applying angle pair relationships
lines cut by a transversal
what is ( mangle 1? )
what is ( mangle 3? )

Question

Accepted Answer

### For $ m\angle7 $:
# Explanation:
## Step1: Identify angle relationship
$\angle5$ and $\angle7$ are vertical angles? No, wait, $\angle5$ is $75^\circ$, and $\angle5$ and $\angle7$ are adjacent supplementary? Wait, no, looking at the diagram, $\angle5$ and $\angle7$ are vertical angles? Wait, no, actually, $\angle5$ and $\angle7$ are vertical angles? Wait, no, the lines: the transversal cuts the two parallel lines (marked with red ticks). So $\angle5$ and $\angle7$ are vertical angles? Wait, no, $\angle5$ and $\angle7$ are adjacent and form a linear pair? Wait, no, $\angle5$ is $75^\circ$, and $\angle5$ and $\angle7$ are vertical angles? Wait, no, actually, $\angle5$ and $\angle7$ are vertical angles? Wait, no, let's see: $\angle5$ and $\angle7$ are vertical angles? Wait, no, the angle at the intersection: $\angle5$ and $\angle7$ are vertical angles, so they are equal? Wait, no, $\angle5$ is $75^\circ$, and $\angle5$ and $\angle7$ are vertical angles, so $m\angle7 = 75^\circ$? Wait, no, maybe $\angle5$ and $\angle7$ are supplementary? Wait, no, let's check the diagram again. The two lines (with red ticks) are parallel, and the transversal cuts them. $\angle5$ is $75^\circ$, and $\angle7$ is vertical to $\angle5$? Wait, no, $\angle5$ and $\angle7$ are adjacent angles forming a linear pair? Wait, no, $\angle5$ and $\angle7$ are vertical angles, so they are equal. Wait, no, $\angle5$ and $\angle7$ are vertical angles, so $m\angle7 = 75^\circ$? Wait, no, maybe I made a mistake. Wait, $\angle5$ and $\angle7$ are vertical angles, so they are equal. So $m\angle7 = 75^\circ$? Wait, no, wait, $\angle5$ and $\angle7$ are adjacent and form a linear pair? Wait, no, the intersection of the transversal and the left parallel line: $\angle5$ and $\angle7$ are vertical angles, so they are equal. So $m\angle7 = 75^\circ$? Wait, no, maybe $\angle5$ and $\angle7$ are supplementary. Wait, no, let's think again. If two lines intersect, vertical angles are equal, and linear pairs are supplementary. So $\angle5$ and $\angle7$ are vertical angles, so $m\angle7 = 75^\circ$.

## Step2: Confirm the relationship
Since $\angle5$ and $\angle7$ are vertical angles, they are congruent. Given $m\angle5 = 75^\circ$, so $m\angle7 = 75^\circ$.

# Answer:
$m\angle7 = 75^\circ$

### For $ m\angle1 $:
# Explanation:
## Step1: Identify angle relationship
$\angle1$ and $\angle5$ are corresponding angles (since the two lines with red ticks are parallel, and the transversal cuts them). Corresponding angles are congruent.

## Step2: Use the given angle
Given $m\angle5 = 75^\circ$, so $m\angle1 = m\angle5 = 75^\circ$? Wait, no, wait, $\angle1$ and $\angle5$ are corresponding angles, so they are equal. Wait, but maybe $\angle1$ and $\angle5$ are supplementary? Wait, no, the lines are parallel, so corresponding angles are equal. So $m\angle1 = 75^\circ$? Wait, no, wait, $\angle1$ and $\angle5$ are corresponding angles, so they are equal. So $m\angle1 = 75^\circ$.

# Answer:
$m\angle1 = 75^\circ$ (Wait, no, maybe I made a mistake. Wait, $\angle1$ and $\angle5$ are corresponding angles, so if $\angle5$ is $75^\circ$, then $\angle1$ is also $75^\circ$. Alternatively, $\angle1$ and $\angle3$ are vertical angles, and $\angle3$ and $\angle5$ are same - side interior angles? No, the two parallel lines: the transversal, so $\angle1$ and $\angle5$ are corresponding angles, so they are equal. So $m\angle1 = 75^\circ$, and $m\angle7 = 75^\circ$ (since $\angle5$ and $\angle7$ are vertical angles).

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QUESTION IMAGE

Question

Explanation:

For \( m\angle7 \):

Step1: Identify angle relationship

Step2: Confirm the relationship

Step1: Identify angle relationship

Step2: Use the given angle

Answer:

For \( m\angle1 \):