Unit 7 Quadrilaterals - Lake · PDF fileTheorem 6-10 If both pairs of ... Unit 7 - Quadrilaterals. Vocabulary A parallelogram is a rhombus if either: ... Kite A quadrilateral in which

Unit 7

Quadrilaterals

2014-15 Geometry

Unit Outline

6-1 The Polygon Angle-Sum Theorems

6-2 Properties of Parallelograms

6-3 Proving that a Quadrilateral is a Parallelogram

6-4 Properties of Rhombuses, Rectangles and Squares

6-5 Conditions for Rhombuses, Rectangles and Squares

6-6 Trapezoids and Kites

6-7 Polygons in the Coordinate Plane

Unit Standards

MAFS.912.G-CO.3.11 (DOK 3)

Prove theorems about parallelograms; use theorems about parallelograms to solve

problems. Theorems include: opposite sides are congruent, opposite angles are

congruent, the diagonals of a parallelogram bisect each other, and conversely,

rectangles are parallelograms with congruent diagonals.

MAFS.912.G-GPE.2.4 (DOK 2)

Use coordinates to prove simple geometric theorems algebraically. For example, prove or

disprove that a figure defined by four given points in the coordinate plane is a rectangle;

prove or disprove that the point (1, √3) lies on the circle centered at the origin and

containing the point (0, 2).

6-1 The Polygon Angle-Sum

Theorems

Unit 7 - Quadrilaterals

Vocabulary

Interior Angles of a Polygon

The angles on the inside of a polygon are called interior

angles.

Exterior Angles of a Polygon

The exterior angles of a polygon are those formed by

extending sides. There is one exterior angle at each vertex.

Vocabulary

Equilateral Polygon

Polygon with all sides congruent

Equiangular Polygon

Polygon with all angles congruent

Regular Polygon

A Polygon that is both equilateral and equiangular.

Polygon Angle-Sum Theorem

The sum of the measures of the angles of an n-gon is (n 2)180.

You can write this as a formula. This formula works for regular and irregular polygons.

Sum of angle measures = (n 2)180


What is the sum of the measures of the angles in a hexagon?

Solution: There are six sides, so n = 6.


= (6 − 2)180 Substitute 6 for n

= 4(180) Subtract.

= 720 Multiply.

The sum of the measures of the angles in a hexagon is 720

Note: You can use the formula to find the measure of one interior angle of a regular polygon

if you know the number of sides.


What is the measure of each angle in a regular pentagon?

Solution: A pentagon has 5 sides, so n = 5.


= (5 − 2)180 Substitute 5 for n

= 3(180) Subtract.

= 540 Multiply.

Divide by the number of angles:

Measure of each angle = 540 5

= 108 Divide.

Each angle of a regular pentagon measures 108.

Your Turn!

Find the sum of the interior angles of each polygon.

Quadrilateral

Solution: Sum of interior angles = 360

Decagon

Solution: Sum of interior angles =1440

Find the measure of an interior angle of each regular polygon.

Decagon

Solution: Interior Angle = 144

32-gon

Solution: Interior Angle = 168.75

Polygon Exterior Angle-Sum Theorem

The sum of the measures of the exterior angles of a

polygon is 360.

Example:

A pentagon has five exterior angles. The sum of

the measures of the exterior angles is always 360,

so each exterior angle of a regular pentagon

measures 72.

Your Turn!

Find the measure of an exterior angle for each regular polygon.

Octagon

Solution: Exterior Angle Measure = 45

Hexagon

Solution: Exterior Angle Measure = 60

Questions?

Instructor Email: [email protected]

mailto:[email protected]

6-2 Properties of

Parallelograms


Vocabulary

Parallelograms

A quadrilateral with both pairs of opposite sides parallel.

The opposite sides are congruent. (Theorem 6-3)

The consecutive angles are supplementary. (Th. 6-4)

The opposite angles are congruent. (Th. 6-5)

The diagonals bisect each other. (Th. 6-6)

Solving Parallelograms

Find the value of x.

Solution: Because the consecutive angles are supplementary,

x + 60 = 180

x = 120


Find the value of x.

Solution: Because opposite sides are congruent,

x + 7 = 15

x = 8


Find the value of x and y.

Solution: Because the diagonals bisect each other, y = 3x and

4x = y + 3.

4x = y + 3

4x = 3x 3 Substitute for y.

x = 3 Subtraction Property of =

y = 3x Given

y = 3(3) Substitute for x.

y = 9 Simplify.

Your Turn!

Find the value of x in each parallelogram.

Solution: x = 8

Solution: x = 3, y = 6

Solution: x = 21

Solution: x = 50

Questions?



6-3 Proving that a Quadrilateral

is a Parallelogram


Vocabulary

Definition of a Parallelogram

If both pairs of opposite sides are parallel, then the

quadrilateral is a parallelogram.

