pairs of angles

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Pairs of AnglesPairs of Angles

Students at West Ridge Middle School are going on a trip to the museum. Nine have never gone before. Twelve have gone once, half as many as that have gone twice, and none have gone more than that. How many students are going on the trip?

27

LESSON 7-1LESSON 7-1

Problem of the Day

7-1

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(For help, go to Lesson 1-6.)

1. Vocabulary Review What is the inverse operation of addition?

Solve each equation.

2. a + 14 = 32

3. b – 5 = 26

4. 10 + c = –31

5. –48 = d – 19

Check Skills You’ll Need


7-1

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Pairs of AnglesPairs of AnglesLESSON 7-1LESSON 7-1

Solutions

1. subtraction

2. 18

3. 31

4. –41

5. –29


7-1

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Find the measure of the supplement of IGJ.


x° + m IGJ = 180°The sum of the measures of supplementary angles is 180º.

x° + 145° – 145° = 180° – 145° Subtract 145º from each side.

x° = 35° Simplify.

The measure of the supplement of m IGJ is 35º. Quick Check

Additional Examples

7-1

Substitute 145º for m DEF.x° + 145° = 180°

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The adjacent angles are HGK and KGJ; KGJ and JGI;

JGI and IGH; IGH and HGK.

The vertical angles are JGI and HGK; HGI and KGJ.

Pairs of AnglesPairs of AnglesLESSON 7-1LESSON 7-1

Name a pair of adjacent angles and a pair of

vertical angles in the figure. Find m HGK.

Since vertical angles are congruent, m HGK = m JGI = 145°.

Quick Check

Additional Examples

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In this figure, if m DKH = 73°, find the measures

of GKJ and JKF.


m DKE + 90° = 180° DKE and FKE are supplementary.

m DKE = 90° Subtract 90º from each side.

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(continued)


m KHE + 73° = 90° KHE and DKH are complementary.

m KHE = 17° Subtract 73º from each side.

GKJ and KHE are vertical angles.m GKJ = m KHE = 17°

JKF and DKH are vertical angles.m JKF = m DKH = 73°

Quick Check

So, the measure of GKJ is 17° is and the measure of JKF is 73°.

Additional Examples

7-1

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Use the diagram to answer Questions 1 – 3.

1. List all pairs of vertical angles.

2. List any angles adjacent to CXD.

3. If m AXB = 110°, find m DXC.

4. An angle measures 57°. What is the measure of its supplement?


123°

110°

AXD and BXC

AXD and BXC; AXB and DXC

Lesson Quiz

7-1

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Angles and Parallel LinesAngles and Parallel Lines

Write the following sentence in mathematical symbols. Then indicate whether the sentence is true or false: Four and one sixth minus five eighths equals three and thirteen twenty fourths.


4 – = 3 ; true16

58

1324

Problem of the Day

7-2

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1. Vocabulary Review Which of the following pairs of angles are supplementary?

Find the measure of the supplement of each angle.

2. 48°

3. 119°

4. 67°

5. 131°

50° and 40°, 100° and 90°, 120° and 60°, 75° and 125°



7-2

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Angles and Parallel LinesAngles and Parallel LinesLESSON 7-2LESSON 7-2

Solutions

1. 120° and 60°

2. 132°

3. 61°

4. 113°

5. 49°


7-2

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1 and 3, 2 and 4, 5 and 7, 6 and 8 are pairs of correspondingangles.

2 and 7, 3 and 6, are pairs of alternate interior angles.


Identify each pair of corresponding angles and

each pair of alternate interior angles.

Quick Check

Additional Examples

7-2

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If p is parallel to q, and m 3 = 56º, find m 6.

m 6 = 56°

Quick Check

m 6 = m 3 = 56° Alternate interior angles are congruent.

Additional Examples

7-2

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p || q because 5 and 7 are congruent alternate interior angles.

s || t because 6 and 7 are congruent corresponding angles.


In the diagram below, m 5 = m 6 = and m 7 = 80º. Explain

why p and q are parallel and why s and t are parallel.


Quick Check

Additional Examples

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Use the diagram to answer the questions.

