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GLENCOE DIVISIONGlencoe/McGraw-Hill8787 Orion PlaceColumbus, Ohio 43240
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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