Please click the speaker symbol on the left to hear the audio that accompanies some of the slides in this presentation. Quadrilaterals

Quadrilateral: A Polygon with 4 sides and 4 angles Quadrilateral: A Polygon with 4 sides and 4 angles. Sides: May or may not be congruent. Angles: The sum of all interior angles in a quadrilateral is 360o. Quad means “4” and lateral means “sides”.

Types of Quadrilaterals

In General …` Quadrilaterals are polygons that have 4 sides and 4 angles.

Parallelograms A parallelogram has two parallel pairs of opposite sides.

Parallelogram Properties- Sides Opposite sides are parallel Opposite sides are congruent

Parallelogram Properties- Angles Opposite angles are congruent Consecutive angles are supplementary m m

Parallelogram Properties- Diagonals Diagonals of a parallelogram meet at the midpoints. Divide the parallelogram into two congruent triangles

Parallelogram- Area A = bh Area equals the base times the height.

Theorems A quadrilateral is a parallelogram if any ONE of the following is true: Both pairs of opposite sides are || Both pairs of opposite sides are ≈ A pair of opposite sides is both || and ≈ Both pairs of opposite angles are ≈ Diagonals bisect each other

Rectangles A rectangle has: two sets of opposite sides parallel, and four right angles. It is also a parallelogram, since it has two pairs of parallel sides.

Rectangle Properties- Sides Opposite sides are parallel Opposite sides are congruent

Rectangle Properties- Angles Four Right Angles Opposite angles are congruent Consecutive angles are supplementary m m

Rectangle Properties- Diagonals Diagonals of a rectangle are congruent. Diagonals bisect each other.

Rectangle - Area A = bh Area equals the base times the height.

Theorems If a parallelogram is a rectangle, then the diagonals are congruent. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Rhombus … A rhombus is a parallelogram with four equal sides. It is not always a rectangle because a rhombus does not have to have 4 right angles.

Rhombus Properties- Sides All four sides are congruent

Rhombus Properties- Angles Same as all other parallelograms: Opposite angles are congruent Consecutive angles are supplementary

Rhombus Properties- Diagonals Diagonals of a rhombus are perpendicular. Each diagonal bisects the opposite angles.

Rhombus- Area A = ½ d1 d2 Area equals ½ times the first diagonal times the second diagonal.

Theorems If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

Squares … A square has: four right angles four congruent sides It is also a rectangle parallelogram rhombus

Square Properties Opposite sides are || All sides are ≈ Opposite angles are ≈ (90°) Consecutive angles are supplementary Diagonals are ≈ Diagonals bisect each other and angles

Square - Area A = s2 Area equals the side times the side.

Trapezoids … Trapezoids only have one pair of parallel sides. A Trapezoid is NOT a parallelogram. `

Parts of a Trapezoid The parallel sides are bases. The nonparallel sides are legs. Pairs of base angles share the same base. M and  H are a pair of base angles.  A and  T are a pair of base angles.

Isosceles Trapezoids Legs are ≈ Both pairs of base angles are ≈ Diagonals are ≈

Median of a Trapezoid Median is a segment that joins the midpoints of the legs. Parallel to the bases Length is ½ the sum of the bases median=½ (b1 + b2)

Trapezoid - Area A = ½ h(b1 + b2) Area equals ½ the height times the sum of the bases.

Quadrilaterals General Quadrilateral Trapezoid 4 sides, 4 angles Only 1 pair of parallel sides

Quadrilaterals General Quadrilateral Parallelogram Trapezoid 4 sides, 4 angles Parallelogram Opposite sides are parallel and congruent Trapezoid Only 1 pair of parallel sides

Quadrilaterals General Quadrilateral Parallelogram Trapezoid Rectangle 4 sides, 4 angles Parallelogram Opposite sides are parallel and congruent Trapezoid Only 1 pair of parallel sides Rectangle A parallelogram with 4 right angles

Quadrilaterals General Quadrilateral Parallelogram Trapezoid Rectangle 4 sides, 4 angles Parallelogram Opposite sides are parallel and congruent Trapezoid Only 1 pair of parallel sides Rectangle A parallelogram with 4 right angles Square A rectangle with 4 congruent sides

Quadrilaterals General Quadrilateral Parallelogram Trapezoid Rectangle 4 sides, 4 angles Parallelogram Opposite sides are parallel and congruent Trapezoid Only 1 pair of parallel sides Rectangle A parallelogram with 4 right angles Rhombus A parallelogram with 4 congruent sides Square A rectangle with 4 congruent sides

Comparison of Quadrilateral Properties Parallelogram Rectangle Square Rhombus Isosceles Trapezoid Opposite sides ≈ # of pairs Opposite sides || Diagonals ≈ Yes /No Diagonals perpendicular Diagonals bisect each other Diagonals bisect opposite angles All angles 90°

Comparison of Quadrilateral Properties Parallelogram Rectangle Square Rhombus Isosceles Trapezoid Opposite sides ≈ 2 pairs 1 pair Opposite sides || Diagonals ≈ No Yes Diagonals perpendicular Diagonals bisect each other Diagonals bisect opposite angles All angles 90°

Venn Diagram of Quadrilaterals Parallelograms Trapezoids Rectangles Squares Rhombi