Question

Complete.

4. In the diagram, $m\angle 1=\underline{?}$and $m\angle 2=\underline{?}$

Short answer

In the diagram, $\mathit{m}\mathbf{\angle }\mathbf{1}\mathbf{=}\mathbf{105}$and$\mathit{m}\mathbf{\angle }\mathbf{2}\mathbf{=}\mathbf{35}$

Step-by-step solution

Step 1. Draw a labelled diagram.

The labelled diagram is:

Step 2. Description of step.

From the diagram it can be noticed that:

$\angle BAD=\angle BAC+\angle CAD$

Therefore, it can be obtained that:

$$\begin{gathered} \angle BAD=\angle BAC+\angle CAD \\ =50°+40° \\ =90° \end{gathered}$$

Therefore, the measure of the angle $\angle BAD$is $90°$.

The sum of the angles of a triangle is $180°$.

Therefore, the sum of the angles of a triangle $\Delta ABD$is $180°$.

Therefore, it can be obtained that:

$\begin{array}{c}\angle BAD+\angle ABD+\angle ADB=180°\\ 90°+55°+\angle 2=180°\\ 145°+\angle 2=180°\end{array}$

$$\begin{gathered} \angle 2=180°-145° \\ \angle 2=35° \end{gathered}$$

Therefore, the measure of the angle $\angle 2$is $35°$.

The sum of the angles of a triangle is $180°$.

Therefore, the sum of the angles of a triangle $\Delta ACD$is $180°$.

Therefore, it can be obtained that:

$$\begin{gathered} \angle CAD+\angle ACD+\angle ADC=180° \\ 40°+\angle 2+\angle 1=180° \\ 40°+35°+\angle 1=180° \\ 75°+\angle 1=180° \\ \angle 1=180°-75° \\ \angle 1=105° \end{gathered}$$

Therefore, the measure of the angle $\angle 1$is $105°$.

Step 3. Write the complete statement.

In the diagram, $m\angle 1=105$and $m\angle 2=35$.