Question

find the measure of each angle indicated.
11)
12)
13)
14)
15)
16)
17)
18)

Explanation:

11) Step1: Identify angle relationship (supplementary)

The given angle \(110^\circ\) and its adjacent angle on the same line are supplementary, so the adjacent angle is \(180 - 110 = 70^\circ\)? No, wait, the two lines are parallel, so the angle we need is equal to the supplementary angle of \(110^\circ\)? Wait, no, the transversal creates corresponding angles or alternate interior angles. Wait, the angle given is \(110^\circ\), and the angle we need is vertical or corresponding? Wait, the two lines are parallel, so the angle above (the "?") and the angle supplementary to \(110^\circ\) (since they are same - side interior? No, wait, the angle \(110^\circ\) and the angle adjacent to "?" are corresponding. Wait, actually, the angle \(110^\circ\) and the angle we need: since the two lines are parallel, the angle we need is equal to \(110^\circ\) if they are corresponding, but wait, no, the \(110^\circ\) angle and the angle adjacent to "?" (on the same transversal) are supplementary? Wait, no, let's correct. The angle \(110^\circ\) and its vertical angle is \(110^\circ\), and the angle we need is equal to that vertical angle because of alternate interior angles. Wait, no, the two horizontal lines are parallel, the transversal cuts them. The angle marked \(110^\circ\) and the angle "?" are corresponding angles? Wait, no, the \(110^\circ\) angle is on the lower line, and the "?" is on the upper line. The angle adjacent to \(110^\circ\) (on the lower line, same transversal) is \(180 - 110=70^\circ\), but that's not right. Wait, no, the correct approach: when two parallel lines are cut by a transversal, corresponding angles are equal, alternate interior angles are equal, same - side interior angles are supplementary. The angle \(110^\circ\) and the angle we need: if we look at the transversal, the angle "?" and the angle supplementary to \(110^\circ\) (since \(110^\circ\) and the angle adjacent to "?" are same - side interior? No, wait, the angle \(110^\circ\) and the angle "?" are actually equal to \(110^\circ\) because they are corresponding angles. Wait, no, I think I made a mistake. Let's start over. The angle given is \(110^\circ\), and the two lines are parallel. The angle "?" and the angle that is supplementary to \(110^\circ\) (because they are same - side interior angles) No, same - side interior angles are supplementary. Wait, the angle \(110^\circ\) and the angle "?" are same - side interior? No, the angle \(110^\circ\) is on the lower line, and "?" is on the upper line, same side of the transversal. So they should be supplementary? Wait, no, same - side interior angles are supplementary. So \(? + 110=180\)? No, that would be \(70\), but that's wrong. Wait, no, the angle \(110^\circ\) and the angle "?" are actually equal because they are alternate exterior angles. Wait, the \(110^\circ\) angle is on the lower line, outside? No, the \(110^\circ\) angle is inside. Wait, I think I messed up. Let's use the fact that vertical angles are equal and corresponding angles are equal. The angle \(110^\circ\) and its vertical angle is \(110^\circ\), and the angle "?" is equal to that vertical angle because of alternate interior angles. So the measure of "?" is \(110^\circ\). Wait, no, that can't be. Wait, no, the correct answer: the angle \(110^\circ\) and the angle "?" are corresponding angles, so they are equal. So \(? = 110^\circ\).

11) Step2: Calculate the angle

Since the two lines are parallel and the transversal creates corresponding angles, the angle "?" is equal to \(110^\circ\).

12) Step1: Identify angle relationship (vertical or corresponding)

The two parallel lines are cut by a transversal. The angle \(84^\circ\) and the angle "?" are vertical angles? No, the \(84^\circ\) angle and the angle "?" are alternate interior angles? Wait, no, the \(84^\circ\) angle and the angle "?" are equal because they are corresponding angles (since the lines are parallel and the transversal is perpendicular? No, the transversal is a straight line. The angle \(84^\circ\) and the angle "?" are equal because they are alternate interior angles. Wait, no, the \(84^\circ\) angle and the angle "?" are vertical angles? No, the \(84^\circ\) angle and the angle "?" are equal because the lines are parallel, so \(? = 84^\circ\).

12) Step2: Determine the angle

Since the two parallel lines are cut by a transversal, the angle "?" is equal to \(84^\circ\) (alternate interior angles or corresponding angles).

13) Step1: Identify angle relationship (supplementary)

The angle \(100^\circ\) and the angle "?" are same - side interior angles? No, the \(100^\circ\) angle and the angle "?" are supplementary? Wait, no, the \(100^\circ\) angle and the angle adjacent to "?" (on the same transversal) are supplementary. Wait, the two parallel lines are cut by a transversal. The angle \(100^\circ\) and the angle "?" are same - side interior angles, so they are supplementary. So \(? = 180 - 100=80^\circ\)? No, wait, the \(100^\circ\) angle and the angle "?" are alternate interior angles? No, the \(100^\circ\) angle is on the upper line, and the angle "?" is on the lower line. The angle adjacent to \(100^\circ\) (on the upper line, same transversal) is \(180 - 100 = 80^\circ\), and the angle "?" is equal to that adjacent angle because of alternate interior angles. So \(? = 80^\circ\).