1. Classify 4 and 7 as 2. Classify 2 and 8 as alternate interior angles, alternate interior angles, corresponding angles, corresponding angles, or neither. or neither.

3. If a || b and m 8 = 80°, 4. Suppose m 5 = 100° and find m 4. m 3 = 100°. What can you

conclude about line a and line b?

neither


alternate interior angles

a || b

80°

Lesson Quiz

7-2

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Congruent PolygonsCongruent Polygons

Solve the proportion: = .


n = 2135

45

n27

Problem of the Day

7-3

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Are the polygons similar? Explain.

2.

1. Vocabulary Review Congruent angles have ? measures.



7-3

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Congruent PolygonsCongruent PolygonsLESSON 7-3LESSON 7-3

Solutions

1. equal

2. ; ; not similar; corresponding sides are not in

proportion.

LNZY

LMXZ

515

=/ 410


7-3

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In the diagram below, list the congruent parts of

the two figures. Then write a congruence statement.

Congruent Angles A M B N C O D P

Congruent SidesAB MNBC NOCD OPDA PM

Since A corresponds to M, B corresponds to N, C corresponds to O, and D corresponds to P, a congruence statement is ABCD MNOP.

Quick Check

Additional Examples

7-3

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SQ VT Side Q E Angle

QP EY Side

Q Y Angle


Show that each pair of triangles is congruent.


QPR EYT by ASA. SQR VTU by SAS.

b.a.

Quick Check

Q T Angle

QR TU Side

Additional Examples

7-3

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A surveyor drew the diagram below to find the distance from

J to I across the canyon. Show that GHI KJI. Then find JK.

J H Both are right angles.

JI HI Both measure 48 ft.

KIJ GIH They are vertical angles.

So ∆GHI ∆ JI by ASA

Corresponding parts of congruent triangles are congruent.JK corresponds to HG, so JK is 36 ft. Quick Check

Additional Examples

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Use ABC and XYZ to answer the questions.

1.Suppose AC= XZ, AB = XY, and BC = YZ. Write a congruence statement for the figures.

2. Suppose ABC and XYZ are congruent. If AB = 5 cm, BC = 8 cm, and AC = 10 cm, find XZ.

3. Suppose B Y, A X, and AB XY. Why is ABC XYZ?


∆ABC ∆XYZ

10 cm

ASA

Lesson Quiz

7-3

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4. Let AB = XY = 9 inches; BC = YZ = 24 inches; and m B = 85°, m Z = 35°, and m Y = 85°. Prove that the triangles are congruent and find m C.

∆ABC ∆XYZ by SAS; m C = 35°

Lesson Quiz

7-3

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Classifying Triangles and QuadrilateralsClassifying Triangles and Quadrilaterals

Ruth was born on February 29, 1984. If she insists on celebrating her birthday only on February 29, when will she celebrate her 12th birthday?

2032


Problem of the Day

7-4

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(For help, go to the Skills Handbook page 640.)

Classify each angle as acute, right, obtuse, or straight.

2. 3.

1. Vocabulary Review How many degrees does a right angle have?



7-4


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Classifying Triangles and QuadrilateralsClassifying Triangles and QuadrilateralsLESSON 7-4LESSON 7-4

Solutions

1. 90°

2. acute

3. obtuse


7-4

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Classify LMN by its sides and angles.

The triangle has two sides that are congruent and three acute angles. It is an isosceles acute triangle.

Quick Check

Additional Examples

7-4

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What is the best name for figure WXYZ? Explainyour choice.

WXYZ has both pairs of opposite sides parallel, but adjacent sides are not equal, so it is a parallelogram.

Quick Check

Additional Examples

7-4

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1. Classify the triangle according to its angles and sides.

3. Determine the best name for the quadrilateral.


obtuse isosceles triangles

rhombus

2. A triangle’s sides are all congruent and its angles all measure 60°. Classify the triangle.

equilateral, acute

Lesson Quiz

7-4

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4. What is the best name for a figure that has four sides congruent, corresponding sides parallel, and all four angles congruent?

square

Lesson Quiz

7-4

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Angles and PolygonsAngles and Polygons

A pound of turkey has 144 g of protein and will serve 4 people. If 4 people are served equal amounts, how many grams of protein will each receive?

