Lesson 14Alternate Interior Angles
Learning Goal
Let’s explore why some angles are always equal.
Learning Targets
If I have two parallel lines cut by a transversal, I can identify alternate interior angles and use that to find missing angle measurements.
Lesson Terms
- alternate interior angles
- transversal
Warm Up: Angle Pairs
Problem 1
Find the measure of angle
. Explain or show your reasoning. Find and label a second
degree angle in the diagram. Find and label an angle congruent to angle .
![Two lines IJ and FH intersecting at point G. Angle FGJ is 30 degrees.](../../../../../../embeds/a98ec4a1--8.1.D1.Image.02.1.png)
Activity 1: Cutting Parallel Lines with a Transversal
Problem 1
Lines
![Parallel lines ABC and DEF cut by transversal HJ at points B and E. The angle adjacent to B is labeled 63 degrees.](../../../../../../embeds/135be478--8.1.D1.Image.07.png)
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 second diagram. Lines and are parallel. The next diagram resembles the first one, but the lines form slightly different angles. Work with your partner to find the six unknown angles with vertices at points
and . What do you notice about the angles in this diagram as compared to the earlier diagram? How are the two diagrams different? How are they the same?
Are you ready for more?
Problem 1
![Parallel lines l and m are cut by two transversals which intersect l in the same point which is 60 degrees. Angle x and a 55 deg angle occur at the intersection of the lines and m.](../../../../../../embeds/73833e96--8.1.D.14.MissingAngle.png)
Parallel lines
Activity 2: Alternate Interior Angles Are Congruent
Problem 1
Lines
![Lines l with point B and k with point A are parallel and m is a transversal intersecting at points P and Q. Point M is the midpoint of segment.](../../../../../../embeds/d8d06278--8.1.D1.Image.14.1.png)
Find a rigid transformation showing that angles
Problem 2
In this picture, lines
![Line l with points B and Q. Line K with points P and A. A third line intersects these at point Q and P and point M which is the midpoint of PQ.](../../../../../../embeds/26860fda--8.1.D1.Image.14.6.png)
Does your argument in the earlier problem apply in this situation? Explain.
Lesson Summary
When two lines intersect, vertical angles are equal and adjacent angles are supplementary, that is, their measures sum to 180
![Two lines intersecting in an X shape. with angles labeled 1 & 3 opposite each other (70 degrees), 2 and 4 opposite each other (110 degrees)](../../../../../../embeds/20293dca--8.1.D1.Image.17.png)
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 parallel lines cut by a transversal with angles 1, 2, 3, 4 as described above and repeated with the 2nd parallel line and angles 5 and 7 (70 deg), 6 and 8 (110 deg).](../../../../../../embeds/dc2d5b83--8.1.D1.Image.19.png)
Alternate interior angles are equal because a
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