Question

ite an equation and solve
2
(2x + 10)°
72°

Explanation:

Step1: Analyze angle sum

The right angle (90°), 72°, and \((2x + 10)^\circ\) sum to 360°? No, wait, it's a full rotation? Wait, no, looking at the diagram, the angles around the point: Wait, actually, the right angle (90°), 72°, and \((2x + 10)^\circ\) – wait, no, maybe it's a straight angle? Wait, no, the diagram has a right angle (90°), 72°, and the angle \((2x + 10)^\circ\) – wait, maybe the sum of the angles around the vertex? Wait, no, perhaps the angle \((2x + 10)^\circ\) plus 72° equals 90°? Wait, no, the right angle is 90°, so maybe \((2x + 10) + 72 = 90\)? Wait, let's check.

Wait, the diagram: there's a right angle (90°), a 72° angle, and the angle \((2x + 10)^\circ\). Wait, maybe the three angles: the right angle, 72°, and \((2x + 10)^\circ\) – no, wait, maybe the angle \((2x + 10)^\circ\) and 72° are complementary to the right angle? Wait, no, let's think again.

Wait, the total around a point is 360°, but here, maybe it's a quadrant? Wait, no, the diagram shows a right angle (90°), a 72° angle, and the angle \((2x + 10)^\circ\). Wait, perhaps the sum of \((2x + 10)^\circ\) and 72° is equal to 90°? Wait, no, that would be if they are complementary. Wait, let's see: the right angle is 90°, so the other two angles (72° and \((2x + 10)^\circ\)) should add up to 90°? Wait, no, maybe the angle \((2x + 10)^\circ\) is equal to 90° - 72°? No, that would be 18°, but then \(2x + 10 = 18\), so \(2x = 8\), \(x = 4\). But that seems too small. Wait, maybe I misread the diagram.

Wait, the diagram has a right angle (90°), a 72° angle, and the angle \((2x + 10)^\circ\). Wait, maybe the angle \((2x + 10)^\circ\) is adjacent to the 72° angle, and together with the right angle, they form a straight line? No, a straight line is 180°. Wait, 90° + 72° + (2x + 10)° = 180°? Let's check: 90 + 72 + 2x + 10 = 180 → 172 + 2x = 180 → 2x = 8 → x = 4. But that's the same as before. Wait, maybe the diagram is a right angle, with one angle 72°, and the other angle is (2x + 10)°, so 72 + (2x + 10) = 90? Let's solve that: 2x + 82 = 90 → 2x = 8 → x = 4. Hmm.

Wait, maybe the problem is that the angle (2x + 10)° and 72° are such that their sum is 90°, because there's a right angle. So:

Step1: Set up equation

\( (2x + 10) + 72 = 90 \)

Step2: Simplify left side

\( 2x + 82 = 90 \)

Step3: Subtract 82 from both sides

\( 2x = 90 - 82 \)
\( 2x = 8 \)

Step4: Divide by 2

\( x = \frac{8}{2} \)
\( x = 4 \)

Wait, but let's confirm. If x = 4, then 2x + 10 = 18, and 18 + 72 = 90, which is a right angle. That makes sense. So the equation is (2x + 10) + 72 = 90, and solving gives x = 4.

Answer:

\( x = 4 \)