Theorem 6-8

If both pairs of opposite sides are congruent, then the


Theorem 6-10

If both pairs of opposite angles are congruent, then

the quadrilateral is a parallelogram.

Vocabulary

Theorem 6-11

If the diagonals bisect each other, then the


Theorem 6-12

If one pair of sides is both congruent and parallel, then

the quadrilateral is a parallelogram.


For what value of x and y must figure ABCD be a parallelogram?

Solution: In a parallelogram, the two pairs of opposite angles are congruent. So, in ABCD, you know that x 2y and 5y + 54 4x. You can use these two expressions to solve for x and y.

Step 1: Solve for y. 5y + 54 4x

5y + 54 4(2y) Substitute 2y for x.

5y + 54 8y Simplify.

54 3y Subtract 5y from each side.

18 y Divide each side by 3.

Step 2: Solve for x. x 2y Opposite angles of a parallelogram are congruent.

x 2(18) Substitute 18 for y.

x 36 Simplify.

For ABCD to be a parallelogram, x must be 36 and y must be 18.

Your Turn!

For what value of x must the quadrilateral be a parallelogram?

Solution: x = 18 Solution: x = 3

Questions?



6-4 Properties of Rhombuses,

Rectangles and Squares


Vocabulary

Rhombus

A parallelogram with four congruent sides.

Special Features:

The diagonals are perpendicular.

The diagonals bisect a pair of opposite angles.

Rectangle

A parallelogram with four congruent angles. These

angles are all right angles.

Special Features:

The diagonals are congruent.

Vocabulary

Square

A parallelogram with four congruent sides and four

congruent angles. A square is both a rectangle and

a rhombus. A square is the only type of rectangle

that can also be a rhombus.

Special Features:

The diagonals are perpendicular.

The diagonals bisect a pair of opposite angles (forming

two 45 angles at each vertex).

The diagonals are congruent.

Finding Angle Measures

Determine the measure of the numbered angles in rhombus DEFG.

Solution: 1 is part of a bisected angle. mDFG = 48, so m1 = 48.

Consecutive angles of a parallelogram are supplementary.

mEFG = 48 + 48 = 96, so mDGF = 180 96 = 84.

The diagonals bisect the vertex angle, so m2 = 84 2 = 42.

Finding Diagonal Length

In rectangle RSBF, SF = 2x + 15 and RB = 5x – 12. What is the length of the diagonal?

Solution: The length of the diagonals of a rectangle are congruent, so SF = RB.

Step 1. Solve for x. SF = 2x + 15 and RB = 5x – 12

SF = RB

2x + 15 = 5x – 12 Substitute values of SF and RB.

15 = 3x – 12

27 = 3x

x = 9 Simplify.

Step 2. Solve for the length of a diagonal.

SF = 2(9) + 15 Substitute the value of x.

SF = 18 + 15

SF = 33 Simplify.

Your Turn!

Determine the measure of the numbered angle.

Solution: 1 = 78 and 2 = 90

TUVW is a rectangle. Find the value of x and the length of each diagonal.

TV = 10x – 4 and UW = 3x + 24

Solution: x = 4; TV = 36; UW = 36

Questions?



6-5 Conditions for Rhombuses,

Rectangles and Squares


Vocabulary

A parallelogram is a rhombus if either:

The diagonals of the parallelogram are perpendicular.

(Theorem 6-16)

A diagonal of the parallelogram bisects a pair of opposite

angles. (Th. 6-17)

A parallelogram is a rectangle if the diagonals of the

parallelogram are congruent.

Using Properties of Special Parallelograms

For what value of x is DEFG a rhombus?

Solution: In a rhombus, diagonals bisect opposite angles.

So, m DGDF = m DEDF.

(4x + 10) = (5x + 6) Set angle measures equal to each other.

10 = x + 6 Subtract 4x from each side.

4 = x Subtract 6 from each side.

Your Turn!

SQ = 14. For what value of x is PQRS a

rectangle? Solve for PT. Solve for PR.

Solution: x = 6

For what value of x is RSTU a rhombus?

What is m SRT? What is m URS?

Solution: x = 48

Questions?



6-6 Trapezoids and Kites


Vocabulary

Trapezoid

A quadrilateral with exactly one pair of parallel

sides. The two parallel sides are called bases.

The two nonparallel sides are called legs.

Midsegment

Parallel to the bases, the length of the

midsegment is half the sum of the lengths of the

bases.

Vocabulary

Isosceles Trapezoid

A trapezoid in which the legs are congruent.

An isosceles trapezoid has some special

properties:

Each pair of base angles is congruent.

The diagonals are congruent

Finding Angle Measures in Trapezoids

CDEF is an isosceles trapezoid and m C = 65. What are m D,

m E, and m F?