36 g


Problem of the Day

7-5

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(For help, go to the Lesson 1-1.)

Evaluate each expression for a = 8.

2. 3(a + 1)

3.

1. Vocabulary Review How do you evaluate an algebraic expression?

5a + 8

a

4. (a – 2)6



7-5


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Angles and PolygonsAngles and PolygonsLESSON 7-5LESSON 7-5

Solutions

1. You replace each variable in the expression with a number and then

simplify.

2. 27

3. 6

4. 36


7-5

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Find the sum of the measures of the interior angles

of an octagon.

An octagon has 8 sides.

(n – 2) 180º = (8 – 2) 180º Substitute 8 for n.

= 1,080º Simplify.

= (6) 180º Subtract.

The sum of the interior angles of an octagon is 1,080.

Quick Check

Additional Examples

7-5

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Find the missing angle measure in the hexagon.

Step 1 Find the sum of the measures of the interior angles of a hexagon.

(n – 2) 180° = (6 – 2) 180° Substitute 6 for n.

Simplify.= 720°

Additional Examples

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(continued)

Step 2 Write an equation.

720° = 120° + 115° + 136° + 80° + 147° + x° Write an equation.

Let x = the missing angle measure.

720° = 598° + x° Add.

122° = x°Subtract 598º fromeach side.

The missing angle measure is 122º.

Quick Check

Additional Examples

7-5

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A design on a tile is in the shape of a regular nonagon. Find

the measure of each angle.


Each angle of a regular nonagon has a measure of 140°.

(n – 2) 180° = (9 – 2) 180° Substitute 9 for n since a nonagon has 9 sides.

= 1,260° Simplify.

1,260° ÷ 9 = 140°Divide the sum by the number of angles in a nonagon.

Quick Check

Additional Examples

7-5

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1. Find the sum of the measures of the interior angles of a polygon having 17 sides.

2. Five angles of a hexagon measure 128°, 190°, 112°, 154°, and 90°. Find the measure of the missing angle.

3. Find the measure of each angle of a regular polygon having 24 sides.


2,700°

46°

165°

4. A regular figure has an interior angle measure of 135°. How many sides does it have?

8

Lesson Quiz

7-5

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Areas of PolygonsAreas of Polygons

Lucy predicted that her final average for math class would be at least a 93. Her test grades were 88, 90, 92, 97. What is the lowest test score she can make on the last test to make this true?

98


Problem of the Day

7-6

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Find the area of each figure.

2. 3.

1. Vocabulary Review What is a formula?



7-6

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Solutions1. A formula is a rule that shows the relationship between two or more quantities.

2. A = • w 3. A = s2

= 10 • 8 = 72

A = 80 cm2 A = 49 ft2

Areas of PolygonsAreas of PolygonsLESSON 7-6LESSON 7-6


7-6

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Find the area of the triangular part of the doghouse.

The area of the triangular part of the doghouse is 378 in.2.

A = bh12 Use the area of a triangle formula.

= • 36 • 2112 Substitute 36 for b and 21 for h.

= 378 Multiply.

Quick Check

Additional Examples

7-6

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Find the area of the trapezoid.


The area of the trapezoid is 35.2 in.2.

A = h (b1 + b2)12

Use the formula.

= (4.4) (6.7 + 9.3)12

Substitute 4.4 for h, 6.7 for b1, and 9.3 for b2.

= 35.2 Simplify.

Quick Check

Additional Examples

7-6

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1. Find the area.


14 cm2

2. An architect is designing a restaurant with a triangular entrance. The base of the triangle is 10 ft wide. The entrance is 14 ft tall. Find the area of the triangle.

70 ft2

Lesson Quiz

7-6

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56 ft2

3. Find the area.

4. If both bases of the figure in Exercise 3 are doubled, what is the new area of the trapezoid?

112 ft2

Lesson Quiz

7-6

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Circumference and Area of a CircleCircumference and Area of a Circle

Opal, Charles, Jean, and Scott had an earthworm-catching contest. Jean caught one fourth as many worms as Opal and twice as many as Charles. Opal caught 3 times as many worms as Scott. How many worms did each of the other three contestants catch if Scott caught 8 worms?

