Question

7 for the rhombus, find the of x. (7x - 17)° (5x + 9)°

Explanation:

Step1: Recall rhombus angle property

In a rhombus, the diagonal bisects the angles. So the angle \((7x - 17)^\circ\) is equal to \(2\times(5x + 9)^\circ\)? Wait, no, wait. Wait, actually, in a rhombus, adjacent angles are supplementary, but also, the diagonal bisects the vertex angles. Wait, looking at the diagram, the diagonal splits the angle \((7x - 17)^\circ\) into two equal angles? Wait, no, maybe the two angles \((7x - 17)^\circ\) and \(2\times(5x + 9)^\circ\) are related? Wait, no, maybe I misread. Wait, the angle marked \((7x - 17)^\circ\) is a vertex angle, and the diagonal splits it into two angles, one of which is \((5x + 9)^\circ\). So in a rhombus, the diagonal bisects the vertex angle, so \((7x - 17)=2\times(5x + 9)\)? Wait, no, that would give a negative x. Wait, maybe the other way: the angle \((7x - 17)^\circ\) and the angle formed by two times \((5x + 9)^\circ\) are equal? Wait, no, maybe the two angles \((7x - 17)^\circ\) and \((5x + 9)^\circ\) are equal? Wait, no, that would be if the diagonal bisects the angle, but maybe the angle \((7x - 17)^\circ\) is equal to \((5x + 9)^\circ\)? Wait, no, let's check. Wait, maybe the correct property is that in a rhombus, the diagonal bisects the vertex angle, so the angle \((7x - 17)^\circ\) is equal to \(2\times(5x + 9)^\circ\)? Wait, solving \(7x - 17 = 2(5x + 9)\) gives \(7x - 17 = 10x + 18\), then \(-3x = 35\), which is negative. That can't be. Wait, maybe the other angle: in a rhombus, adjacent angles are supplementary, but maybe the angle \((7x - 17)^\circ\) and the angle \(2\times(5x + 9)^\circ\) are supplementary? Wait, no, let's think again. Wait, the diagram shows a rhombus with a diagonal, splitting one angle into two angles, one of which is \((5x + 9)^\circ\), and the original angle is \((7x - 17)^\circ\). Wait, maybe the diagonal bisects the angle, so \((7x - 17) = 2(5x + 9)\) is wrong. Wait, maybe the angle \((7x - 17)^\circ\) is equal to \((5x + 9)^\circ\)? Let's try that: \(7x - 17 = 5x + 9\), then \(2x = 26\), so \(x = 13\). Ah, that works. So the correct equation is \(7x - 17 = 5x + 9\) because the diagonal bisects the angle, so the two angles are equal? Wait, no, if the diagonal bisects the angle, then the angle is split into two equal parts, so the original angle is twice the bisected angle. Wait, but in this case, maybe the angle \((7x - 17)^\circ\) is equal to the angle formed by two times \((5x + 9)^\circ\)? No, that gave a negative. Wait, maybe the angle \((7x - 17)^\circ\) and \((5x + 9)^\circ\) are equal because of some other property? Wait, let's check with x=13: \(7(13)-17=91-17=74\), \(5(13)+9=65+9=74\). Oh! So they are equal. So that means the diagonal bisects the angle, so the two angles are equal. So the equation is \(7x - 17 = 5x + 9\).

Step2: Solve the equation

\(7x - 17 = 5x + 9\)
Subtract \(5x\) from both sides: \(2x - 17 = 9\)
Add 17 to both sides: \(2x = 26\)
Divide by 2: \(x = 13\)

Answer:

13