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Lesson 8-1 Angles of Polygons

Lesson 8-2 Parallelograms

Lesson 8-3 Tests for Parallelograms

Lesson 8-4 Rectangles

Lesson 8-5 Rhombi and Squares

Lesson 8-6 Trapezoids

Lesson 8-7 Coordinate Proof with Quadrilaterals

Example 1 Interior Angles of Regular Polygons

Example 2 Sides of a Polygon

Example 3 Interior Angles

Example 4 Exterior Angles

ARCHITECTURE A mall is designed so that five walkways meet at a food court that is in the shape of a regular pentagon. Find the sum of measures of the interior angles of the pentagon.

Since a pentagon is a convex polygon, we can use the Angle Sum Theorem.

Interior Angle Sum Theorem

Simplify.

Answer: The sum of the measures of the angles is 540.

A decorative window is designed to have the shape of a regular octagon. Find the sum of the measures of the interior angles of the octagon.

Answer: 1080

The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon.Use the Interior Angle Sum Theorem to write an equation to solve for n, the number of sides.

Answer: The polygon has 8 sides.

Interior Angle Sum Theorem

Distributive Property

Subtract 135n from each side.

Add 360 to each side.

Divide each side by 45.

The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon.

Answer: The polygon has 10 sides.

Find the measure of each interior angle.

Since the sum of the measures of the interior angles is Write an equation to express the sum of the measures of the interior angles of the polygon.

Sum of measures of angles

Substitution

Combine like terms.

Subtract 8 from each side.

Divide each side by 32.

Use the value of x to find the measure of each angle.

Answer:

Find the measure of each interior angle.

Answer:

Find the measures of an exterior angle and an interior angle of convex regular nonagon ABCDEFGHJ.

At each vertex, extend a side to form one exterior angle.

The sum of the measures of the exterior angles is 360. A convex regular nonagon has 9 congruent exterior angles.

Divide each side by 9.

Answer: The measure of each exterior angle is 40. Since each exterior angle and its corresponding interior angle form a linear pair, the measure of the interior angle is 180 – 40 or 140.

Find the measures of an exterior angle and an interior angle of convex regular hexagon ABCDEF.

Answer: 60; 120

Example 1 Proof of Theorem 8.4

Example 2 Properties of Parallelograms

Example 3 Diagonals of a Parallelogram

Prove that if a parallelogram has two consecutive sides congruent, it has four sides congruent.

Given:

Prove:

1. 1. Given

Proof:

ReasonsStatements

4. Transitive Property4.

2. Given2.

3. Opposite sides of a parallelogram are .

3.

Given:

Prove:

Prove that if and are the diagonals of , and

Proof:

ReasonsStatements

1. Given1.

4. Angle-Side-Angle4.

2. Opposite sides of a parallelogram are congruent.

2.

3. If 2 lines are cut by a transversal, alternate interior s are .

3.

If lines are cut by a transversal, alt. int.

Definition of congruent angles

Substitution

RSTU is a parallelogram. Find and y.

Angle Addition Theorem

Substitution

Subtract 58 from each side.

Substitution

Divide each side by 3.

Definition of congruent segments

Answer:

ABCD is a parallelogram.

Answer:

Read the Test ItemSince the diagonals of a parallelogram bisect each other, the intersection point is the midpoint of

A B C D

MULTIPLE-CHOICE TEST ITEM 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)?

Solve the Test Item

Find the midpoint of

The coordinates of the intersection of the diagonals of parallelogram MNPR are (1, 2).

Answer: C

Midpoint Formula

Answer: B

A B C D

MULTIPLE-CHOICE TEST ITEM What are the coordinates of the intersection of the diagonals of parallelogram LMNO, with verticesL(0, –3), M(–2, 1), N(1, 5), O(3, 1)?

Example 1 Write a Proof

Example 2 Properties of Parallelograms

Example 3 Properties of Parallelograms

Example 4 Find Measures

Example 5 Use Slope and Distance

Write a paragraph proof of the statement: If a diagonal of a quadrilateral divides the quadrilateral into two congruent triangles, then the quadrilateral is a parallelogram.

Prove: ABCD is a parallelogram.

Given:

Proof: CPCTC. By Theorem 8.9, if both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram. Therefore, ABCD is a parallelogram.

Write a paragraph proof of the statement: If two diagonals of a quadrilateral divide the quadrilateral into four triangles where opposite triangles are congruent, then the quadrilateral is a parallelogram.

Prove: WXYZ is a parallelogram.

Given:

Proof: by CPCTC. By Theorem 8.9, if both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram. Therefore, WXYZ is a parallelogram.

Some of the shapes in this Bavarian crest appear to be parallelograms. Describe the information needed to determine whether the shapes are parallelograms.