13) Step2: Calculate the angle

Using the supplementary relationship (same - side interior angles), \(?=180 - 100 = 80^\circ\).

14) Step1: Identify angle relationship (supplementary)

The angle \(111^\circ\) and the angle "?" are same - side interior angles? No, the angle \(111^\circ\) and the angle "?" are supplementary? Wait, the two parallel lines are cut by a transversal. The angle \(111^\circ\) and the angle "?" are same - side interior angles, so they are supplementary. So \(? = 180 - 111=69^\circ\)? No, wait, the angle \(111^\circ\) and the angle "?" are corresponding angles? No, the angle \(111^\circ\) is on the lower line, and the angle "?" is on the upper line. The angle adjacent to \(111^\circ\) (on the lower line, same transversal) is \(180 - 111 = 69^\circ\), and the angle "?" is equal to that adjacent angle because of alternate interior angles. So \(? = 69^\circ\).

14) Step2: Calculate the angle

Using the supplementary relationship, \(? = 180-111 = 69^\circ\).

15) Step1: Identify angle relationship (supplementary)

The angle \(125^\circ\) and the angle "?" are same - side interior angles? No, the angle \(125^\circ\) and the angle "?" are supplementary? Wait, the two parallel lines are cut by a transversal. The angle \(125^\circ\) and the angle "?" are same - side interior angles, so they are supplementary. So \(?=180 - 125 = 55^\circ\)? No, wait, the angle \(125^\circ\) and the angle "?" are alternate interior angles? No, the angle \(125^\circ\) is on the upper line, and the angle "?" is on the lower line. The angle adjacent to \(125^\circ\) (on the upper line, same transversal) is \(180 - 125=55^\circ\), and the angle "?" is equal to that adjacent angle because of alternate interior angles. So \(? = 55^\circ\).

15) Step2: Calculate the angle

Using the supplementary relationship, \(?=180 - 125=55^\circ\).

16) Step1: Identify angle relationship (corresponding)

The angle \(47^\circ\) and the angle "?" are corresponding angles. Since the two parallel lines are cut by a transversal, corresponding angles are equal. So \(? = 47^\circ\).

16) Step2: Determine the angle

By the corresponding angles postulate, \(? = 47^\circ\).

17) Step1: Identify angle relationship (corresponding)

The angle \(53^\circ\) and the angle "?" are corresponding angles. Since the two parallel lines are cut by a transversal, corresponding angles are equal. So \(? = 53^\circ\).

17) Step2: Determine the angle

By the corresponding angles postulate, \(? = 53^\circ\).

18) Step1: Identify angle relationship (supplementary)

The angle \(113^\circ\) and the angle "?" are same - side interior angles? No, the angle \(113^\circ\) and the angle "?" are supplementary? Wait, the two parallel lines are cut by a transversal. The angle \(113^\circ\) and the angle "?" are same - side interior angles, so they are supplementary. So \(?=180 - 113 = 67^\circ\)? No, wait, the angle \(113^\circ\) and the angle "?" are vertical angles? No, the angle \(113^\circ\) and the angle "?" are equal because of alternate interior angles? Wait, no, the angle \(113^\circ\) and the angle "?" are supplementary. Wait, the angle \(113^\circ\) and the angle adjacent to "?" (on the same transversal) are supplementary, and the angle "?" is equal to that adjacent angle. So \(? = 180 - 113=67^\circ\).

18) Step2: Calculate the angle

Using the supplementary relationship, \(? = 180-113 = 67^\circ\).

Answers:

11) \(\boldsymbol{110^\circ}\)
12) \(\boldsymbol{84^\circ}\)
13) \(\boldsymbol{80^\circ}\)
14) \(\boldsymbol{69^\circ}\)
15) \(\boldsymbol{55^\circ}\)
16) \(\boldsymbol{47^\circ}\)
17) \(\boldsymbol{53^\circ}\)
18) \(\boldsymbol{67^\circ}\)

Answer:

s:

1. \(\boldsymbol{110^\circ}\)
2. \(\boldsymbol{84^\circ}\)
3. \(\boldsymbol{80^\circ}\)
4. \(\boldsymbol{69^\circ}\)
5. \(\boldsymbol{55^\circ}\)
6. \(\boldsymbol{47^\circ}\)
7. \(\boldsymbol{53^\circ}\)
8. \(\boldsymbol{67^\circ}\)