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Math 367 Homework Assignment 7 due Thursday, April 14 1. Lines `1 and `2 in the figure below are parallel. Let x = µ(∠1), y = µ(∠2), and z = µ(∠3). If z = 72 and x = y + 18, find x and y. A A A `1 A A A 2 A A 1 A `2 A A A A A A A 3 A 2. Prove that for any triangle, the measure of an exterior angle is the sum of the measures of its two remote interior angles. 3. Prove Theorem 5.1.6, that is, prove that if ` and `0 are parallel lines and t is a transversal such that t ⊥ `, then t ⊥ `0 . (Hint: Use Theorem 5.1.1.) 4. Prove Theorem 5.1.10, Part 1, that is, prove that either diagonal divides a parallelogram into two congruent triangles. (Hint: Find alternate interior angles and use the ASA Theorem.) 5. Prove Theorem 5.1.10, Part 2, that is, prove that the opposite sides of a parallelogram are congruent. (Hint: Use #4 above.) 6. Prove Theorem 5.1.10, Part 3, that is, prove that the opposite angles of a parallelogram are congruent. (Hint: Use #4 above.) 7. Recall that a rectangle is defined to be a quadrilateral each of whose angles is a right angle. ←→ ←→ ←→ ←→ A parallelogram is defined to be a quadrilateral ABCD for which AB k CD and AD k BC. Let ABCD be a rectangle. Prove that ABCD is a parallelogram. (Hint: Consider the lines and transversals obtained by extending some of the sides of the rectangle, and apply the Alternate Interior Angles Theorem (Theorem 4.4.2).)