Question

Find the values of x and y.

Short answer

The values of x and y are $40°$and role="math" localid="1638176591854" $30°$respectively.

Step-by-step solution

Step 1. Name the figure and highlight angle DCO and angle BCO .

Step 2. Apply an angle sum theorem.

Consider a $\Delta AOB$, such that,

$$\begin{gathered} m\angle BAO+m\angle AOB+m\angle ABO=180 \\ 90+m\angle AOB+40=180 \\ m\angle AOB=50 \end{gathered}$$

Step 3. Apply vertically opposite angles.

From the given figure, it can be observed that $\angle AOB$and $\angle DOC$are vertically opposite angles. Therefore, $m\angle AOB=m\angle DOC=50°$.

Step 4. Apply an angle sum theorem.

Consider a $\Delta BAC$, such that,

$$\begin{gathered} m\angle BAC+m\angle ACB+m\angle ABC=180 \\ 90+y+60=180 \\ y=30 \end{gathered}$$

Step 5. Apply an angle sum theorem.

Consider a $\Delta DOC$, such that,

$$\begin{gathered} m\angle DOC+m\angle DCO+m\angle ODC=180 \\ 50+x+90=180 \\ x=40 \end{gathered}$$

Therefore, the values of x and y are $40°$and $30°$respectively.