Question

Find the value of x and y for each diagram.

Short answer

The values of x and y are 47 and 75 respectively.

Step-by-step solution

Step 1. Label the given diagram.

Label the figure as shown:

Step 2. Write the relation between the angles AOD and AOC.

From the given diagram it can be noticed that the angles AOD and AOC lies on the straight line CD.

Therefore, the sum of the angles AOD and AOC is 180°.

Therefore,$\angle AOD+\angle AOC=180°$

Step 3. Find the value of x.

Find the value of x by using the fact that $\angle AOD+\angle AOC=180°$.

Therefore,

$$\begin{gathered} \angle AOD+\angle AOC=180° \\ x+3x-8=180 \\ 4x=180+8 \\ 4x=188 \\ x=\frac{188}{4} \\ x=47 \end{gathered}$$

Therefore, the value of x is 47.

Step 4. Find the measure of the angle AOC.

As, the value of x is 47 and the measure of the angle AOC is $3x-8$.

Therefore, the measure of the angle AOC is:

$3x-8=3\left(47\right)-8=141-8=133$

Therefore, the measure of the angleAOC is 133°.

Step 5. Write the relation between the angles AOC and BOD.

From the diagram it can be noticed that the angles AOC and BOD are vertically opposite angles.

Therefore, $\angle AOC=\angle BOD$.

Step 6. Find the value of y.

As, $\angle AOC=\angle BOD$.

Therefore,

$$\begin{gathered} \angle AOC=\angle BOD \\ 133=2y-17 \\ 133+17=2y \\ 150=2y \\ \frac{150}{2}=y \\ 75=y \end{gathered}$$

Therefore, the value of y is 75.