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CP1 Math 2 Unit 1: Deductive Geometry: Day 16 Name_________________________ Parallel Lines & PAI Objectives: • Observe relationships among the angles formed when two parallel lines are cut by a transversal • Apply the PAI postulate Vocabulary: • Transversal: a line that intersects two or more coplanar lines at different points • Parallel lines: lines that have the same slope; they never intersect The PAI Postulate: Note: In many textbooks, PAI is a proved theorem instead of an assumed postulate. If you are curious about proving PAI, see page 470 of your book, but this is not expected that you prove it in this course. Practice: Try applying PAI in the following problems. Complete your work on a separate sheet of paper. 1. Answer the following questions about the given diagram in which 𝑚 ∥ 𝑛 (line m is parallel to line n). a. There are two pairs of alternate interior angles that are congruent because of the PAI Postulate. One pair is ∠4 ≅ ∠7. What is the other pair of congruent alternate interior angles? b. Use PAI to prove that ∠4 ≅ ∠8. c. Determine the relationship between ∠2 and ∠7, and then prove it. (FYI: These angles are called “consecutive angles” or “same-­‐side interior angles”). d. Suppose that ∠3 = 132°. What is the measure of ∠6? Write a proof to support your answer. 2. The goal of this problem is to prove some relationships between the angle measures of a parallelogram. a. Which of the following appears to be the relationship between the measures of ∠1 and ∠2? Circle one. congruent complementary (add to 90°) supplementary (add to 180°) b. Which of the following appears to be the relationship between the measures of ∠1 and ∠3? Circle one. congruent complementary supplementary The key to proving these relationships is to add some transversals to the diagram so that the PAI Postulate can be used. Here’s the same parallelogram with some lines extended and angle number labels added. c. Why is ∠1 ≅ ∠5? d. Why is ∠5 ≅ ∠6? e. Why is ∠6 ≅ ∠3? Hint: If you’re not sure, try rotating the paper. f. Using your ideas from parts c, d, and e, write a proof that ∠1 ≅ ∠3. g. How are the measures of ∠5 and ∠2 related? Explain why. h. Write a proof that ∠1 and ∠2 are supplementary. Homework: GeoGebra Investigation In class, we established that when two parallel lines are cut by a transversal, the alternate interior angles are congruent. Your job is to discover the other angle relationships that result. Diagram: Line l is the transversal cutting m and n. m and n are parallel lines. m n l b d f a c h e g Part 1: Step 1: Construct two parallel lines Step 2: Construct a transversal (this can be any line that goes through both parallel lines) Step 3: Use the angle measurement tool to measure each of the 8 angles created. 1. Which angles are congruent to each other? Be specific. Don’t just refer to them by letter, but describe how they are spatially related (example: “alternate interior angles” instead of just “Angle D and E”). 2. Move the transversal (using the point) and the parallel lines (by clicking on the lines). Which relationships from #1 are still true? 3. Now test out various ways to sum the angles. Which sums of angle measures are invariant? Part 2: Step 1: Construct two intersecting lines cut by a transversal. Step 2: Measure the angles created. 1. Which angles are congruent to each other? You may find it helpful to sketch a picture of your GeoGebra diagram and label the angles. 2. Move the transversal and the intersecting lines. Which angles remain congruent? 3. Which sums of angle measures are invariant? 4. How are the angles of intersecting lines cut by a transversal the same as the angles of parallel lines cut by a transversal? 5. How are the angles of intersecting lines cut by a transversal different from the angles of parallel lines cut by a transversal?