Question

geometry
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parallel lines & transversal practice
name_____________________
date_____
find the measure of the indicated angle that makes lines u and v parallel.
1)
diagram of two parallel lines u and v cut by a transversal, with 114° angle and? angle
2)
diagram of two parallel lines u and v cut by a transversal, with 55° angle and? angle

Explanation:

Problem 1:

Step1: Identify Angle Relationship

The given angle (114°) and the unknown angle are same - side interior angles? Wait, no. Wait, when two lines are parallel and cut by a transversal, same - side interior angles are supplementary? Wait, no, actually, the angle given (114°) and the unknown angle: let's see, the angle of 114° and the unknown angle, if lines u and v are parallel, then the unknown angle and 114° are either alternate interior angles (if they are congruent) or same - side interior angles (supplementary). Wait, looking at the diagram, the 114° angle and the unknown angle: if we consider the transversal, the 114° angle and the unknown angle, when lines u and v are parallel, the unknown angle should be equal to 114°? Wait, no, wait, maybe they are alternate interior angles. Wait, no, let's re - examine. The angle of 114° and the unknown angle: if the lines u and v are parallel, then the angle adjacent to 114° (supplementary) and the unknown angle would be related? Wait, no, maybe the 114° angle and the unknown angle are alternate interior angles. Wait, no, let's think again. The sum of same - side interior angles is 180°? Wait, no, same - side interior angles are supplementary. Wait, the 114° angle and the unknown angle: if they are same - side interior angles, then they should be supplementary? Wait, no, maybe I made a mistake. Wait, the angle of 114° and the unknown angle: looking at the diagram, the two lines u and v are parallel, cut by a transversal. The 114° angle and the unknown angle: if the 114° angle is an interior angle, and the unknown angle is also an interior angle on the same side? No, wait, maybe the 114° angle and the unknown angle are alternate interior angles, so they are equal. Wait, no, let's calculate. Wait, 180 - 114 = 66? No, that can't be. Wait, maybe the 114° angle and the unknown angle are corresponding angles. Wait, no, the diagram: the transversal cuts u and v. The 114° angle is above line u, and the unknown angle is below line v. Wait, maybe the 114° angle and the unknown angle are equal. Wait, no, let's check the correct relationship. If two parallel lines are cut by a transversal, alternate interior angles are congruent, corresponding angles are congruent, same - side interior angles are supplementary. Let's assume that the 114° angle and the unknown angle are same - side interior angles? No, same - side interior angles add up to 180. Wait, 180 - 114 = 66? No, that doesn't seem right. Wait, maybe the 114° angle and the unknown angle are alternate interior angles, so they are equal. Wait, maybe I misread the diagram. Wait, the problem says "Find the measure of the indicated angle that makes lines u and v parallel". So we need to find the angle such that when we set it to a certain value, lines u and v are parallel. So for the first problem, the angle of 114° and the unknown angle: if we consider that the 114° angle and the unknown angle are same - side interior angles, then they should be supplementary? No, same - side interior angles are supplementary when lines are parallel. Wait, no, the converse: if same - side interior angles are supplementary, then lines are parallel. So if we want lines u and v to be parallel, then the sum of the 114° angle and the unknown angle should be 180°? Wait, no, 180 - 114 = 66? No, that can't be. Wait, maybe the 114° angle and the unknown angle are alternate interior angles, so they are equal. Wait, maybe the diagram shows that the 114° angle and the unknown angle are alternate interior angles, so the unknown angle is 114°? Wait, no, that doesn't make sense. Wait, let's start over. For two parallel lines cut by a transversal:

- Corresponding angles: equal.
- Alternate interior angles: equal.
- Same - side interior angles: supplementary (sum to 180°).

In the first diagram, the 114° angle and the unknown angle: if we look at the positions, the 114° angle is above line u, and the unknown angle is below line v. Wait, maybe the 114° angle and the unknown angle are same - side interior angles. So if lines u and v are parallel, then 114°+x = 180°, so x = 180 - 114 = 66°? No, that seems wrong. Wait, maybe I have the angle relationship reversed. Wait, maybe the 114° angle and the unknown angle are alternate interior angles, so they are equal. So x = 114°? Wait, no, that would mean same - side interior angles are equal, which is only true if they are 90°, which is not the case here. Wait, I think I made a mistake. Let's check the second problem first.

Problem 2:

Step1: Identify Angle Relationship

The given angle is 55°, and the unknown angle. When lines u and v are parallel, the 55° angle and the unknown angle are corresponding angles or alternate interior angles. So if lines u and v are parallel, the unknown angle should be equal to 55°, because they are alternate interior angles (or corresponding angles).

Problem 1 Re - examination:

Wait, maybe in problem 1, the 114° angle and the unknown angle are same - side interior angles, so they should be supplementary. So the unknown angle is 180 - 114 = 66°? No, that can't be. Wait, no, maybe the 114° angle and the unknown angle are alternate interior angles, so they are equal. Wait, the diagram: the two lines u and v are horizontal, and the transversal is a line that crosses them. The 114° angle is between the transversal and line u, and the unknown angle is between the transversal and line v. If they are alternate interior angles, then they are equal. So the unknown angle is 114°? But that would mean same - side interior angles are 114° and 114°, which sum to 228°, which is not 180°. So that's wrong. Wait, I think I messed up the angle types. Let's use the consecutive interior angles (same - side interior angles) theorem: if two parallel lines are cut by a transversal, then consecutive interior angles are supplementary. So in problem 1, the 114° angle and the unknown angle are consecutive interior angles, so 114 + x = 180, so x = 66°. In problem 2, the 55° angle and the unknown angle are alternate interior angles, so x = 55°.

Problem 1 Solution:

Step1: Determine Angle Relationship

The angle of 114° and the unknown angle are same - side interior (consecutive interior) angles. For lines u and v to be parallel, same - side interior angles must be supplementary (sum to 180°).

Step2: Calculate the Unknown Angle

Let the unknown angle be \( x \). Then \( 114^{\circ}+x = 180^{\circ} \).
Solving for \( x \), we get \( x=180^{\circ}- 114^{\circ}=66^{\circ} \).

Problem 2 Solution:

Step1: Determine Angle Relationship

The angle of 55° and the unknown angle are alternate interior angles. For lines u and v to be parallel, alternate interior angles are congruent.

Step2: Determine the Unknown Angle

Since alternate interior angles are congruent when lines are parallel, the unknown angle is equal to 55°.

Answer:

Problem 1: The measure of the indicated angle is \( \boldsymbol{66^{\circ}} \)
Problem 2: The measure of the indicated angle is \( \boldsymbol{55^{\circ}} \)