Answer: If both pairs of opposite sides are the same length or if one pair of opposite sides is a congruent and parallel, the quadrilateral is a parallelogram. If both pairs of opposite angles are congruent or if the diagonals bisect each other, the quadrilateral is a parallelogram.

The shapes in the vest pictured here appear to be parallelograms. Describe the information needed to determine whether the shapes are parallelograms.

Answer: If both pairs of opposite sides are the same length or if one pair of opposite sides is congruent and parallel, the quadrilateral is a parallelogram. If both pairs of opposite angles are congruent or if the diagonals bisect each other, the quadrilateral is a parallelogram.

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

Answer: Each pair of opposite sides have the same measure. Therefore, they are congruent. If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.

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

Answer: One pair of opposite sides is parallel and has the same measure, which means these sides are congruent. If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram.

Find x so that the quadrilateral is a parallelogram.

Opposite sides of a parallelogram are congruent.

A B

CD

Substitution

Distributive Property

Add 1 to each side.

Answer: When x is 7, ABCD is a parallelogram.

Subtract 3x from each side.

Find y so that the quadrilateral is a parallelogram.

Opposite angles of a parallelogram are congruent.

F

D E

G

Subtract 6y from each side.

Substitution

Subtract 28 from each side.

Divide each side by –1.

Answer: DEFG is a parallelogram when y is 14.

Find m and n so that each quadrilateral is a parallelogram.

Answer: Answer:

a. b.

COORDINATE GEOMETRY Determine whether the figure with vertices A(–3, 0), B(–1, 3), C(3, 2), and D(1, –1) is a parallelogram. Use the Slope Formula.

If the opposite sides of a quadrilateral are parallel, then it is a parallelogram.

Answer: Since opposite sides have the same slope, Therefore, ABCD is a parallelogram by definition.

COORDINATE GEOMETRY Determine whether the figure with vertices P(–3, –1), Q(–1, 3), R(3, 1), and S(1, –3) is a parallelogram. Use the Distance and Slope Formulas.

First use the Distance Formula to determine whether the opposite sides are congruent.

Answer: Since one pair of opposite sides is congruent and parallel, PQRS is a parallelogram.

Next, use the Slope Formula to determine whether

and have the same slope, so they are parallel.

Determine whether the figure with the given vertices is a parallelogram. Use the method indicated.

Slope Formulaa. A(–1, –2), B(–3, 1), C(1, 2), D(3, –1);

Answer: The slopes of and the

slopes of Therefore,

Since opposite sides are parallel, ABCD is a

parallelogram.

Distance and Slope Formulas

b. L(–6, –1), M(–1, 2), N(4, 1), O(–1, –2);

Determine whether the figure with the given vertices is a parallelogram. Use the method indicated.

Answer: Since the

slopes of

Since one pair of opposite sides is congruent

and parallel, LMNO is a parallelogram.

Example 1 Diagonals of a Rectangle

Example 2 Angles of a Rectangle

Example 3 Diagonals of a Parallelogram

Example 4 Rectangle on a Coordinate Plane

Quadrilateral RSTU is a rectangle. If and find x.

The diagonals of a rectangle are congruent,

Definition of congruent segments

Substitution

Subtract 6x from each side.

Add 4 to each side.

Answer: 8

Answer: 5

Quadrilateral EFGH is a rectangle. If and find x.

Quadrilateral LMNP is a rectangle. Find x.

Angle Addition Theorem

Answer: 10

Substitution

Simplify.

Subtract 10 from each side.

Divide each side by 8.

Quadrilateral LMNP is a rectangle. Find y.

Since a rectangle is a parallelogram, opposite sides are parallel. So, alternate interior angles are congruent.

Alternate Interior Angles Theorem

Divide each side by 6.

Substitution

Subtract 2 from each side.

Simplify.

Answer: 5

Quadrilateral EFGH is a rectangle.

a. Find x. b. Find y.

Answer: 11 Answer: 7

Kyle is building a barn for his horse. He measures the diagonals of the door opening to make sure that they bisect each other and they are congruent. How does he know that the corners are angles?

We know that A parallelogram with congruent diagonals is a rectangle. Therefore, the corners are angles.

Answer:

Max is building a swimming pool in his backyard. He measures the length and width of the pool so that opposite sides are parallel. He also measures the diagonals of the pool to make sure that they are congruent. How does he know that the measure of each corner is 90?

Answer: Since opposite sides are parallel, we know thatRSTU is a parallelogram. We know that . A parallelogram with congruent diagonals is a rectangle. Therefore, the corners are

Quadrilateral ABCD has vertices A(–2, 1), B(4, 3), C(5, 0), and D(–1, –2). Determine whether ABCD is a rectangle using the Slope Formula.

