8.6 Trapezoids

Objectives Recognize and apply properties of trapezoids Solve problems using the medians of trapezoids

Trapezoids A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called the bases. A trapezoid has two pairs of base angles. In trapezoid ABCD, D and C are one pair of base angles. The other pair is A and B. The nonparallel sides of the trapezoid are called the legs.

Trapezoids If the legs of a trapezoid are ≅, then it is called an isosceles trapezoid. Theorem 8.18: Both pairs of base s of an isosceles trapezoid are ≅. (A ≅ B and C ≅ D) Theorem 8.19: The diagonals of an isosceles trapezoid are ≅. (AC ≅ BD)

Example 1: Write a flow proof. Given: KLMN is an isosceles trapezoid. Prove:

Example 1: Proof:

Your Turn: Write a flow proof. Given: ABCD is an isosceles trapezoid. Prove:

Your Turn: Proof:

Example 2: The top of this work station appears to be two adjacent trapezoids. Determine if they are isosceles trapezoids. Each pair of base angles is congruent, so the legs are the same length. Answer: Both trapezoids are isosceles.

Your Turn: The sides of a picture frame appear to be two adjacent trapezoids. Determine if they are isosceles trapezoids. Answer: yes

Example 3a: ABCD is a quadrilateral with vertices A(5, 1), B(–3, –1), C(–2, 3), and D(2, 4). Verify that ABCD is a trapezoid. A quadrilateral is a trapezoid if exactly one pair of opposite sides are parallel. Use the Slope Formula.

Example 3a: slope of slope of slope of slope of Answer: Exactly one pair of opposite sides are parallel, So, ABCD is a trapezoid.

Example 3b: ABCD is a quadrilateral with vertices A(5, 1), B(–3, 1), C(–2, 3), and D(2, 4). Determine whether ABCD is an isosceles trapezoid. Explain.

Example 3b: First use the Distance Formula to show that the legs are congruent. Answer: Since the legs are not congruent, ABCD is not an isosceles trapezoid.

Your Turn: QRST is a quadrilateral with vertices Q(–3, –2), R(–2, 2), S(1, 4), and T(6, 4). a. Verify that QRST is a trapezoid. Answer: Exactly one pair of opposite sides is parallel. Therefore, QRST is a trapezoid. b. Determine whether QRST is an isosceles trapezoid. Explain. Answer: Since the legs are not congruent, QRST is not an isosceles trapezoid.

Medians of Trapezoids BC || AD MN = ½ (BC + AD) median The segment that joins the midpoints of the legs of a trapezoid is called the median (MN). It is also referred to as the midsegment. Theorem 8.20: The median of a trapezoid is || to the bases and its measure is ½ the sum of the measures of the bases. BC || AD MN = ½ (BC + AD) median

Example 4a: DEFG is an isosceles trapezoid with median Find DG if and

Example 4a: Theorem 8.20 Substitution Multiply each side by 2. Subtract 20 from each side. Answer:

Example 4b: DEFG is an isosceles trapezoid with median Find , and if and Because this is an isosceles trapezoid,

Example 4b: Consecutive Interior Angles Theorem Substitution Combine like terms. Divide each side by 9. Answer: Because

Your Turn: WXYZ is an isosceles trapezoid with median a. Answer: b. Answer: Because

Assignment Pre-AP Geometry Pg. 442 #9 – 19, 22 – 28, 32, 34