Warm up: Can you conclude that the parallelogram is a rhombus, a rectangle, or a square? Explain. For what value of x is parallelogram ABCD a rectangle? Given WRST is a parallelogram and WS is congruent to RT, how can you classify WRST? Explain.

6.6/6.7 - Trapezoids and Kites AND polygons in the coordinate plane I can verify and use properties of trapezoids and kites. I can classify polygons in the coordinate plane.

The nonparallel sides are called legs. Trapezoids A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides of a trapezoid are called bases. The nonparallel sides are called legs. The two angles that share a base of a trapezoid are called base angles. A trapezoid has two pairs of base angles.

An isosceles trapezoid is a trapezoid with legs that are congruent. ABCD below is an isosceles trapezoid. The angles of an isosceles trapezoid have some unique properties.

Theorem 6 – 19 Theorem 6 – 19 Theorem If… Then… If a quadrilateral is an isosceles trapezoid, then each pair of base angles is congruent.

Problem: Finding Angle Measures in Trapezoids CDEF is an isosceles trapezoid and m<C = 65. What are m<D, m<E, and m<F?

Problem: Finding Angle Measures in Trapezoids In the diagram, PQRS is an isosceles trapezoid and m<R = 106. What are m<P, m<Q, and m<S?

Problem: Finding Angle Measures in Trapezoids RSTU is an isosceles trapezoid and m<S = 75. What are m<R, m<T, and m<U?

Theorem 6 – 20 Theorem 6 – 20 Theorem If… Then… If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.

Trapezoid Midsegment Theorem In Chapter 5 – 1, we learned about midsegments of triangles. Trapezoids also have midsegments. The midsegment of a trapezoid is the segment that joins the midpoints of its legs. The midsegment has two unique properties. Trapezoid Midsegment Theorem Theorem If… Then… If a quadrilateral is a trapezoid, then The midsegment is parallel to the bases, and The length of the midsegment is half the sum of the lengths of the bases.

Problem: Using the Midsegments of a Trapezoid QR is the midsegment of trapezoid LMNP. What is x?

Problem: Using the Midsegments of a Trapezoid MN is the midsegment of trapezoid PQRS. What is x? What is MN?

Problem: Using the Midsegments of a Trapezoid TU is the midsegment of trapezoid WXYZ. What is x?

The angles, sides, and diagonals of a kite have certain properties. A kite is a quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent. The angles, sides, and diagonals of a kite have certain properties. Theorem 6 – 22 Theorem If… Then… If a quadrilateral is a kite, then its diagonals are perpendicular.

Problem: Finding Angle Measures in Kites Quadrilateral DEFG is a kite. What are m<1, m<2, and m<3?

Problem: Finding Angle Measures in Kites Quadrilateral KLMN is a kite. What are m<1, m<2, and m<3?

Problem: Finding Angle Measures in Kites Quadrilateral ABCD is a kite. What are m<1 and m<2?

Concept Summary: Relationships Among Quadrilaterals

After: Lesson Check What are the measures of the numbered angles? What is the length of the midsegment of a trapezoid with bases of length 14 and 26?

After: Lesson Check Is a kite a parallelogram? Explain. How is a kite similar to a rhombus? How is it different? Explain. Since a parallelogram has two pairs of parallel sides, it certainly has one pair of parallel sides. Therefore, a parallelogram must also be a trapezoid. What is the error in this reasoning? Explain.

Homework: Page 394, #8-34 even AND Page 403, #5-7 all, 10