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Warm up: Identify the best name for the given traits of each quadrilateral. 1.) ABCD has two sets of parallel sides and a right angle. 2.) HGDT has 2 sets of opposite congruent sides and the left side is congruent to the top 3.) YHBG has 2 sets of opposite congruent angles and its diagonals are perpendicular. 6.5 TRAPEZOIDS AND KITES LEQ: What are the special properties of trapezoids and kites? Using properties of trapezoids Theorem 6-15 If a trapezoid is isosceles, then each pair of base angles is congruent. A ≅ B, C ≅ D C A B D Proof of part of thm. 6-15 Prove that m<C=m<D A B Construct BE so that it’s parallel to AC 1 C 1.) AC=BD, AB//CD 1.) Given 2.) AC=BE 2.) Opposite sides of parallelograms are congruent 3.) BE=BD 3.) Transitive Property/substitution 4.) m<D=m<1 4.) Isosceles Triangle Thm. 5.) m<1=m<C 5.) Corresponding Angles 6.) m<C=m<D 6.) Transitive/Substitution E D Ex. 1: Using properties of Isosceles Trapezoids PQRS is an isosceles trapezoid. Find mS, mQ, mR and QR. P Q 50° 2.6 S R LAYER CAKE A baker is making a cake like the one at the right. The top layer has a diameter of 8 inches and the bottom layer has a diameter of 20 inches. How big should the middle layer be? (the top of the middle layer must be exactly halfway between the top of the first and 3rd layers) Recall midsegment of a triangle Half the distance of the base Theorem 6-18: Midsegment of a trapezoid Different!! The median of a trapezoid is the segment that connects the midpoints of its legs. Thm 6-18 says the median: Is parallel to each base: MN║AD, MN║BC Has a length equal to the average of the base lengths MN = ½ (AD + BC) B M C N A D Example: Both are medians Find the value of x. Find the value of x. 4 x 5.8 x+3 8.6 22 Theorem 6-16 The diagonals of an isosceles trapezoid are congruent. Why??*Bonus* proof due Monday Ex: Find the value of x if AC = 2x + 10 and BD = 3x + 4 A D B C Kites Kites Theorem 6-17: Recap: What are the The diagonals of a kite requirements in are order to be a kite? _____________. What other quadrilaterals share this characteristic? *note*: one pair of opp. <‘s is bisected (the diag. that splits the congruent legs) Example: Find the measures of the numbered angles. Find the measures of the numbered angles. 32˚ 46˚ What rules can we derive for kites? 1.) diagonals are perpendicular 2.)adjacent sides congruent, but not all sides congruent