Solution:

m C + m D = 180 Same-side interior angles are

supplementary

65 + m D = 180

m D = 115 Simplify

m C = m F = 65 Base Angles are Congruent

m D = m E = 115

C F

D E

Vocabulary

Kite

A quadrilateral in which two pairs of

consecutive sides are congruent and no

opposite sides are congruent.

In a kite, the diagonals are perpendicular.

(Theorem 6-22)

Notice that the sides of a kite are the

hypotenuses of four right triangles whose legs

are formed by the diagonals.

Proving Congruent Triangles in a Kite

Statement Reasoning

1) 𝐹𝐺 ≅ 𝐹𝐽 Given

2) mFKG = mGKH = mHKJ = mJKF = 90 Th. 6-22

3) 𝐹𝐾 ≅ 𝐹𝐾 Reflexive Property

4) ∆𝐹𝐾𝐺 ≅ ∆𝐹𝐾𝐽 HL Theorem

5) 𝐽𝐾 ≅ 𝐾𝐺 CPCTC

6) 𝐾𝐻 ≅ 𝐾𝐻 Reflexive Property

7) ∆𝐽𝐾𝐻 ≅ ∆𝐺𝐾𝐻 SAS Postulate

8) 𝐽𝐻 ≅ 𝐺𝐻 Given

9) 𝐹𝐻 ≅ 𝐹𝐻 Reflexive Property

10) ∆𝐹𝐽𝐻 ≅ ∆𝐹𝐺𝐻 SSS Postulate

H

Your Turn!

In kite FGHJ in the problem, m JFK = 38 and m KGH = 63. Find the following angle and

side measures.

m FKJ

Solution: 90

m GHK

Solution: 27

If FG = 4.25, what is JF?

Solution: 4.25

If HG is 5, what is JH?

Solution: 5

H

Questions?



6-7 Polygons in the Coordinate

Plane


Key Concepts

Distance Formula

𝑑 = 𝑥2 − 𝑥12 + 𝑦2 − 𝑦1

2

Midpoint Formula

𝑥1 + 𝑥22

,𝑦1 + 𝑦2

2

Slope Formula

𝑚 =𝑦2 − 𝑦1𝑥2 − 𝑥1

Classifying a Triangle

Is ∆𝐴𝐵𝐶 scalene, isosceles or right?

Solution: Find the lengths of the sides using the Distance

Formula.

𝑑 = 𝑥2 − 𝑥12 + 𝑦2 − 𝑦1

2

𝐵𝐴 = 6 − 0 2 + 4 − 3 2 = 6 2 + 1 2 = 36 + 1 = 37

𝐵𝐶 = 6 − 4 2 + 4 − 0 2 = 2 2 + 4 2 = 4 + 16 = 20

𝐶𝐴 = 4 − 0 2 + 0 − 3 2 = 4 2 + −3 2 = 16 + 9 = 25 = 5

The sides are all different lengths. So, ABC is scalene.

Classifying a Parallelogram

Is quadrilateral GHIJ a parallelogram?

Solution: Find the slopes of the opposite sides.

𝑚𝐺𝐻 =4 − 3

0 − (−3)=

1

3

𝑚𝐽𝐼 =−1 − (−2)

4 − 1=

1

3

𝑚𝐻𝐼 =−1 − 4

4 − 0=

−5

4

𝑚𝐺𝐽 =−2 − 3

1 − (−3)=

−5

4

So, 𝐽𝐼 ∥ 𝐺𝐻 and 𝐻𝐼 ∥ 𝐺𝐽. Therefore, GHIJ is a parallelogram.

Your Turn!

∆𝐽𝐾𝐿 has vertices at 𝐽 −2,4 , 𝐾 1, 6 𝑎𝑛𝑑 𝐿 4,4 . Determine whether ∆𝐽𝐾𝐿 is scalene, isosceles

or equilateral. Explain.

Solution: The triangle is isosceles because the measure of two sides of the triangle are the

same.

Trapezoid ABCD has vertices at 𝐴 2,1 , 𝐵 12,1 , 𝐶 9,4 𝑎𝑛𝑑 𝐷(5,4). Which formula would help

you find out if this trapezoid is isosceles? Is this an isosceles trapezoid? Explain.

Solution: The slope formula can be used to determine the slopes of each base. The distance

formula can be used to determine the length of the legs.

Yes, trapezoid ABCD is an isosceles trapezoid.

(Determined from calculating the slopes of AB and CD as well as the lengths of the legs AD an BC.)

Questions?



Unit 7 Quadrilaterals - Lake · PDF fileTheorem 6-10 If both pairs of ... Unit 7 - Quadrilaterals. Vocabulary A parallelogram is a rhombus if either: ... Kite A quadrilateral in which

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