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Name ________________________________________ Date ___________________ Class __________________ LESSON 6-6 Reteach Properties of Kites and Trapezoids A kite is a quadrilateral with exactly two pairs of congruent consecutive sides. If a quadrilateral is a kite, such as FGHJ, then it has the following properties. Properties of Kites FH ⊥ GJ ∠G ≅ ∠J Exactly one pair of opposite angles is congruent. The diagonals are perpendicular. A trapezoid is a quadrilateral with exactly one pair of parallel sides. If the legs of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. Each nonparallel side is called a leg. Base angles are two consecutive angles whose common side is a base. Each � side is called a base. Isosceles Trapezoid Theorems • In an isosceles trapezoid, each pair of base angles is congruent. • If a trapezoid has one pair of congruent base angles, then it is isosceles. • A trapezoid is isosceles if and only if its diagonals are congruent. In kite ABCD, m∠BCD = 98°, and m∠ADE = 47°. Find each measure. 1. m∠DAE _________________________________________ 2. m∠BCE _________________________________________ 3. m∠ABC _________________________________________ 4. Find m∠J in trapezoid JKLM. 5. In trapezoid EFGH, FH = 9. Find AG. _________________________________________ ________________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 6-46 Holt McDougal Geometry Name ________________________________________ Date ___________________ Class __________________ LESSON 6-6 Reteach Properties of Kites and Trapezoids continued Trapezoid Midsegment Theorem The midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs. • The midsegment of a trapezoid is parallel to each base. AB & MN and AB & LP • The length of the midsegment is one-half the sum of the length of the bases. 1 AB = (MN + LP) 2 AB is the midsegment of LMNP. Find each value so that the trapezoid is isosceles. 7. AC = 2z + 9, BD = 4z − 3. Find the value of z. 6. Find the value of x. _________________________________________ ________________________________________ Find each length. 8. KL 9. PQ _________________________________________ 10. EF ________________________________________ 11. WX _________________________________________ ________________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 6-47 Holt McDougal Geometry found from BA and AE by using the Pythagorean Theorem. BD is the sum of 1 BE and ED. The area is (AC)(BD). 2 3. No; possible answer: there is no way to use the Pythagorean Theorem to find the length of AE, and thus AC, with the information provided. 4. Possible answer: It is given that QU is perpendicular to PT and SR is perpendicular to SR , so ∠PTU and ∠QUT are right angles. It is also given that PQ is parallel to SR , so the same-side interior angles ∠PTU and ∠TPQ are supplementary, as are ∠QUT and ∠PQU. An angle supplementary to a right angle must be a right angle, so ∠TPQ and ∠PQU are right angles. All four interior angles of PQUT are right angles, so PQUT is a rectangle by the definition of a rectangle. 5. Possible answer: It is given that JKLM is an isosceles trapezoid, so JK ≅ ML and KL & JM . Base angles in an isosceles trapezoid are congruent, so ∠J ≅ ∠M. Corresponding angles are equal, so ∠NKL ≅ ∠J and ∠NLK ≅ ∠M. By the Transitive Property of Congruence, ∠NKL ≅ ∠NLK. ∠NKL and ∠NLK are the base angles of UNKL, so it is an isosceles triangle. Thus NK ≅ NL . By the Segment Addition Postulate, JN = JK + NK and MN = ML + NL. By the Addition Property of Equality and the definition of congruent segments, JN = MN. Because JN and MN have the same length, they are congruent. So UJNM is isosceles. Challenge 1. Possible answer: A dart is a concave quadrilateral with exactly two pairs of congruent consecutive sides. 2. Darts and kites both have exactly two pairs of congruent consecutive sides. Darts are concave, and kites are convex. 3. x = 11; y = 2 4. Possible answer: The lines are perpendicular. The line containing the first diagonal bisects the diagonal that joins the fin angles. 5. Possible answer: The triangles are congruent and obtuse. 6. Possible answer: It is given that AB ≅ AD and BC ≅ DC. By the Reflex. Prop. of ≅, AC ≅ AC. This means that UABC ≅ UADC by SSS. So ∠B ≅ ∠D by CPCTC. Problem Solving 1. 3. 5. 7. 23.1 124° C B 2. 74.0 4. 18 in. 6. G Reading Strategies 1. 2. 3. 4. 5. 6. Reteach 1. 43° 2. 55° 3. 70° 4. 56° 5. 2.8 6. x = ±4 7. z = 6 8. 21 9. 9.5 7. 10. 9.15 11. 10.8 Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. A66 Holt McDougal Geometry