Question

Find the measures of the numbered angles.

Short answer

The measures of$\angle 1,\angle 2,\angle 3,\angle 4,\angle 5$ and$\angle 6$ is$\mathbf{58}\mathbf{°}\mathbf{,}\mathbf{90}\mathbf{°}\mathbf{,}\mathbf{32}\mathbf{°}\mathbf{,}\mathbf{87}\mathbf{°}\mathbf{,}\mathbf{35}\mathbf{°}$ and$\mathbf{55}\mathbf{°}$ respectively.

Step-by-step solution

Step 1. Label the given diagram. 

Step 2. Description of step.

Consider $\mathrm{\Delta }BEC$, by an angle sum theorem,

$$\begin{gathered} m\angle 1+32+90=180 \\ m\angle 1=180-122 \\ =58° \end{gathered}$$

Step 3. Description of step.

$AB$is parallel$CD$then$\angle 5$ and$\angle ACD$are alternate interior angles then

$m\angle 5=m\angle ACD=35°$.

$AD$is parallel$BC$then$\angle 3$ and$\angle ACB$ are alternate interior angles then

$m\angle 3=m\angle ACB=32°$.

Step 4. Description of step.

From the given figure, it can be observed that$m\angle 2=90°$

Step 5. Description of step.

Consider $\mathrm{\Delta }ADC$, by an angle sum theorem,

$$\begin{gathered} m\angle 3+m\angle 4+35=180 \\ 58+m\angle 4+35=180 \\ m\angle 4=180-93 \\ =87° \end{gathered}$$

Step 6. Description of step.

Consider $\mathrm{\Delta }AEB$, by an angle sum theorem,

$$\begin{gathered} m\angle 2+m\angle 5+m\angle 6=180 \\ 90+35+m\angle 6=180 \\ m\angle 6=180-125 \\ =55° \end{gathered}$$

Therefore, the measures of$\angle 1,\angle 2,\angle 3,\angle 4,\angle 5$ and$\angle 5$is$58°,90°,32°,87°,35°$ and$55°$respectively.