Lesson 12Alternate Interior Angles
Learning Goal
Let’s explore why some angles are always equal.
Learning Targets
I can find unknown angle measures by reasoning about complementary or supplementary angles.
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
- adjacent angles
- alternate interior angles
- complementary
- straight angle
- supplementary
- transversal
- vertical angles
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 .
Activity 1: Cutting Parallel Lines with a Transversal
Problem 1
Lines
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
Activity 2: Alternate Interior Angles Are Congruent
Problem 1
Lines
Find a rigid transformation showing that angles
Problem 2
In this picture, lines
Does your argument in the earlier problem apply in this situation? Explain.
Activity 3: Info Gap: Angle Finding
Problem 1
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the problem card:
Silently read your card and think about what information you need to be able to answer the question.
Ask your partner for the specific information that you need.
Explain how you are using the information to solve the problem.
Continue to ask questions until you have enough information to solve the problem.
Share the problem card and solve the problem independently.
Read the data card and discuss your reasoning.
If your teacher gives you the data card:
Silently read your card.
Ask your partner “What specific information do you need?” and wait for them to ask for information.
If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information.
Before sharing the information, ask “Why do you need that information?” Listen to your partner’s reasoning and ask clarifying questions.
Read the problem card and solve the problem independently.
Share the data card and discuss your reasoning.
Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.
Lesson Summary
When two lines intersect, vertical angles are equal and adjacent angles are supplementary, that is, their measures sum to 180
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.
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