6.4 6.5 6.6: Quadrilaterals and Their Properties Objectives: Be able to use properties of sides and angles of rhombuses, rectangles, squares, trapezoids and kites. Be able to use properties of diagonals of rhombuses, rectangles and squares. Be able to identify quadrilaterals based on limited information

Quadrilaterals Rhombus Rectangle Square A parallelogram with four congruent sides. Rhombus A parallelogram with four right angles. Rectangle A parallelogram with four congruent sides, and four right angles. Square

Corollaries Rhombus Corollary: A quadrilateral is a rhombus if and only if it has four congruent sides. Rectangle Corollary: A quadrilateral is a rectangle if and only if it has four right angles. Square Corollary: A quadrilateral is a square if and only if it is a rhombus and a rectangle. You can use these to prove that a quadrilateral is a rhombus, rectangle or square without proving first that the quadrilateral is a parallelogram.

Example: 1) Decide whether the statement is always, sometimes, or never. A. A rectangle is a square. B. A square is a rhombus.

Theorems Theorem 6.11 Theorem 6.12 Theorem 6.13 A parallelogram is a rhombus if and only if its diagonals are perpendicular. A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. Theorem 6.12 Theorem 6.13 A parallelogram is a rectangle if and only if its diagonals are congruent.

Examples: 2) Which of the following quadrilaterals have the given property? All sides are congruent. All angles are congruent. The diagonals are congruent. Opposite angles are congruent. Parallelogram Rectangle Rhombus Square

Example: 3) In the diagram at the right, PQRS is a rhombus. What is the value of y?

Trapezoids A trapezoid is a quadrilateral with exactly one pair of parallel sides. Bases: The parallel sides of a trapezoid. Legs: The nonparallel sides of the trapezoid. Isosceles Trapezoid: A trapezoid whose legs are congruent. Midsegment: A segment that connects the midpoints of the legs and that is parallel to each base. Its length is one half the sum of the lengths of the bases. Base Midsegment Leg Leg Base Angles Base

Isosceles Trapezoids A trapezoid that has congruent legs.

Theorem 6.14 Theorem 6.15 Theorem 6.16 A B If a trapezoid is isosceles, then each pair of base angles is congruent. D C A B If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. D C A B A trapezoid is isosceles if and only if its diagonals are congruent. C D

Example C D E F

Theorem 6.17: Midsegment of a trapezoid The midsegment of a trapezoid is the segment that connects the midpoints of its legs. Theorem 6.17: Midsegment of a trapezoid The midsegment of a trapezoid is parallel to each base and its length is one half the sums of the lengths of the bases. MN║AD, MN║BC MN = ½ (AD + BC)

Example: 5) Find the length of the midsegment RT.

Definition A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.

Kite Theorems Theorem 6.18 If a quadrilateral is a kite, then its diagonals are perpendicular. AC  BD Theorem 6.19 If a quadrilateral is a kite, then exactly one pair of opposite angles is congruent. A ≅ C, B ≅ D

Example 6) Find the lengths of all four sides of the kite.

Example 7) Find mG and mJ in the diagram at the right. 132° 60°