Question

Extended Problem Solving: In the bicycle frame shown, $\angle 1$is called the head angle, and $\angle 2$ is called the seat angle. Lines $m$and $n$are parallel.

1. Identify: Are $\angle 1$ and $\angle 2$ are corresponding, alternate interior, or alternate exterior angles.
2. Explain: $\angle 1$ and $\angle 2$ have the same measure?
3. Calculate: $\angle 1$ and $\angle 2$.Explain how you got your answer.

Short answer

a. The angles $\angle 1$ and $\angle 2$ arealternate interior angles.

b. Yes, the measures are equal $m\angle 1=m\angle 2$.

c. The measures of the angles are $m\angle 1=72°$ and $m\angle 2=72°$.

Step-by-step solution

1. Step 1. Given Information.

Lines$m$ and $n$ are paralleland $t$is the transversal. The provided angles are $\angle 1$ and $\angle 2$, and the diagram 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 $\angle 1$ and $\angle 2$ are located inside the parallel lines and in opposite corners, $\angle 1$ and $\angle 2$ are alternate interior angles.

Step 4. Conclusion.

The angles $\angle 1$ and $\angle 2$ arealternate interior angles.

2.Step 1. Given Information.

Lines $m$ and $n$ are paralleland $t$is the transversal. The provided angles are $\angle 1$ and $\angle 2$, and the diagram is,

Step 2. Concepts used.

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 measures of alternate interior angles are equal.

Step 3. Explanation.

Since $\angle 1$and $\angle 2$ are located inside the parallel lines and in opposite corners, $\angle 1$ and $\angle 2$ are alternate interior angles.

So, the measures must be equal, that is, $m\angle 1=m\angle 2$.

Step 4. Conclusion.

Yes, the measures are equal $m\angle 1=m\angle 2$.

3.Step 1. Given Information.

Lines $m$ and $n$ are paralleland $t$is the transversal. The provided angles are $\angle 1$ and $\angle 2$, and the diagram is,

Step 2. Find the measure of angle 1.

The sum of the linear paired angles is $180°$, $m\angle 2+108°=180°$.

The measure of $\angle 2$ will be,

$\begin{array}{c}m\angle 2+108°=180°\\ m\angle 2=180°-108°\\ m\angle 2=72°\end{array}$

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

$\begin{array}{c}m\angle 1=m\angle 2\\ =72°\end{array}$

Step 3. Conclusion.

The measures of the angles are $m\angle 1=72°$ and $m\angle 2=72°$.