Unit 6

Special Quadrilaterals

Polygon Angle Sum

• The sum of the measures of the interior angles of an n-gon is (n-2)180, where n represents the number of sides

• Example – – What is the sum of the interior angles of a 13

sided figure• (13-2) 180• (11)(180)• 1980°

You Try

• Sum of the interior angles of a Heptagon– 900°

• Sum of the interior angles of a 17-gon?– 2700°

• Sum of the interior angles of a Quadrilateral? – 360°

Types of Polygons

• Equilateral Polygon – All sides are congruent

• Equiangular Polygon – All angles are congruent

• Regular polygon – All angles and all sides are congruent

Finding One Interior Angle of a Regular Polygon

• The measure of each interior angle of a regular n-gon is [(n-2)180] ⁄ n, where n represents the number of sides

• Example:– Find the measure of one interior angle of a regular

hexagon• [(6-2)180]/6• [(4)180]/6• 720/6• 120°

You Try• Find the measure of one interior angle of a regular 16 –

gon.• 157.5°

• Find the measure of one interior angle of a regular nonagon.• 140°

• Find the measure of one interior angle of a regular 11 – gon.• 147.2727272727

Polygon Exterior Angle Theorem

• The sum of the measures of the exterior angles of a polygon, with one angle at each vertex, is always 360°.

• Example:– What is one exterior angle of a regular octagon? • 360/8 • 45°

Parallelograms

Parallelogram

• A parallelogram is a special quadrilateral with both pair of opposite sides parallel.

Properties of a parallelogram

• Both pairs of opposite sides are congruent

• Both pairs of opposite angles are congruent

• Consecutive angles are supplementary

• Diagonals Bisect Each other

How to prove a quadrilateral is a parallelogram

• There are 6 ways to prove that a quadrilateral is a parallelogram

• By the Definition of a parallelogram that

states if both pairs of opposite sides are parallel then a quadrilateral is a parallelogram

• If both pairs of opposite sides are congruent then quadrilateral is a parallelogram

• If both pairs of opposite angles are congruent then quadrilateral is a parallelogram

• If consecutive angles are supplementary then quadrilateral is a parallelogram

• If diagonals bisect each other then quadrilateral is a parallelogram

• If one pair of opposite sides are congruent and parallel then quadrilateral is a parallelogram

Rectangle, Rhombus, Square

Rectangle

• A rectangle is a parallelogram with four right angles.

• If a parallelogram is a rectangle then all parallelogram properties apply.

Properties of Rectangles

• If a parallelogram is a rectangle then the diagonals are congruent.

Rhombus

• A Rhombus is a parallelogram with four congruent sides

• If a parallelogram is a rhombus then all parallelogram properties apply.

Properties of a Rhombus

• If a parallelogram is a rhombus, then its diagonals are perpendicular

• If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles

Square

• A square is a parallelogram with four congruent sides and four right angles.

• A square is a parallelogram, rectangle, and rhombus so all the properties apply!

Proving a Quadrilateral is a Rectangle

• To prove a quadrilateral is a rectangle you must first prove it is a parallelogram.

• Then:– If all angles are right angles then parallelogram is

a rectangle.

– If the diagonals of the parallelogram are congruent then it is a rectangle.

Proving a Quadrilateral is a Rhombus

• To prove a quadrilateral is a rhombus you must first prove it is a parallelogram.

• Then:– If all sides are congruent then parallelogram is a rhombus.

– If the diagonals of the parallelogram are perpendicular then it is a rhombus.

– If the diagonals of a parallelogram bisect opposite angles then it is a rhombus.

Proving a Quadrilateral is a Square

• To prove a quadrilateral is a square you must first prove it is a parallelogram.

• Then: – Prove that parallelogram is a rhombus.

• Then:– Prove that parallelogram is a rectangle.

Then quadrilateral is a Square!

Trapezoid / Kite

Trapezoid

• A trapezoid is a quadrilateral with one pair of parallel sides, know as the bases. The non-parallel sides are know as the legs.

Isosceles Trapezoid

• A trapezoid with legs that are congruent.

Properties of Isosceles Trapezoid

• If a quadrilateral is an isosceles trapezoid then each pair of base angles are congruent

• If a quadrilateral is an isosceles trapezoid then diagonals are congruent

Trapezoid Midsegment Theorem

• The midsegment of a trapezoid is parallel to the bases and is half the sum of the lengths of the bases.

Kite

• A kite is a quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent.

• If a quadrilateral is a kite then the diagonals are perpendicular.