Question

The diagram shown is a side view of a chair designed in the 1930s. In the diagram, $m\angle 1=95°$and $m\angle 3=45°$. Tell whether the statement is true or false. Explain your reasoning.

1. $\angle 2$ and $\angle 3$are corresponding angles.
2. $\angle 2$ and $\angle 3$ have equal measures.
3. $\angle 2$ and $\angle 3$are complementary angles.
4. The sum of the measures of $\angle 1,\angle 2$and $\angle 3$is $180°$.

Short answer

1. False. The angles $\angle 2$and $\angle 3$ arenot corresponding angles.
2. True. The angles $\angle 2$and $\angle 3$have same measures.
3. True. The angles $\angle 2$and $\angle 3$are complementary angles.
4. False. The sum of measures of angle $\angle 1$, $\angle 2$ and $\angle 3$ is not $180°$.

Step-by-step solution

a.Step 1. Given Information.

The provided measures are $m\angle 1=95°$and $m\angle 3=45°$, and thediagram is,

Step 2. Definition of alternate interior angles.

When two lines are crossed by another line, it is known as transversal.

The angles internal to the two parallel lines, in alternate corners of the transversal are called alternate interior angles.

Step 3. Explanation.

In the given figure, two parallel lines are intersected by a transversal. Since $\angle 2$ and $\angle 3$ are located inside the parallel lines and in opposite corners, $\angle 2$ and $\angle 3$ are alternate interior angles, not corresponding angles.

Step 4. Conclusion.

The given statement is false. The angles $\angle 2$and $\angle 3$ arenot corresponding angles.

b.Step 1. Given Information.

The provided measures are $m\angle 1=95°$and $m\angle 3=45°$, and thediagram is,

Step 2. Definition of alternate interior angles.

When two lines are crossed by another line, it is known as transversal.

The angles internal to the two parallel lines, in alternate corners of the transversal are called alternate interior angles. The measure of alternate interior angles are equal.

Step 3. Explanation.

In the given figure, two parallel lines are intersected by a transversal. Since $\angle 2$and $\angle 3$ are located inside the parallel lines and in opposite corners, $\angle 2$ and $\angle 3$ are alternate interior angles.

So, the measures must be equal, that is, $m\angle 2=m\angle 3$.

Step 4. Conclusion.

The given statement is true. The angles $\angle 2$ and $\angle 3$ have same measures.

c.Step 1. Given Information.

The provided measures are $m\angle 1=95°$and $m\angle 3=45°$, and thediagram is,

Step 2. Definition used.

Complementary angles

If the sum of two angles is $90°$, then they are known as Complementary angles.

Step 3. Explanation.

Since $\angle 2$ and $\angle 3$ are alternate interior angles,the measures must be equal, that is, $m\angle 2=m\angle 3$.

It is given that $m\angle 3=45°$. So, $m\angle 2=45°$ . The sum of measure of $\angle 2$ and $\angle 3$ is,

$\begin{array}{c}m\angle 2+m\angle 3=45°+45°\\ =90°\end{array}$

Step 4. Conclusion.

The given statement is true. The angles $\angle 2$ and $\angle 3$ are complementary angles.

d.Step 1. Given Information.

The provided measures are $m\angle 1=95°$and $m\angle 3=45°$, and thediagram is,

Step 2. Explanation.

It is given that $m\angle 1=95°$, $m\angle 3=45°$.

Since $\angle 2$ and $\angle 3$ are alternate interior angles,the measures must be equal, that is, $m\angle 2=m\angle 3$. So, $m\angle 2=45°$ . The sum of measure of $\angle 1$, $\angle 2$ and $\angle 3$ is,

$\begin{array}{c}m\angle 1+m\angle 2+m\angle 3=95°+45°+45°\\ =185°\end{array}$

Step 4. Conclusion.

The given statement is false. The sum of measures of angle $\angle 1$, $\angle 2$ and $\angle 3$ is not $180°$.