Do Now: List all you know about the following parallelograms. 1.) Rectangle 2.) Rhombus 3.) Square

Geometry 8.4: Properties of Rhombuses, Rectangles, and Squares 8.5: e Properties of Trapezoids and Kites

Rectangles Parallelogram with 4 right angles

Rhombus Parallelogram with four congruent sides.

Square A square is a parallelogram with four congruent sides and four right angles. A square is a rhombus and a rectangle.

Theorems about Diagonals Diagonals of a rhombus are perpendicular (also true for a square- remember a square is a rhombus.)

Theorems about Diagonals Diagonals of a rhombus bisect the opposite angles (also true for a square - remember a square is a rhombus.)

Theorems about Diagonals Diagonals of a rectangle are congruent (also true for a square- remember a square is a rectangle.)

Examples

Examples

Examples

Trapezoids A trapezoid is a quadrilateral with exactly one pair of parallel sides. (not parallelogram) The parallel sides are called bases. The other two sides are called legs. A trapezoid has two pairs of base angles. base A B One pair of base Angles: A & B. Another pair: D and C. leg leg C D base

Isosceles Trapezoids If the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid. Theorem 8.14: If a trapezoid is isosceles, then each pair of base angles is congruent. A B D C

ABCD is an isosceles trapezoid. Theorem 8.15 If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. A B D C ABCD is an isosceles trapezoid. (AD is congruent to BC).

Examples: Find the missing angles.

Theorem 8.16 A trapezoid is isosceles if and only if its diagonals are congruent. A B D C ABCD is an isosceles trapezoid if and only if .

Midsegment of a Trapezoid Midsegment connects the midpoints of the legs. Theorem 8.17: The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. (average) A B Midsegment M N D C

Examples Find the length of the other base.

Examples

Kites A kite is a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent. (not parallelogram)

Kites Theorem 8.18: If a quadrilateral is a kite, then its diagonals are perpendicular. Theorem 8.19: If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

Examples

Example: Find the side lengths of the kite.