Question

explore parallel lines cut by a transversal

1. a room in an attic has a sloping wall that makes an angle of 55° with the floor, which is parallel to the ceiling. what is the measure of an angle that forms a pair of same - side interior angles with that angle?
2. math on the spot line n is parallel to line p. find the measure of each angle.
3. oak street runs parallel with maple street. the angle between north main street and west oak street where the turn is shown measures 65°. what is the measure of the angle between the roads at the turn shown from south main street onto west maple street? explain.
4. two parallel lines are cut by a transversal and two of the same - side interior angles formed have measures of (4x + 3)° and (x + 2)°.

a. how are the angles related?
b. write and solve an equation to find the value of x.
c. what are the two angle measures?

1. reason in the diagram, two parallel lines are cut by a transversal, and ∠1 and ∠6 do not form any of the kinds of angle pairs that you have learned. what is a way you can find the measure of ∠6 if you know that the measure of ∠1 is 132? explain.

Explanation:

1.

Step1: Recall same - side interior angles property

Same - side interior angles are supplementary when two parallel lines are cut by a transversal. Let the given angle be $\alpha = 55^{\circ}$. Let the same - side interior angle be $\beta$.

Step2: Use the supplementary - angle formula

$\alpha+\beta = 180^{\circ}$. Substitute $\alpha = 55^{\circ}$ into the formula: $\beta=180^{\circ}- 55^{\circ}$.

Step1: Identify vertical and corresponding angles

$\angle1$ and the $40^{\circ}$ angle are vertical angles, so $\angle1 = 40^{\circ}$. $\angle1$ and $\angle5$ are corresponding angles, so $\angle5=\angle1 = 40^{\circ}$. $\angle1$ and $\angle3$ are vertical angles, so $\angle3 = 40^{\circ}$. $\angle2$ and $\angle1$ are supplementary, so $\angle2=180^{\circ}-\angle1 = 140^{\circ}$. $\angle4$ and $\angle2$ are vertical angles, so $\angle4 = 140^{\circ}$. $\angle6$ and $\angle2$ are corresponding angles, so $\angle6=\angle2 = 140^{\circ}$. $\angle7$ and $\angle3$ are corresponding angles, so $\angle7 = 40^{\circ}$.

Step1: Recognize corresponding angles

Since Oak Street and Maple Street are parallel, the angle between North Main Street and West Oak Street and the angle between South Main Street and West Maple Street are corresponding angles.

Step2: State the angle measure

Corresponding angles are equal when two parallel lines are cut by a transversal. So the measure of the angle between South Main Street and West Maple Street is $65^{\circ}$.

Answer:

$125^{\circ}$

2.