TRAPEZOIDS AND KITES Lesson 6-6

VOCABULARY: Trapezoid: A quadrilateral with exactly one pair of parallel sides. Bases: The parallel sides of a trapezoid. Legs of a Trapezoid: The non parallel sides of a trapezoid. Base Angles: Angles formed by a base and a leg of a trapezoid. They come in pairs of two. Isosceles Trapezoid: If the legs of a trapezoid are congruent, it forms a special trapezoid. Midsegment of a Trapezoid: the segment of a trapezoid that connects the midpoints of its legs. Kite: A quadrilateral with exactly two pairs of consecutive congruent sides. Opposite sides of a kite are not parallel.

ISOSCELES TRAPEZOID PROPERTIES **ERROR ALERT: Base angles of a trapezoid are only congruent IF the trapezoid is an isosceles.

EXAMPLE: Step 1: graph the quadrilateral Step 2: determine what we need to know and what we are given. Given: Quadrilateral ABCD Prove: that it is an isosceles trapezoid. Step 3: We know…..an isosceles trapezoid has to have two congruent legs and one set of parallel sides, so we have to prove that AD ∥ BC and we have to prove that AB = DC. We can prove parallel lines through slope formula and we can prove congruence through distance formula.

CONT. EXAMPLE:

THE MIDSEGMENT OF A TRAPEZOID IS THE SEGMENT THAT CONNECTS THE MIDPOINTS OF THE LEGS OF THE TRAPEZOID. IT IS ALWAYS PARALLEL TO EACH BASE. ITS FORMULA IS:

EXAMPLE: We can use the midsegment of a trapezoid formula to find this. Substitution 15 =.5 (x ) Distributive Property 15 =.5x  5.9 =.5x.5.5  11.8 = x

A KITE IS A QUADRILATERAL WITH EXACTLY TWO PAIRS OF CONSECUTIVE CONGRUENT SIDES. UNLIKE A PARALLELOGRAM, THE OPPOSITE SIDES OF A KITE ARE NOT CONGRUENT OR PARALLEL.

PROPERTIES NOT IN OUR BOOK:

First of all, the problem DOES NOT TELL US THIS IS A KITE, but according to the definition of a kite, it does have exactly two pairs of consecutive congruent sides, so it must be a kite…therefore the kite property theorems will apply. So, according to 6.26, kites have exactly ONE pair of opposite angles that are congruent. If m ∠ BAD ≠ m ∠ BCD, then m ∠ ABC and m ∠ ACD must be equal. And quadrilaterals have interior measure of 360 degrees. m ∠ BAD + m ∠ BCD + m ∠ ABC + m ∠ ADC = x + x = x = x = 272 (dividing both sides by 2) x = 136