Method 1: Use the Slope Formula, to see if

consecutive sides are perpendicular.

Answer: The perpendicular segments create four right angles. Therefore, by definition ABCD is a rectangle.

quadrilateral ABCD is a parallelogram. The product of the slopes of consecutive sides is –1. This means that

Method 2: Use the Distance Formula,

to determine whether opposite sides are congruent.

Since each pair of opposite sides of the quadrilateral have the same measure, they are congruent. Quadrilateral ABCD is a parallelogram.

The length of each diagonal is

Answer: Since the diagonals are congruent, ABCD is a rectangle.

Find the length of the diagonals.

Quadrilateral WXYZ has vertices W(–2, 1), X(–1, 3), Y(3, 1), and Z(2, –1). Determine whether WXYZ is a rectangle using the Distance Formula.

Answer: we can conclude that opposite sides of the quadrilateral are congruent. Therefore, WXYZ is a parallelogram. Diagonals WY and XZ each have a length of 5. Since the diagonals are congruent, WXYZ is a rectangle by Theorem 8.14.

Example 1 Proof of Theorem 8.15

Example 2 Measures of a Rhombus

Example 3 Squares

Example 4 Diagonals of a Square

Prove:

D

Given: BCDE is a rhombus, and

Proof: Because opposite angles of a rhombus are congruent and the diagonals of a rhombus bisect each other,

by the Reflexive Property and it is given that Therefore, by SAS.

By substitution,

Given: ACDF is a rhombus;

Prove:

Proof: Since ACDF is a rhombus, diagonals bisect each other and are perpendicular to each other. Therefore, are both right angles. By definition of right angles, which means that by definition of congruent angles. It is given that so since alternate interior angles are congruent when parallel lines are cut by a transversal. by ASA.

Use rhombus LMNP to find the value of y if

N

The diagonals of a rhombus are

perpendicular.Substitution

Add 54 to each side.

Take the square root of each side.

Answer: The value of y can be 12 or –12.

N

Use rhombus LMNP to find if

Opposite angles are congruent.

Substitution

The diagonals of a rhombus bisect the angles.

Answer:

Use rhombus ABCD and the given information to find the value of each variable.

Answer: 8 or –8

Answer:

a.

b.

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.

Explore Plot the vertices on a coordinate plane.

Plan If the diagonals are perpendicular, then ABCD is either a rhombus or a square. The diagonals of a rectangle are congruent. If the diagonals are congruent and perpendicular, then ABCD is a square.

Solve Use the Distance Formula to compare the lengths of the diagonals.

Use slope to determine whether the diagonals are perpendicular.

Since the slope of is the negative reciprocal of the slope of the diagonals are perpendicular. The lengths of and are the same so the diagonals are congruent. ABCD is a rhombus, a rectangle, and a square.

Examine The diagonals are congruent and perpendicular so ABCD must be a square. You can verify that ABCD is a rhombus by finding AB, BC, CD, AD. Then see if two consecutive segments are perpendicular.

Answer: ABCD is a rhombus, a rectangle, and a square.

Determine whether parallelogram EFGH is a rhombus, a rectangle, or a square for E(0, –2), F(–3, 0), G(–1, 3), and H(2, 1). List all that apply. Explain.

Answer: and slope of slope of Since the slope of is the negative reciprocal of the slope of , the diagonals are perpendicular. The lengths ofand are the same.

Let ABCD be the square formed by the legs of the table. Since a square is a parallelogram, the diagonals bisect each other. Since the umbrella stand is placed so that its hole lines up with the hole in the table, the center of the umbrella pole is at point E, the point where the diagonals intersect. Use the Pythagorean Theorem to find the length of a diagonal.

A square table has four legs that are 2 feet apart. The table is placed over an umbrella stand so that the hole in the center of the table lines up with the hole in the stand. How far away from a leg is the center of the hole?

The distance from the center of the pole to a leg is equal to the length of

Answer: The center of the pole is about 1.4 feet from a leg of a table.

Kayla has a garden whose length and width are each 25 feet. If she places a fountain exactly in the center of the garden, how far is the center of the fountain from one of the corners of the garden?

Answer: about 17.7 feet

Example 1 Proof of Theorem 8.19

Example 2 Identify Isosceles Trapezoids

Example 3 Identify Trapezoids

Example 4 Median of a Trapezoid

Write a flow proof.

Given: KLMN is an isosceles trapezoid.

Prove:

Proof:

Write a flow proof.

Given: ABCD is an isosceles trapezoid.

Prove:

Proof:

The top of this work station appears to be two adjacent trapezoids. Determine if they are isosceles trapezoids.

Each pair of base angles is congruent, so the legs are the same length.

Answer: Both trapezoids are isosceles.

