Chapter 6

Quadrilaterals

Types of Polygons

Triangle – 3 sides Quadrilateral – 4 sides Pentagon – 5 sides Hexagon – 6 sides Heptagon – 7 sides Octagon – 8 sides Nonagon – 9 sides Decagon – 10 sides Dodecagon – 12 sides All other polygons = n-gon

Lesson 6.1 : Angles of Polygons

Interior Angle Sum Theorem The sum of the

measures of the interior angles of a polygon is found by S=180(n-2)

Ex: Hexagon

Exterior Angle Sum Theorem The sum of the

measures of the exterior angles of a polygon is 360 no matter how many sides.

Lesson 6.1 : Angles of Polygons

Find the measure of an interior and an exterior angle for each polygon.

24-gon

3x-gon

Find the measure of an exterior angle given the number of sides of a polygon

260 sides

Lesson 6.1: Angles of Polygons

The measure of an interior angle of a polygon is given. Find the number of sides.

175

168.75

A pentagon has angles (4x+5), (5x-5), (6x+10), (4x+10), and 7x. Find x.

180-175=5360/5= 72

A. Find the value of x in the diagram.

Lesson 6.2: Parallelograms

Properties of Parallelograms

Opposite sides of a parallelogram are congruent

Opposite angles in a parallelogram are congruent

Consecutive angles in a parallelogram are supplementary

If a parallelogram has 1 right angle, it has 4 right angles.

The diagonals of a parallelogram split it into 2 congruent triangles

The diagonals of a parallelogram bisect each other

A parallelogram is a quadrilateral with both pairs of opposite sides parallel

____?

?

?

A. ABCD is a parallelogram. Find AB.

B. ABCD is a parallelogram. Find mC.

C. ABCD is a parallelogram. Find mD.

A. If WXYZ is a parallelogram, find the value of r, s and t.

What are the coordinates of the intersection of the diagonals of parallelogram MNPR, with vertices M(–3, 0), N(–1, 3), P(5, 4), and R(3, 1)?

Lesson 6.3 : Tests for Parallelograms If…

Both pairs of opposite sides are parallelBoth pairs of opposite sides are congruentBoth pairs of opposite angles are congruentThe diagonals bisect each otherOne pair of opposite sides is congruent and

parallel Then the quadrilateral is a parallelogram

Determine whether the quadrilateral is a parallelogram. Justify your answer.

Which method would prove the quadrilateral is a parallelogram?

Determine whether the quadrilateral is a parallelogram.

Determine whether the quadrilateral is a parallelogram.

Find x and y so that the quadrilateral is a parallelogram.

COORDINATE GEOMETRY Graph quadrilateral QRST with vertices Q(–1, 3), R(3, 1), S(2, –3), and T(–2, –1). Determine whether the quadrilateral is a parallelogram. Justify your answer by using the Slope Formula.

Given quadrilateral EFGH with vertices E(–2, 2), F(2, 0), G(1, –5), and H(–3, –2). Determine whether the quadrilateral is a parallelogram. (The graph does not determine for you)

6.4-6.6 Foldable

Fold the construction paper in half both length and width wise

Unfold the paper and hold width wise Fold the edges in to meet at the center

crease Cut the creases on the tabs to make 4

flaps

Lesson 6.4 : Rectangles

Characteristics of a rectangle: Both sets of opp. Sides are

congruent and parallel Both sets opp. angles are

congruent Diagonals bisect each other Diagonals split it into 2

congruent triangles Consecutive angles are

supplementary If one angle is a right angle

then all 4 are right angles

In a rectangle the diagonals are congruent.

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

Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ.

Quadrilateral RSTU is a rectangle. If mRTU = 8x + 4 and mSUR = 3x – 2, find x.

Quadrilateral EFGH is a rectangle. If mFGE = 6x – 5 and mHFE = 4x – 5, find x.

Quadrilateral JKLM has vertices J(–2, 3), K(1, 4), L(3, –2), and M(0, –3). Determine whether JKLM is a rectangle using the Distance Formula.

6.5: Squares (special type of parallelogram)

A quadrilateral with 4 congruent sides

Characteristics of a square: Both sets of opp. sides are

congruent and parallel Both sets of opp. angles are

congruent Diagonals bisect each other Diagonals split it into 2 congruent

triangles Consecutive angles are

supplementary If an angle is a right angle then all

4 angles are right angles Diagonals bisect the pairs of

opposite angles Diagonals are perpendicular

A square is a rhombus and a rectangle.

Lesson 6.5 : Rhombi (special type of parallelogram)

A quadrilateral with 4 congruent sides

Characteristics of a rhombus: Both sets of opp. sides are

congruent and parallel Both sets of opp. angles are

congruent Diagonals bisect each other Diagonals split it into 2 congruent

triangles Consecutive angles are

supplementary If an angle is a right angle then all

4 angles are right angles

In a rhombus: Diagonals are perpendicular Diagonals bisect the pairs of

opposite angles

A. The diagonals of rhombus WXYZ intersect at V.If mWZX = 39.5, find mZYX.

B. The diagonals of rhombus WXYZ intersect at V. If WX = 8x – 5 and WZ = 6x + 3, find x.

A. ABCD is a rhombus. Find mCDB if mABC = 126.

B. ABCD is a rhombus. If BC = 4x – 5 and CD = 2x + 7, find x.

QRST is a square. Find n if mTQR = 8n + 8.

QRST is a square. Find QU if QS = 16t – 14 and QU = 6t + 11.

Determine whether parallelogram ABCD is a rhombus, a rectangle, or a square for A(–2, –1), B(–1, 3), C(3, 2), and D(2, –2). List all that apply. Explain.

Kite

Two sets of consecutive sides are congruent

Diagonals are perpendicular

6.6: Trapezoids

A quadrilateral with exactly 1 pair of opposite parallel sides (bases), 2 pairs of base angles, and 1 pair of non-parallel sides (legs)

Isosceles Trapezoid: A trapezoid with congruent legs

and congruent base angles

Diagonals of an isosceles trapezoid are congruent

Median (of a trapezoid): The segment that connects

the midpoints of the legs

The median is parallel to the bases

base

base

leg leg

Base angle Base angle

Median = ½ (base + base)

A B

CD

AC = BD

A. Each side of the basket shown is an isosceles trapezoid. If mJML = 130, KN = 6.7 feet, and LN = 3.6 feet, find mMJK.

B. Each side of the basket shown is an isosceles trapezoid. If mJML = 130, KN = 6.7 feet, and JL is 10.3 feet, find MN.

In the figure, MN is the midsegment of trapezoid FGJK. What is the value of x.

WXYZ is an isosceles trapezoid with medianFind XY if JK = 18 and WZ = 25.

A. If WXYZ is a kite, find mXYZ.

A. If WXYZ is a kite, find mXYZ.