Angle Properties
A transversal is a line that intersects two or more coplanar lines in two different points. In the diagram below, line t, is the transversal of lines a and b. The angles formed by the transversal and the two lines are given special names. See the chart below.
A transversal is a line that intersects two or more coplanar lines in two different points. In the diagram below, line t, is the transversal of lines a and b. The angles formed by the transversal and the two lines are given special names. See the chart below.
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Exploration:
When the two lines that the transversal crosses are parallel, then these angles have a special relationship. Let’s see how their relationship changes.
When the two lines that the transversal crosses are parallel, then these angles have a special relationship. Let’s see how their relationship changes.
Same Side Interior Angles Theorem or Consecutive Interior Angles Theorem.
If two parallel lines are cut by a transversal, then the pairs of same side interior or consecutive interior angles are supplementary. If parallel lines, then same side interior angles are supplementary. or If parallel lines, then consecutive interior angles are supplementary. |
\(\angle 4\) & \(\angle 5\) are supplementary
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Example 1:
Name the angle pair using one of the following choices:
-Vertical Angles
-Linear Pair of Angles
-Corresponding Angles
-Alternate Interior Angles
-Alternate Exterior Angles
-Same Side Interior Angles
-Same Side Exterior Angles
*Recall: the angle pairs still have these relationships even when the lines are NOT parallel!
Name the angle pair using one of the following choices:
-Vertical Angles
-Linear Pair of Angles
-Corresponding Angles
-Alternate Interior Angles
-Alternate Exterior Angles
-Same Side Interior Angles
-Same Side Exterior Angles
*Recall: the angle pairs still have these relationships even when the lines are NOT parallel!
Solution:
a) Vertical Pair of Angles
The angles are opposite each other in the same intersection.
b) Consecutive Exterior Angles
The angles are on the same side of the transversal, line \(t\), and are outside of lines \(m\) and \(n\).
c) Alternate Exterior Angles
The angles are on different (alternating) sides of the transversal, line \(t\), and are outside of lines \(m\) and \( n\).
d) Consecutive Interior Angles
The angles are on the same side of the transversal, line \(t\), and are in between, or in the interior of, lines \(m\) and \(n\).
Example 2:
Name the angle pair, then tell their relationship (congruent, supplementary, or neither).
a) Vertical Pair of Angles
The angles are opposite each other in the same intersection.
b) Consecutive Exterior Angles
The angles are on the same side of the transversal, line \(t\), and are outside of lines \(m\) and \(n\).
c) Alternate Exterior Angles
The angles are on different (alternating) sides of the transversal, line \(t\), and are outside of lines \(m\) and \( n\).
d) Consecutive Interior Angles
The angles are on the same side of the transversal, line \(t\), and are in between, or in the interior of, lines \(m\) and \(n\).
Example 2:
Name the angle pair, then tell their relationship (congruent, supplementary, or neither).
Solution:
a) Alternate Interior Angles; Neither -
the angles are alternate interior angles, however since lines m and n are not parallel, the angles are not known to be
congruent.
b) Same Side Exterior Angles; Supplementary
Since lines \(m\) and \(n\) are parallel, same side exterior angles are supplementary
c) Alternate Interior Angles; Congruent
Since lines \(m\) and n are parallel, alternate interior angles are congruent
d) Vertical Angles; Congruent
Vertical angles are ALWAYS congruent
e) No Angle-Pair Name; Supplementary
Even though these angles do not have an angle-pair name, we can use vertical angles with \(a\), which are always
congruent. Then that angle and \(c\) would be same side interior angles, which are supplementary because of lines
\(m\) and \(n\) being parallel.
Example 3:
Solve for the variable using angle-pair relationships. The find the measure of each labeled angle.
a) Alternate Interior Angles; Neither -
the angles are alternate interior angles, however since lines m and n are not parallel, the angles are not known to be
congruent.
b) Same Side Exterior Angles; Supplementary
Since lines \(m\) and \(n\) are parallel, same side exterior angles are supplementary
c) Alternate Interior Angles; Congruent
Since lines \(m\) and n are parallel, alternate interior angles are congruent
d) Vertical Angles; Congruent
Vertical angles are ALWAYS congruent
e) No Angle-Pair Name; Supplementary
Even though these angles do not have an angle-pair name, we can use vertical angles with \(a\), which are always
congruent. Then that angle and \(c\) would be same side interior angles, which are supplementary because of lines
\(m\) and \(n\) being parallel.
Example 3:
Solve for the variable using angle-pair relationships. The find the measure of each labeled angle.
Example 5:
Crook Problem
Find the measure of angle \(x\).
Crook Problem
Find the measure of angle \(x\).
Solution:
To solve a crook problem, draw a line parallel to the other parallel lines through the crook. If you look at the upper two parallel lines, there are two consecutive interior angles or same side interior angles.
To solve a crook problem, draw a line parallel to the other parallel lines through the crook. If you look at the upper two parallel lines, there are two consecutive interior angles or same side interior angles.
One of these angle measures is given - \(140^{\circ}\). You can find the other measure because these angles are supplementary. So, the other angle measure is \(40^{\circ}\). This \(40^{\circ}\) measure is part of the angle measure \(x\) that we are looking for.
If you look at the lower two parallel lines, the given angle, \(47^{\circ}\), is an alternate interior angle with the lower part of angle measure\( x\). Since alternate interior angles are congruent when lines are parallel, the lower part of angle measure \(x\) is \(47^{\circ}\).
If you add \(40^{\circ} + 47^{\circ}\), angle \(x\) has a measure of \(87^{\circ}\).
If you look at the lower two parallel lines, the given angle, \(47^{\circ}\), is an alternate interior angle with the lower part of angle measure\( x\). Since alternate interior angles are congruent when lines are parallel, the lower part of angle measure \(x\) is \(47^{\circ}\).
If you add \(40^{\circ} + 47^{\circ}\), angle \(x\) has a measure of \(87^{\circ}\).
Solution:
To find the \(m\angle D\), notice that the line containing \(\overline{QU}\)and the line containing \(\overline{DA}\) are parallel. Then the line containing \(\overline{QD}\) is a transversal cutting parallel lines, and the blue highlighted angles are same side interior angles. Since the lines are parallel, the same side interior angles are supplementary. So we know that: |
Due to angle-pair relationships with parallel lines, specifically same side interior or consecutive interior angles, one given angle in a parallelogram enables us to find the value of each angle measure.
Quick Check
1) Solve for the variable using angle-pair relationships. The find the measure of each labeled angle
1) Solve for the variable using angle-pair relationships. The find the measure of each labeled angle
Crook Problem
2) Find the measure of angle \(x\).
2) Find the measure of angle \(x\).