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Name: Chapter 6: Quadrilaterals

Lesson 6-1: Angles of Polygons Date:

A of a polygon is a segment that connects any two nonconsecutive

vertices.

Example:

Triangle Quadrilateral Pentagon Hexagon

Since the sum of the angle measures of a triangle is , we can make a table and look for a

pattern to find the sum of the angle measures for any convex polygon.

Polygon Number of

Sides

Number of

Triangles Sum of Interior Angle Measures

Triangle

Quadrilateral

Pentagon

Hexagon

n-gon

This leads us to:


Number

of Sides Polygon

3

4

5

6

7

8

9

10

11

12

n

Example 1:

A. Find the sum of the measures of the interior angles of a convex nonagon.

B. Find the measure of each interior angle of parallelogram 𝑅𝑆𝑇𝑈.

Use a protractor to measure each angle to

the nearest degree. Then find the sum of

these angles.


Example 2: A mall is designed so that five walkways meet at a food court

that is in the shape of a regular pentagon. Find the measure of one of the

interior angles of the pentagon.

Example 3: The measure of an interior angle of a regular polygon is 150.

Find the number of sides in the polygon.

Example 4:

A. Find the value of 𝑥 in the diagram.

B. Find the measure of each exterior angle of a regular decagon.


Lesson 6-2: Parallelograms Date:

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

Example

Example 1: In parallelogram 𝐴𝐵𝐶𝐷, suppose 𝑚∠𝐵 = 32, 𝐶𝐷 = 80, and 𝐵𝐶 = 15.

A. Find 𝐴𝐷

B. Find 𝑚∠𝐶

C. Find 𝑚∠𝐷


Example 2: 𝑊𝑋𝑌𝑍 is a parallelogram.

A. Find the value of 𝑟.

B. Find the value of 𝑠.

C. Find the value 𝑡.

Example 3: 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)?

x

y


Lesson 6-3: Tests for Parallelograms Date:

If a quadrilateral has each pair of opposite sides , then by definition, it is a parallelogram.

Other ways to determine if a quadrilateral is a parallelogram include:

Example 1: Determine whether each quadrilateral is a parallelogram. Justify your answer.

a. b. c.

d. e.

8 cm

8 cm

2 cm 2 cm


Example 2: MULTIPLE CHOICE The diagram shows a car jack used to raise a car from the ground. In the

diagram, 𝐴𝐷̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅ and 𝐴𝐵̅̅ ̅̅ ≅ 𝐷𝐶̅̅ ̅̅ . Based on this information, which statement will be true, regardless of

the height of the car jack.

A. ∠𝐴 ≅ ∠𝐵

B. ∠𝐴 ≅ ∠𝐶

C. 𝐴𝐵̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅

D. 𝑚∠𝐴 + 𝑚∠𝐶 = 180

Example 3: Find x and y so that the quadrilateral is a parallelogram.

You may need to use the distance, slope, and midpoint formulas to determine whether a quadrilateral in

the coordinate plane is a parallelogram.

Use the distance formula to show that the pairs of opposite sides are congruent.

Use the slope formula to show that the pairs of opposite sides have the same slope.

Use the midpoint formula to show that each diagonal has the same midpoint (they intersect at the

same point).

Example 4: Quadrilateral QRST has 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.

x

y


Lesson 6-4: Rectangles Date:

A rectangle is a parallelogram with four angles.

By definition, a rectangle has the following properties:

• All four angles are angles.

• Opposite sides are and .

• Opposite angles are .

• Consecutive angles are .

• Diagonals each other.

A rectangle is a parallelogram, but a parallelogram isn’t necessarily a rectangle!

Example 1: A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If

𝐽𝐾 = 12 feet, and 𝐿𝑁 = 6.5 feet, find 𝐾𝑀.

Example 2: Quadrilateral 𝑅𝑆𝑇𝑈 is a rectangle. If 𝑚∠𝑅𝑇𝑈 = 8𝑥 + 4 and 𝑚∠𝑆𝑈𝑅 = 3𝑥 – 2, find 𝑥.


Example 3: Some artists stretch their own canvas over wooden frames. This allows them to customize the

size of a canvas. In order to ensure that the frame is rectangular before stretching the canvas, an artist

measures the sides and the diagonals of the frame. If 𝐴𝐵 = 12 inches, 𝐵𝐶 = 35 inches, 𝐶𝐷 = 12

inches, 𝐷𝐴 = 35 inches, 𝐵𝐷 = 37 inches, and 𝐴𝐶 = 37 inches, explain how an artist can be sure that

the frame is rectangular.

Example 4: 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.

x

y


Lesson 6-5: Rhombi & Squares Date:

A is a parallelogram with all four sides congruent.

It has all the properties of a parallelogram in addition to:

Example 1:

A. The diagonals of rhombus 𝑊𝑋𝑌𝑍 intersect at 𝑉. If

𝑚∠𝑊𝑍𝑋 = 39.5, find 𝑚∠𝑍𝑌𝑋.

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


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

A parallelogram that is both a rectangle and a rhombus is also a square.

A square is a rhombus, but a rhombus isn’t necessarily a square.

A square is a rectangle, but a rectangle isn’t necessarily a square.

Example 2:

A. If LMNP is a parallelogram, 1 2, and 2 6, is LMNP is a rhombus?

B. If ABCD is a parallelogram and 𝐴𝐷̅̅ ̅̅ 𝐷𝐶̅̅ ̅̅ , is ADCD is a rhombus?


Example 3: Hector is measuring the boundary of a new garden. He wants the garden to be square. He has

set each of the corner stakes 6 feet apart. What does Hector need to know to make sure that the garden is

square?

Since a rhombus has four congruent sides, one diagonal separates the rhombus into two congruent isosceles

triangles. Both diagonals would separate the rhombus into four congruent right triangles.

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

If the diagonals of a parallelogram are perpendicular, then it is a

If the diagonals of a parallelogram are both congruent and perpendicular, then it is a ,

, and .

Example 4: 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.

x

y


Lesson 6-6: Trapezoids and Kites Date:

A is a quadrilateral with exactly one pair of parallel sides.

The parallel sides are called the . The nonparallel sides are called the .

The are formed by the base and one of the legs.

If the legs are congruent, then it is called an .

Example:

Example 1: Each side of the basket shown is an isosceles trapezoid.

A. If 𝑚∠𝐽𝑀𝐿 = 130, 𝐾𝑁 = 6.7 feet, and 𝐿𝑁 = 3.6 feet, find 𝑚∠𝑀𝐽𝐾.

B. If 𝑚∠𝐽𝑀𝐿 = 130, 𝐾𝑁 = 6.7 feet, and 𝐽𝐿 is 10.3 feet, find 𝑀𝑁.


Example 2: Quadrilateral ABCD has vertices A(5, 1), B(–3, –1), C(–2, 3), and D(2, 4). Show that ABCD is a

trapezoid and determine whether it is an isosceles trapezoid.

The of a trapezoid is the segment that connects the midpoints of the

legs of the trapezoid.

Example:

Example 3: In the figure, 𝑀𝑁̅̅ ̅̅ ̅ is the midsegment of trapezoid 𝐹𝐺𝐽𝐾. What is the value of 𝑥?

x

y


A is a quadrilateral with exactly two pairs of consecutive congruent sides. (unlike a

parallelogram, the opposite sides are not parallel or congruent)

Example:

Example 4:

A. If 𝑊𝑋𝑌𝑍 is a kite, find 𝑚∠𝑋𝑌𝑍.

B. If 𝑀𝑁𝑃𝑄 is a kite, find 𝑁𝑃.

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