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0.3.11 Alternate Interior Angles Lesson Objectives Required Materials Show that alternate interior angles are congruent. geometry toolkits Setup: Angle Pairs (5 minutes) Students in groups of 2. Access to protractors. Statement 1. and Anticipated Responses are lines that intersect at . Estimate 1. Answers vary. Sample response: I think that angle the measures of angles and in the diagram without using a protractor. Share your estimate with a partner and revise if needed. measures and angle measures . 2. Answers vary. Sample response: I was close. My angle sum was correct. 3. . 4. Angle measures measures . and angle 5. Sketch should show a right angle, bisected. Each angle measures 2. Use a protractor to measure angles and . . How close were your estimates? Was the sum of your two estimates accurate? 3. Find the measure of angle your reasoning. . Explain or show 4. Find and label a second degree angle in the diagram. Find and label an angle congruent to angle . 5. Sketch a diagram that shows two angles that are both congruent to each other and complementary. What are their angle measures? Setup: Cutting Parallel Lines With a Transversal (20 minutes) Students in groups of 2, 1 minute quiet time, partners to complete task. Statement Lines and transversal Anticipated Responses are parallel. They are cut by . 1. Explanations vary. Sample response: With a protractor and supplementary and vertical angles. 1. Work with your partner to find the seven unknown angle measures in the diagram. Explain your reasoning. 2. What do you notice about the eight angles at points and ? 3. Using what you noticed, find the measures of the 2. Answers vary. Sample response:The angles in the same place relative to the transversal have the same measure. four angles at point in the second diagram. Lines and are parallel. 3. Answers vary. Sample response: Angle is a 34 degree angle and angle is a 146 degree angle. 4. At vertex at vertex the angles measure they measure and and . and 5. Answers vary. Sample response: In both pictures, the two pair of vertical angles at each vertex are congruent. Also adjacent angles at each vertex are supplementary. In the first picture, the angle measures at the two vertices are the same while in the second picture they are different. 4. The next diagram resembles the first one but the lines form slightly different angles. Work with your partner to find the 6 unknown angles with vertices at points and . 5. What do you notice about the angles in this diagram as compared to the previous diagram? How are the two diagrams different? How are they the same? Setup: Alternate Interior Angles are Congruent (10 minutes) Access to ruler and tracing paper. Statement Anticipated Responses Suppose and are parallel lines. The point is on , and the point is on , as shown in the diagram. 1. If I rotate the picture 180 degrees with center , line angles and trade places. This means that they are congruent. 2. If and are not parallel, then I can not show that angles and are congruent with rigid motions. is the midpoint of segment , which is contained in line , which is a transversal line to and . 1. Find a rigid transformation showing that angles and are congruent. 2. In this picture, lines and are no longer parallel. is still the midpoint of segment . Does your argument in the previous problem apply in this situation? Explain. Setup: All The Rest (5 minutes) Cool-down (5 minutes) Anticipated Responses The diagram shows two parallel lines cut by a transversal. One angle measure is shown. a: , b: , c: , d: , e: , e: , g: . Find the values of , , , , , and . Lesson Summary (5 minutes) What are alternate interior angles? When we have two parallel lines cut by a transversal, how can we find the measures of other angles if we know the measure of one of them? How did we use rigid transformations to help find these angle measures?