Opal, 24; Jean, 6; Charles, 3

Problem of the Day

7-7

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Circumference and Area of a CircleCircumference and Area of a Circle


Find the area of the figure below.

2.

1. Vocabulary Review Explain the difference between perimeter and area.



7-7

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Solutions

1. Perimeter is the distance around a figure. Area is the number of square units a figure encloses.

2. A = bh

= • 13 • 7

Circumference and Area of a CircleCircumference and Area of a CircleLESSON 7-7LESSON 7-7

12

12

A = 45.5 ft²


7-7

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Circumferences and Areas of CirclesCircumferences and Areas of CirclesLESSON 7-7LESSON 7-7

The diameter of a tractor tire is 125 cm. Find the

circumference and area. Round to the nearest tenth.

C = d Use the formula for circumference.

= (125) Substitute 125 for d.

The circumference is about 392.7 cm.

Use a calculator.125 392.6990817

Additional Examples

7-7

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A = r 2 Use the formula for the area of a circle.

= (62.5) 2 The radius is 125 ÷ 2, or62.5. Substitute 62.5 for r.

The area is about 12,271.8 cm2.

(continued)

Quick Check

Use a calculator.62.5 12271.8463

Additional Examples

7-7

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Circumferences and Areas of CirclesCircumferences and Areas of Circles

Find the area of the unshaded region of the square tile with a

circle inside of it, as shown below. Round to the nearest tenth.


You can separate the figure into a circle and a square.

Additional Examples

7-7

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(continued)


Area of square = s2

Step 2 Find the area of the circle.

Area of circle = r2

A = (12)2 Substitute 12 for s.

= 144 Simplify.

113.1 Multiply. Round to the nearest tenth.

A = (6)2 Substitute 6 for r.

Step 1 Find the area of the square.

Additional Examples

7-7

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(continued)


Step 3 Subtract the area of the circle from the area of the square.

The area of the shaded region is about144 cm2 – 113.1 cm2 = 30.9 cm2.

Quick Check

Additional Examples

7-7

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1. Find the circumference and area of the circle. Round to the nearest tenth.

2. The diameter of the lid of a can of soup is 5 cm. Find its circumference and area. Round to the nearest tenth.


18.8 km; 28.3 km2

15.7 cm; 19.6 cm2

Lesson Quiz

7-7

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273.0 in.2

3. Find the area. Round to the nearest tenth.

Lesson Quiz

7-7

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ConstructionsConstructions

Estimate in years the age of someone who is a million minutes old.

about 2 yr

Problem of the Day

7-8

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ConstructionsConstructions


List the congruent parts of each pair of congruent figures.

2. ∆PQR ∆TUV

1. Vocabulary Review How are congruent polygons the same?

3. ABCD LMNO



7-8

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Solutions

1. Congruent polygons have the same size and shape.

2. P T; Q U; R V;

PQ TU; QR UV; RP VT

ConstructionsConstructionsLESSON 7-8LESSON 7-8

3. A L; B M; C N;

AB LM; BC MN; CD NO;

D O;

DA OL


7-8

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Use a protractor to draw a 40º angle and label it B.

Construct N congruent to B.

Step 1 Draw a ray with endpoint N.

Step 3 Keep the compass open to the same width. Put the compass tip at N. Draw an arc that intersects the ray at a point O.

Step 2 Put the compass tip at B and draw an arc that intersects the sides of B. Label the points of intersection A and C.

Additional Examples

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(continued)

Step 4 Adjust the compass widthSo that the tip is at C and the pencil is at A.

Step 5 Keep the compass open to the same width. Put the compass tip on O and draw an arc that intersects the first arc at point M.

N is congruent to B.

Step 6 Draw NM.

Quick Check

Additional Examples

7-8

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Draw a line f. Construct a line parallel to line f.

Step 1 Draw line f.

Step 2 Draw line k that intersects line f at D. Label the angle formed 1. Then label point E on line k.

Step 3 Construct an angle at E that is congruent to 1.

EC is parallel to line f.

Quick Check

Additional Examples

7-8

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For Exercises 1–2, use the diagram below.

1. Construct an angle that is congruent to R.

2. Construct a line parallel to RS.

Lesson Quiz

7-8

pairs of angles

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