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Find the measure of angle . Explain or show your reasoning.
Find and label a second angle in the diagram. Find and label an angle congruent to angle .
Angle is a right angle. Find the measure of angle .
Lines and are parallel. They are cut by transversal .
Line A C contains point B. Line D F contains point E. Line J H contains points B and E. Angle A B J is labeled question mark. Angle A B E is labeled 63 degrees. Angle E B C is labeled question mark. Angle C B J is labeled question mark. Angle D E H is labeled question mark. Angle H E F is labeled question mark. Angle F E B is labeled question mark. Angle B E D is labeled question mark.
With your partner, find the seven unknown angle measures in the diagram. Explain your reasoning.
What do you notice about the angles with vertex and the angles with vertex ?
Using what you noticed, find the measures of the four angles at point in the diagram. Lines and are parallel.
Line A C contains point B. Line D F contains point E. Line H G contains points B and E. Angle A B H is labeled with a question mark. Angle A B E is labeled with a question mark. Angle E B C is labeled with a question mark. Angle C B H is labeled with a question mark. Angle G E F is labeled 34 degrees.
Lines and are parallel and is a transversal. Point is the midpoint of segment .
Find a rigid transformation showing that angles and are congruent.
Lines and are not parallel in this image.
Line A C contains point B. Line D F contains point E. Line J H contains points B and E. Angle A B J is labeled question mark. Angle A B E is labeled 63 degrees. Angle E B C is labeled question mark. Angle C B J is labeled question mark. Angle D E H is labeled question mark. Angle H E F is labeled 108 degrees. Angle F E B is labeled question mark. Angle B E D is labeled question mark.
Find the missing angle measures around point and point .
What do you notice about the angles in this diagram?
Point is the midpoint of line segment .
Can you find a rigid transformation that shows angle is congruent to angle ? Explain your reasoning.
If two angle measures add up to , then we say the angles are complementary. Here are three examples of pairs of complementary angles.
If two angle measures add up to , then we say the angles are supplementary. Here are three examples of pairs of supplementary angles.
When two lines intersect, vertical angles are congruent, and adjacent angles are supplementary, so their measures sum to 180. For example, in this figure angles 1 and 3 are congruent, angles 2 and 4 are congruent, angles 1 and 4 are supplementary, and angles 2 and 3 are supplementary.
Two intersecting lines. Angle 1 is labeled 70 degrees. Angle 2 is labeled 110 degrees. Angle 3 is labeled 70 degrees. Angle 4 is labeled 110 degrees.
When two parallel lines are cut by another line, called a transversal, two pairs of alternate interior angles are created. (“Interior” means on the inside, or between, the two parallel lines.) For example, in this figure angles 3 and 5 are alternate interior angles and angles 4 and 6 are also alternate interior angles.
Two lines that do not intersect. A third line intersects with both lines. At the first intersection, angle 1 is labeled 70 degrees. Angle 2 is labeled 110 degrees. Angle 3 is labeled 70 degrees. Angle 4 is labeled 110 degrees. At the second intersection, angle 5 is marked 70 degrees. Angle 6 is marked 110 degrees. Angle 7 is marked 70 degrees. Angle 8 is marked 110 degrees.
Alternate interior angles are equal because a rotation around the midpoint of the segment that joins their vertices takes each angle to the other. Imagine a point halfway between the two intersections. Can you see how rotating about takes angle 3 to angle 5?
Using what we know about vertical angles, adjacent angles, and alternate interior angles, we can find the measures of any of the eight angles created by a transversal if we know just one of them. For example, starting with the fact that angle 1 is we use vertical angles to see that angle 3 is , then we use alternate interior angles to see that angle 5 is , then we use the fact that angle 5 is supplementary to angle 8 to see that angle 8 is since . It turns out that there are only two different measures. In this example, angles 1, 3, 5, and 7 measure , and angles 2, 4, 6, and 8 measure .
Alternate interior angles are created when 2 parallel lines are crossed by another line. This line is called a transversal. Alternate interior angles are inside the parallel lines and on opposite sides of the transversal.
This diagram shows 2 pairs of alternate interior angles:
There are two horizontal parallel lines, and a third diagonal line drawn from the bottom left to the upper right, intersecting both horizontal lines. The diagonal line is labeled transversal. There are four angles created by the diagonal line inside the parallel lines. The upper left angle is labeled a, upper right is b, lower left is c, and lower right is d.
Complementary angles have measures that add up to 90.
For example, a angle and a angle are complementary.
Supplementary angles have measures that add up to 180.
For example, a angle and a angle are supplementary.
A transversal is a line that crosses parallel lines.
This diagram shows a transversal line intersecting parallel lines and .