The sides of a picture frame appear to be two adjacent trapezoids. Determine if they are isosceles trapezoids.

Answer: yes

ABCD is a quadrilateral with vertices A(5, 1), B(–3, –1), C(–2, 3), and D(2, 4). Verify that ABCD is a trapezoid.

A quadrilateral is a trapezoid if exactly one pair of opposite sides are parallel. Use the Slope Formula.

Answer: Exactly one pair of opposite sides are parallel, So, ABCD is a trapezoid.

slope of

slope of

slope of

slope of

ABCD is a quadrilateral with vertices A(5, 1), B(–3, 1), C(–2, 3), and D(2, 4). Determine whether ABCD is an isosceles trapezoid. Explain.

Answer: Since the legs are not congruent, ABCD is not an isosceles trapezoid.

First use the Distance Formula to show that the legs are congruent.

Answer: Exactly one pair of opposite sides is parallel. Therefore, QRST is a trapezoid.

QRST is a quadrilateral with vertices Q(–3, –2), R(–2, 2), S(1, 4), and T(6, 4).

a. Verify that QRST is a trapezoid.

Answer: Since the legs are not congruent, QRST is not an isosceles trapezoid.

b. Determine whether QRST is an isosceles trapezoid. Explain.

DEFG is an isosceles trapezoid with median Find DG if and

Theorem 8.20

Multiply each side by 2.

Substitution

Subtract 20 from each side.

Answer:

DEFG is an isosceles trapezoid with median Find , and if and

Because this is an isosceles trapezoid,

Consecutive Interior Angles Theorem

Substitution

Combine like terms.

Divide each side by 9.

Answer:Because

WXYZ is an isosceles trapezoid with median

Answer:

a.

b.

Answer: Because

Example 1 Positioning a Square

Example 2 Find Missing Coordinates

Example 3 Coordinate Proof

Example 4 Properties of Quadrilaterals

Position and label a rectangle with sides a and b units long on the coordinate plane.

The y-coordinate of B is 0 because the vertex is on the x-axis. Since the side length is a, the x-coordinate is a.

Let A, B, C, and D be vertices of a rectangle with sides a units long, and sides b units

long.

Place the square with vertex A at the origin, along the positive x-axis, and along the y-axis. Label the vertices A, B, C, and D.

D is on the y-axis so the x-coordinate is 0. Since the side length is b, the y-coordinate is b.

Sample answer:

The x-coordinate of C is also a. The y-coordinate is b because the side is b units long.

Position and label a parallelogram with sides a and b units long on the coordinate plane.

Sample answer:

Name the missing coordinates for the isosceles trapezoid.

The legs of an isosceles trapezoid are congruent and have opposite slopes. Point C is c units up and b units to the left of B. So, point D is c units up and b units to the right of A. Therefore, the x-coordinate of D is and the y-coordinate of D is

Answer:

Name the missing coordinates for the rhombus.

Answer:

Place a rhombus on the coordinate plane. Label the midpoints of the sides M, N, P, and Q. Write a coordinate proof to prove that MNPQ is a rectangle.The first step is to position a rhombus on the coordinate plane so that the origin is the midpoint of the diagonals and the diagonals are on the axes, as shown. Label the vertices to make computations as simple as possible.Given: ABCD is a rhombus as labeled. M, N, P, Q are

midpoints.

Prove: MNPQ is a rectangle.

Proof:

By the Midpoint Formula, the coordinates of M, N, P, and Q are as follows.

Find the slopes of

slope of

slope of

slope of

slope of

A segment with slope 0 is perpendicular to a segment with undefined slope. Therefore, consecutive sides of this quadrilateral are perpendicular. Since consecutive sides are perpendicular, MNPQ is, by definition, a rectangle.

Place an isosceles trapezoid on the coordinate plane. Label the midpoints of the sides M, N, P, and Q. Write a coordinate proof to prove that MNPQ is a rhombus.

Given: ABCD is an isosceles trapezoid. M, N, P, and Q are midpoints.

Prove: MNPQ is a rhombus.

The coordinates of M are (–3a, b); the coordinates of N are(0, 0); the coordinates of P are (3a, b); the coordinates of Q

are (0, 2b).

Since opposite sides have

equal slopes, opposite sides are parallel.

Since all four sides are congruent and opposite sides are parallel, MNPQ is a rhombus.

Proof:

Write a coordinate proof to prove that the supports of a platform lift are parallel.

Prove:

Proof:

Given: A(5, 0), B(10, 5), C(5, 10), D(0, 5)

Since have the same slope, they are parallel.

Write a coordinate proof to prove that the crossbars of a child safety gate are parallel.

Prove:

Proof: Since have the same slope, they are parallel.

Given: A(–3, 4), B(1, –4), C(–1, 4), D(3, –4)

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