9-6: Rhombus, Rectangle, and Square Proof Geometry

Quadrilateral Family

What is a Quadrilateral? All quadrilaterals have four sides. They also have four angles. The sum of the four angles totals 360°. These properties are what make quadrilaterals alike, but what makes them different?

Parallelogram Two sets of parallel sides . (Definition) Two sets of congruent sides. (Theorem) The angles that are opposite each other are congruent . (Theorem) The diagonals bisect each other. (Theorem) Consecutive angles are supplementary. (Theorem)

Rectangle A rectangle is a parallelogram all of whose angles are right angles. (Def.) Has all properties of quadrilateral and parallelogram A rectangle also has four right angles. (Theorem) The diagonals are congruent. (Theorem)

To Prove a quadrilateral is a rEctangle Prove it is a parallelogram with 1 right angle Theorem: If a parallelogram has one right angle then it is a rectangle. Prove that the diagonals are congruent to each other and bisect each other. Theorem: If the diagonals of a quadrilateral are congruent and bisect each other, then it is a rectangle.

Rhombus A rhombus is a parallelogram all of whose sides are congruent. (Definition) All properties of parallelogram All sides are congruent (Theorem) The diagonals bisect the angles. (Theorem) The diagonals are perpendicular bisectors of each other (Theorem)

To Prove a quadrilateral is a rhombus Prove it is a parallelogram with all 4 sides congruent. Prove it is a parallelogram with each diagonal bisecting a pair of opposite angles. Prove it is a quadrilateral with diagonals that bisect each other and are perpendicular to each other.

Square To Prove a quadrilateral is a Square A square is a rectangle all of whose sides are congruent.(Definition) All the properties of a parallelogram apply All the properties of a rectangle apply All the properties of a rhombus apply To Prove a quadrilateral is a Square Prove it is a rectangle and a rhombus

Trapezoid To Prove a quadrilateral is a Trapezoid Trapezoid has one and only one set of parallel sides. To Prove a quadrilateral is a Trapezoid Prove a set of sides is parallel Technically, we would need to show the other pair of sides are not parallel. We won’t do that.

Isosceles Trapezoid An isosceles trapezoid has two equal sides. These equal sides are called the legs of the trapezoid, which are the non- parallel sides of the trapezoid. Never assume a trapezoid is isosceles unless it is given Both pair of base angles in an isosceles trapezoid are also congruent. The diagonals are congruent

To Prove a quadrilateral is an isosceles trapezoid Prove that is a trapezoid with the non parallel set of sides congruent Prove that it is a trapezoid with congruent diagonals Prove that it is a trapezoid with base angles congruent.

Kite Two adjacent sets of consecutive congruent sides (Definition) Diagonals are perpendicular to each other. (Theorem) One of the diagonals bisects the other. The angles included between the sets congruent sides are congruent—i.e. one pair of congruent opp. Angles. (Theorem)

To Prove a quadrilateral is a Kite Prove that it is a quadrilateral with two sets of consecutive congruent sides. Prove that one diagonal is the perpendicular bisector of the other.

HOmework p. 289: #4-6, 8-10