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Lesson 14Alternate Interior Angles

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.

14.1 Angle Pairs

  1. Find the measure of angle JGH .  Explain or show your reasoning.

    Two lines are in intersecting. One of the angles created is 30 degrees.
  2. Find and label a second 30^\circ degree angle in the diagram. Find and label an angle congruent to angle JGH .

14.2 Cutting Parallel Lines with a Transversal

Lines AC and DF are parallel. They are cut by transversal HJ .

Two parallel lines are itersected by one other line.

  1. With your partner, find the seven unknown angle measures in the diagram. Explain your reasoning.

  2. What do you notice about the angles with vertex  B and the angles with vertex E ?

  3. Using what you noticed, find the measures of the four angles at point B in the second diagram. Lines AC and DF are parallel.

    Two parallel lines are itersected by one other line.

  4. 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 B and E .

  5. 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?

Two parallel lines are itersected by two other lines.

Parallel lines \ell and m are cut by two transversals which intersect \ell in the same point. Two angles are marked in the figure. Find the measure x of the third angle.

14.3 Alternate Interior Angles Are Congruent

  1. Lines \ell and k are parallel and t is a transversal. Point M is the midpoint of segment PQ .
    Two parallel lines are itersected by one other line.

    Find a rigid transformation showing that angles MPA and MQB are congruent.
  2. In this picture, lines \ell and k are no longer parallel. M is still the midpoint of segment PQ .
    Three lines are intersecting with several points

    Does your argument in the earlier problem apply in this situation? Explain.

Lesson 14 Summary

When two lines intersect, vertical angles are equal and adjacent angles are supplementary, that is, their measures sum to 180 ^\circ . For example, in this figure angles 1 and 3 are equal, angles 2 and 4 are equal, angles 1 and 4 are supplementary, and angles 2 and 3 are supplementary.

A circle is cut into 40 parts. The angles are 70, 70, 110, and 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.

Intersecting lines create angles within two circles. Both circle have angles of 70, 70, 110, and 110 degrees.

Alternate interior angles are equal because a 180^\circ rotation around the midpoint of the segment that joins their vertices takes each angle to the other. Imagine a point M halfway between the two intersections—can you see how rotating 180^\circ about M 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 70^\circ we use vertical angles to see that angle 3 is 70^\circ , then we use alternate interior angles to see that angle 5 is 70^\circ , then we use the fact that angle 5 is supplementary to angle 8 to see that angle 8 is   110^\circ since 180 -70 = 110 . It turns out that there are only two different measures. In this example, angles 1, 3, 5, and 7 measure 70^\circ , and angles 2, 4, 6, and 8 measure 110^\circ .

Glossary Terms

alternate interior angles

Alternate interior angles are created when two parallel lines are crossed by another line called a transversal. Alternate interior angles are inside the parallel lines and on opposite sides of the transversal.

This diagram shows two pairs of alternate interior angles. Angles a and d are one pair and angles b and c are another pair.

Alternate interior angles are shown
transversal

A transversal to two parallel lines is a line that cuts across them, intersecting each one.

This diagram shows a transversal line k intersecting parallel lines m and \ell .

diagram shows a transversal line "k" intersecting parallel lines "m" and "l".

Lesson 14 Practice Problems

  1. Use the diagram to find the measures of each angle. Explain your reasoning.

    1. m{\angle ABC}
    2. m{\angle EBD}
    3. m{\angle ABE}
    Two lines are in intersecting. One of the angles created is 50 degrees.

  2. Lines k and \ell are parallel, and the measure of angle ABC is 19 degrees.

    Points F, C, and D lie on line k, where point F is to the left of point C and point D is to the right of point C. Points A and B lie on line l, where point B is to the right of point A. Lines k and l are parallel, where line k is above line l, and both lines slant upward and to the right. A third line, labeled m, intersects lines k and l at point C and point B and has a point labeled E located to the left of point C.
    1. Explain why the measure of angle ECF is 19 degrees. If you get stuck, consider translating line \ell by moving B to C .
    2. What is the measure of angle BCD ? Explain.

  3. The diagram shows three lines with some marked angle measures.

    Intersecting lines create angles within two circles. One circle has an angle of 70 degrees ad the other circle has an angle of 53 degrees. The other angles are undetermined.

    Find the missing angle measures marked with question marks.

  4. The two figures are scaled copies of each other. 

    1. What are some ways that you can tell they are scaled copies?
    2. What is the scale factor that takes Figure 1 to Figure 2?
    3. What is the scale factor that takes Figure 2 to Figure 1?
    Figure 1 is transformed to become Figure 2.