Question

As shown in the diagram, the two structures of the Puerta de Europa (Door of Europe) in Madrid, Spain, lean toward each other at an angle of $15°$from the vertical.

1. In the diagram, $\angle 1$ and the angle that measures $15°$ are complementary angles. Find $m\angle 1$.
2. Tell whether $\angle 1$and $\angle 2$ arecomplementary or supplementary angles. Then use your answer to part (a) to find $m\angle 2$.

Short answer

1. The measure of angle $m\angle 1$ is $75°$.
2. The measure of angle $m\angle 1$ is $105°$.

Step-by-step solution

a.Step 1. Given Information.

$\angle 1$ and $15°$ are complementary angles.

Step 2. Concept used.

If the sum of two angles is $90°$ then the angles are called complementary angles and if the sum is $180°$ then the angles are called supplementary angles.

Step 3. Calculation.

Since $\angle 1$ and $15°$ are complementary angles.

$\begin{array}{c}m\angle 1+15°=90°\\ m\angle 1=90°-15°\\ =75°\end{array}$

Step 4. Conclusion.

The measure of the angle that is complementary to an angle measuring $15°$ is $75°$.

b.Step 1. Given Information.

$\angle 1$ and $15°$are complementary angles.

Step 2. Concept used.

If the sum of two angles is $90°$ then the angles are called complementary angles and if the sum is $180°$ then the angles are called supplementary angles.

Step 3. Calculation.

Since linear angle measures $180°$. Thus, $\angle 1$ and $\angle 2$ are supplementary angles.

$\begin{array}{c}m\angle 1+m\angle 2=180°\\ 75°+m\angle 2=180°\\ m\angle 2=180°-75°\\ =105°\end{array}$

Step 4. Conclusion.

The measure of the angle that is supplementary to an angle measuring $75°$ is $105°$.