Question

the image shows two lines cut by a transversal. the measure of ∠1 is 90.5°, and the measure of ∠7 is 89.5°. determine whether each statement is true. select true or false for each statement. ∠4 is congruent to ∠5 because they are alternate interior angles. the two lines are parallel. the measure of ∠6 is 90.5°.

Explanation:

Step1: Check angle relationship

$\angle7$ and $\angle5$ are supplementary ($\angle7+\angle5 = 180^{\circ}$). Given $\angle7=89.5^{\circ}$, then $\angle5=180 - 89.5=90.5^{\circ}$. $\angle5$ and $\angle6$ are vertical angles. Vertical angles are equal, so $\angle6=\angle5 = 90.5^{\circ}$.

Step2: Check parallel lines

If two lines are cut by a transversal and corresponding angles are equal, then the lines are parallel. $\angle7$ and $\angle3$ are corresponding angles. $\angle7 = 89.5^{\circ}$, $\angle3=180 - 90.5=89.5^{\circ}$ (since $\angle1 = 90.5^{\circ}$ and $\angle1+\angle3=180^{\circ}$). So the two lines are parallel.

Step3: Check $\angle4$ and $\angle5$

$\angle4$ and $\angle5$ are alternate interior angles. When two parallel lines are cut by a transversal, alternate interior angles are equal.

Answer:

• The measure of $\angle6$ is $90.5^{\circ}$: True
• The two lines are parallel: True
• $\angle4$ is congruent to $\angle5$ because they are alternate interior angles: True