Parallel Lines and a Transversal
Vocabulary Parallel lines: Lines in the same plane that have the same slope and never intersect. Transversal: A line that intersects two or more parallel lines.
Vocabulary Interior Angles: Angles that lie between the parallel lines. Same Side Interior Angles: Interior angles on the same side of the transversal. Alternate Interior Angles: Angles that lie between the parallel lines and on opposite sides of the transversal, and are congruent.
Vocabulary Exterior Angles: Angles that lie outside the parallel lines. Same Side Exterior Angles: Exterior angles on the same side of the transversal. Alternate Exterior Angles: Angles that lie outside the parallel lines and on opposite sides of the transversal, and are congruent.
Vocabulary Corresponding Angles: Angles that are in the same relative position and are congruent. Supplementary Angles: Angles that have a sum of 180. Linear Pair: Two angles with a common side that are supplementary. Vertical Angles: Formed when two lines cross, they are on opposite sides of both lines.
Theorem When parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent, the pairs of alternate interior angles are congruent, and the pairs of alternate exterior angles are congruent.
Converse of the Theorem When corresponding angles or alternate interior angles or alternate exterior angles are congruent, then the lines that are cut by the transversal to form the angles are parallel.
Corresponding Angles Observe where the angles from the top intersection end up when it is “cut out” and slid down the transversal…
Identifying Angles. In the figure below, 𝐿 1 ∥ 𝐿 2 , and 𝑚 is a transversal. Parallel lines: Transversal: Interior Angles: Exterior Angles: Corresponding Angles: Alt. Int. Angles: Alt. Ext. Angles: Supplementary: Vertical Angles: Linear Pairs:
Discussion 1 2 Consider the relationships between the angles… How many angles do we need to know the measure of in order to determine the measure of the rest? 1 How many different angle measures should we expect to see amongst the angles? 2 They are corresponding angles because they are on the same side of the transversal and in corresponding positions, (i.e., above each of 𝐿 1 and 𝐿 2 or below each of 𝐿 1 and 𝐿 2 ). We can show their congruence using translation.
What if… What if L1 and L2 were not parallel? Which angles are corresponding angles? Are they congruent? Why or why not? L1 L2 m 1 2 3 4 5 6 7 8 What about alternate interior? Alternate exterior? What has to be true for these angles to be congruent?
Example 1 1. If ∠1≅∠3 is k parallel to l? Be prepared to Explain. 𝑛 𝑚 𝑙 𝑘 16 15 14 13 12 11 10 9 7 6 5 4 3 2 1 8 1. If ∠1≅∠3 is k parallel to l? Be prepared to Explain. y: yes n: no 2. If ∠9≅∠6 is m parallel to n? Be prepared to Explain. y: yes n: no
Example 2 Using the diagram below, determine if angles 1 and 9 are: a) interior b) exterior c) vertical d) corresponding 𝑛 𝑚 𝑙 𝑘 16 15 14 13 12 11 10 9 7 6 5 4 3 2 1 8
Example 3 Using the diagram below, determine if angles 10 and 15 are: a) Alternate Interior b) Alternate Exterior c) Vertical d) Corresponding 𝑛 𝑚 𝑙 𝑘 16 15 14 13 12 11 10 9 7 6 5 4 3 2 1 8
Example 4 Using the diagram below, determine if angles 1 and 4 are: a) interior b) exterior c) vertical d) corresponding 𝑛 𝑚 𝑙 𝑘 16 15 14 13 12 11 10 9 7 6 5 4 3 2 1 8
Example 5 Using the diagram below, determine if angles 5 and 2 are: a) interior b) exterior c) vertical d) corresponding 𝑛 𝑚 𝑙 𝑘 16 15 14 13 12 11 10 9 7 6 5 4 3 2 1 8