Lesson 5-3 Theorems Involving Parallel Lines (page 177) Essential Question How can the properties of quadrilaterals be used to solve real life problems?

Properties of Parallelograms Parallelogram has both pairs of opposite sides parallel. Parallelogram has both pairs of opposite sides congruent. Parallelogram has both pairs of opposite angles congruent. Parallelogram ’ s diagonals bisect each other. The consecutive angles of a parallelogram are supplementary.

5 Ways to Prove that a Quadrilateral is a Parallelogram SS how that both pairs of opposite sides are parallel. SS how that both pairs of opposite sides are congruent. SS how that one pair of opposite sides are both ≅ & ||. SS how that both pairs of opposite angles are congruent. SS how that the diagonals bisect each other.

If two lines are parallel, then all points on one line are equidistant from the other line. Theorem 5-8

If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. Theorem 5-9 Proof:For an outline of this proof see page 177. Note that parallelograms and congruent triangles are used in the proof. We will do an informal proof using the parallel lines on notebook paper. B C XA ➤ ➤ ➤ Y Z

A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side. Theorem 5-10 Given:M is the midpoint of Prove:N is the midpoint of BC M A ➤ ➤ N D ● ➤

The segment that joins the midpoints of two sides of a triangle: (1) is parallel to the third side; (2) is half as long as the third side. Theorem 5-11 Given:M is the midpoint of N is the midpoint of BC M A N

The segment that joins the midpoints of two sides of a triangle: (1) is parallel to the third side; (2) is half as long as the third side. Theorem 5-11 Given:M is the midpoint of N is the midpoint of BC M A N ➤ ➤

The segment that joins the midpoints of two sides of a triangle: (1) is parallel to the third side; Given:M is the midpoint of N is the midpoint of Prove: (1) BC M A N ➤ ➤ Proof:To prove this part of the theorem, use Theorem reasoning indirectly to prove this part of the theorem.

The segment that joins the midpoints of two sides of a triangle: (1) is parallel to the third side; (2) is half as long as the third side. Theorem 5-11 BC M A N

The segment that joins the midpoints of two sides of a triangle: (2) is half as long as the third side. BC M A N ● L

Example #1: If NT = x + 6 and NZ = 3 x - 8, find the value of x, NT, TZ, and NZ. 20 x = ______ NT = ______ TZ = ______ NZ = ______ M N T S Z Y x+6+x+6 = 3x-8 (x+6) (3x-8) 2x+12 = 3x-8 20 = x NT = TZ = x+6 = 20+6 = 26 NZ = 3x-8 = 3(20)-8 = 60-8 = 52 NT + TZ = NZ

Example #2 Name the points shown that must be midpoints of the sides of the large triangle. Y X C A B Z 66 A & C

Example #3: If ST = 4 x + 4 and AB = x + 40, find x, ST, and AB. S R A B T x = ______ AB = ______ ST = ______

Assignment Written Exercises on pages 180 & 181 RECOMMENDED: 1, 3, 17, 19 REQUIRED: 5 to 15 odd numbers, 20 Prepare for Quiz on Lessons 5-1 to 5-3 Worksheet on Lessons 5-1 to 5-3: Parallelograms How can the properties of quadrilaterals be used to solve real life problems?