Lesson 1 1 Core Focus on Geometry Alternate

Lesson 1. 1 Core Focus on Geometry Alternate Exterior and Interior Angles

Warm-Up Find the value of x. 1. 2. x=9 x = 10

Lesson 1. 1 Alternate Exterior and Interior Angles Recognize and apply properties of alternate exterior and alternate interior angles.

Good to Know! There are many special angle pairs, including complementary angles, linear pairs, supplementary angles and vertical angles. Complementary Angles Two angles with a sum of 90. Linear Pair Two adjacent angles with a sum of 180. C 41 5 4 49 A G H m 5 + m 4 = 90 m G + m H = 90 Supplementary Angles Two angles with a sum of 180. 1 2 m 1 + m 2 = 180 100 80 120 60 B m ABC + m CBD = 180 Vertical Angles Two congruent, non-adjacent angles formed by intersecting lines. 2 M K m M + m K = 180 D m 1 = m 3 1 3 4 m 2 = m 4

Vocabulary Transversal A line that intersects two or more lines in the same plane. Alternate Exterior Angles Two angles on the outside of two lines and on opposite sides of a transversal. Alternate Interior Angles Two angles on the inside of two lines and on opposite sides of a transversal.

Explore! Alternate Exterior & Alternate Interior Step 1 Draw two parallel lines intersected by a transversal. Make the diagram large enough to easily measure the angles with a protractor. Label your angles 1 through 8. Step 2 Use a protractor to measure the eight angles on the drawing from Step 1. Record their measures. Step 3 Which pairs of angles in your diagram are alternate exterior angles? What do you notice about the measure of each pair of alternate exterior angles? Step 4 Which pairs of angles are alternate interior angles? What do you notice about the measure of each pair of alternate interior angles? Step 5 Trace the figure from the top of this page onto a piece of paper.

Explore! Alternate Exterior & Alternate Interior Step 6 Slide your tracing until angles 1, 2, 3 and 4 are on top of angles 5, 6, 7 and 8. a. What do you observe? Explain. b. Does this confirm your conclusion in Steps 3 and 4? Step 7 Draw two lines that are not parallel. Draw a transversal that intersects both lines. Step 8 Use a protractor to measure the eight angles created in Step 7. Step 9 Write a conclusion about the alternate interior angles and alternate exterior angles formed by parallel lines compared to those formed by non -parallel lines.

Alternate Interior and Alternate Exterior Angles Alternate Interior Angles If two parallel lines are intersected by a transversal, then the alternate interior angles are congruent. Angles 3 and 5 are alternate interior angles. Angles 4 and 6 are alternate interior angles. 1 2 4 3 5 6 8 7 Alternate Exterior Angles If two parallel lines are intersected by a transversal, then alternate exterior angles are congruent. Angles 1 and 7 are alternate exterior angles. Angles 2 and 8 are alternate exterior angles.

Example 1 Name the angle relationship between 1 and 2. Determine whether 1 and 2 are congruent. a. b. c. a. Alternate interior angles. The angles are congruent because the lines are parallel. b. Vertical angles are congruent to one another. c. Alternate exterior angles. The lines are not parallel, so the angles are not congruent.

Example 2 Identify the special angle pair Relationship between the angles shown. Solve for x. (8 x + 4)° and 86° are alternate exterior angles. Lines l and m are parallel so the alternate exterior angles are congruent. 8 x + 4 = 86 a – 4 a 8 x = 82 a 8 8 a x = 10. 25 Subtract 4 from each side. Divide by 8 on each side. Check the solution. Substitute 10. 25 for x. Multiply. Add. (8 x + 4) = 86 a (8(10. 25) + 4) = 86 a (82 + 4) = 86 a 86 = 86 a The angles are alternate exterior angles and x = 10. 25.

Example 3 Use the figure at right. a. Solve for x. The lines are parallel so alternate interior angles are congruent. Subtract 2 x from each side of the equation. a Add 58 to each side of the equation. Divide each side of the equation by 7. 9 x – 58 = 2 x + 5 – 2 x a 7 x – 58 = 5 a + 58 +58 a 7 x = 63 a 7 7 a x=9 a

Example 3 Continued… Use the figure at right. b. Find the measure of each identified angle. Write the given expression for each angle. Substitute 9 for x. Multiply. Simplify. (9 x – 58) (2 x + 5) (9(9) – 58) (81 – 58) 23 (2(9) + 5) (18 + 5) 23 Each angle measures 23. Alternate interior angles are congruent so part b verifies the solution for x.

Communication Prompt What are the similarities and differences between alternate interior angles and alternate exterior angles?

Exit Problems Find the value of x. 1. 2. x = 22 x=8