Question

1. find the values of x and y. (5x - 17)° (2y + 5)° (3x - 11)°

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. So, \(5x - 17=3x - 11\).
\[

$$\begin{align*} 5x-3x&=- 11 + 17\\ 2x&=6\\ x&=3 \end{align*}$$

\]

Step2: Use the fact that the sum of angles around a point is \(360^{\circ}\) or use the right - angle relationship

Since there is a right - angle (\(90^{\circ}\)) and we know the vertical - angle relationship, we consider the sum of the angles at the intersection. Let's assume the angles are part of a set of angles that add up to \(180^{\circ}\) (a straight - line or a right - angle related set).
We have another angle \((2y + 5)^{\circ}\). If we consider the right - angle and the other non - right angles, we can set up an equation. But first, substitute \(x = 3\) into one of the angle expressions. Let's assume the sum of two non - right angles and the right - angle is \(180^{\circ}\).
The angle \(5x-17=5\times3 - 17=15 - 17=-2\) (this is wrong, let's assume the sum of the two non - vertical non - right angles and the right - angle is \(180^{\circ}\)).
We know that \((5x - 17)+(2y + 5)+90 = 180\). Substitute \(x = 3\) into it:
\[

$$\begin{align*} (5\times3-17)+(2y + 5)+90&=180\\ (15 - 17)+(2y+5)+90&=180\\ -2+2y + 5+90&=180\\ 2y+93&=180\\ 2y&=180 - 93\\ 2y&=87\\ y& = 43.5 \end{align*}$$

\]

Answer:

\(x = 3,y = 43.5\)