait, no, that would be if they are alternate angles, but the diagram doesn't show parallel lines. Wait, maybe the problem is that \( \angle UVT = (x + 8)^\circ \)? Wait, no, the diagram has \( \angle UVT \) labeled as \( 26^\circ \) (maybe a handwritten note), so if \( \angle UVT = 26^\circ \), then \( x + 8 = 26 \), so \( x = 18 \). Ah, that makes sense. Maybe the handwritten \( 26 \) is the measure of \( \angle UVT \), so \( x + 8 = 26 \).

Step3: Solve for x

If \( x + 8 = 26 \), then subtract 8 from both sides: \( x = 26 - 8 = 18 \).

Answer:

\( x = 18 \)