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Answers

Geometry Copyright © Big Ideas Learning, LLC Answers All rights reserved. A8

Chapter 2 2.1 Start Thinking

Sample answer: If an animal is a horse, then it is a mammal; If an animal is not a mammal, then it cannot be a horse. Any fact stated in the form of an "if-then" statement could be used, as long as it is factual and leads the reader to believe the original statement as a result.

2.1 Warm Up

1. hexagon 2. right

3. complementary 4. straight

2.1 Cumulative Review Warm Up

1. ( )6, 1 2. ( )1, 1− 3. ( )72

2, 4. ( )5, 1−

2.1 Practice A

1. If you like the ocean, then you are a good swimmer.

2. If it is raining outside, then it is cold.

3. If you are a child, then you must attend school.

4. If angles are congruent, then they have equal angle measures.

5. a. conditional: If an animal is a puppy, then it is a dog; true

b. If an animal is a dog, then it is a puppy; false

c. If an animal is not a puppy, then it is not a dog; false

d. If an animal is not a dog, then it is not a puppy; true

6. true; By definition, the sum of two complementary angles is 90 .°

7. false; The sides are not congruent.

8. An angle is obtuse if and only if the angle measure is greater than 90° and less than 180 .°

9. Two angles are supplementary if and only if the sum of their angle measures is 180 .°

10. yes; By definition, the negation of a true sentence is false, and the negation of a false sentence is true.

11. Sample answer: If two angles are not complementary, then the sum of their angle measures is 180 .°

2.1 Practice B

1. If you like to eat, then you are a good cook.

2. If an animal is a bear, then it is a mammal.

3. a. If a tree is an oak tree, then it is a deciduous tree; true

b. If a tree is a deciduous tree, then it is an oak tree; false

c. If a tree is not an oak tree, then it is not a deciduous tree; false

d. If a tree is not a deciduous tree, then it is not an oak tree; true

4. true; Vertical angles share opposite rays.

5. false; The angles of a parallelogram are not always perpendicular.

6. A quadrilateral is a rectangle if and only if it has all perpendicular sides.

7. yes; By definition, true statements always have true contrapositives.

8. If 7, then 3 2 23.x x= + =

9. If 38 , then 52m ILH m GLH∠ = ° ∠ = ° because they are complementary angles. If 38 ,m ILH∠ = ° then 38m FLK∠ = ° because they are vertical angles. If 52 , then 52m GLH m KLJ∠ = ° ∠ = ° because they are vertical angles.

2.1 Enrichment and Extension

2.1 Puzzle Time

A SPELLING BEE

p q ∨p q p q ∧ p q

T T T F F F

T F T F T F

F T T T F F

F F F T T T

∨ p q ( )∨ ∧ p q p ( )∧ ∨ p q q

F F F

T F T

T T F

T T T

Answers

Copyright © Big Ideas Learning, LLC Geometry All rights reserved. Answers

A9

2.2 Start Thinking

yes; Sample answer:

Statement 1: “If I can go sledding, then there is snow on the ground.”

Statement 2: “If there is snow on the ground, then it is cold outside.”

Statement 3: “If I can go sledding, then it is cold outside.”; true

2.2 Warm Up

1. 0.06; 0.27, 0.33 2. 2.22; 3.3, 5.52− − −

3. 5; 7, 12− − − 4. 0.5; 3.9, 4.4

5. 2; 8, 10 6. 7; 5, 12− − −


1. 7.5 square units 2. 6 square units

3. 12 square units 4. 24 square units

2.2 Practice A

1. The next number is one more than twice the preceding number; 95, 191

2. The list items are letters in alphabetical order followed by letters in reverse alphabetical order; X, D

3. The difference of any two even integers is always even. Sample answer: 36 16 20− =

4. The product of three negative numbers is always negative. Sample answer: ( )( )( )2 3 5 30− − − = −

5. The bisector of a straight angle creates two right angles.

6. You got wet. 7. not possible

8. If you study, then you will pass the class.

9. If a straight angle is bisected, then each angle is a right angle.

10. Law of Syllogism

11. inductive reasoning; The conjecture is based on the assumption that the weather pattern will continue.

12. deductive reasoning; The conjecture is based on the fact that 92 14 1288,× = which is even.

13. The Rocky Mountains are taller than the Appalachian Mountains.

14. P ns=

2.2 Practice B

1. The list items are letters in alphabetical order followed by numbers in decreasing numerical order starting with 26; D, 23

2. The pattern is a sequence of spider webs, each web having one more row of webs than the previous web.

3. The sum of two absolute values is always positive; Sample answer: 3 7 3 7 10− + = + =

4. The product of a number and its square is the number to the third power; Sample answer:

( )( ) ( )2 35 5 5 25 125 5= = =

5. If the angles are right, obtuse, or any acute angle other than 45 ,° then they will not be complementary.

6. not possible

7. and AOB DOB∠ ∠ share a common ray.

8. not possible

9. If it is Tuesday, then you water the flowers.

10. deductive reasoning; The facts of mammals and laws of logic were used to draw the conclusion.

11. inductive reasoning; The conjecture is based on the assumption that a pattern will continue.

12. no; Based on the Law of Syllogism, a series of true conditional statements will always be true.

13. Using inductive reasoning, you can make the conjecture that organic produce costs more than nonorganic produce because this was true in all of the specific cases listed in the table.


1. 7 guesses

2. lengths in ft: Stage 1: 1, Stage 2: 43, Stage 3: 16

9,

Stage 4: 6427

; The expression that models the

pattern of the length at a given stage is ( ) 143

.n −

Answers


3. areas in 2ft : Stage 1: 1, Stage 2: 89, Stage 3: 64

81,

Stage 4: 512729

; The expression that models that

pattern of the shaded area at a given stage is

( ) 189

.n −

2.2 Puzzle Time

LADY BUGS

2.3 Start Thinking

Sample answer: doors, windows, scale, stairs, water lines

2.3 Warm Up

1. 148° 2. 25° 3. 31° 4. 72°


1. 22 , 22m ABD m CBD∠ = ° ∠ = °

2. 85 , 85m ABD m CBD∠ = ° ∠ = °

3. 33 , 33m ABD m CBD∠ = ° ∠ = °

4. 64 , 64m ABD m CBD∠ = ° ∠ = °

2.3 Practice A

1. Sample answer: There is exactly one line through points C and H.

2. Sample answer: Line contains points G and D.

3. Sample answer: CH

and GE

intersect at point D.

4. Sample answer: Points B, H, and E are noncollinear and define plane M.

5. Sample answer: Plane M contains the noncollinear points B, H, and E.

6. Sample answer: Points G and E lie in Plane M so,

GE

lies in plane M.

7. Sample answer:

8. Sample answer:

9. Sample answer:

10. yes 11. no 12. yes

13. no 14. yes

15. If three points are noncollinear, then there exists exactly one plane that contains them; converse: If there exists exactly one plane that contains three points, then the three points are noncollinear. inverse: If three points are collinear, then there are multiple planes that contain all three points. contrapositive: If there are multiple planes that contain three points, then the three points are collinear; The converse, inverse, and contrapositive are true.

16. no; Three lines must intersect each other at three points.

2.3 Practice B

1. Sample answer: There is exactly one line through points C and G.

2. Sample answer: EF

contains points E and F.

3. Sample answer: and CG EF intersect at point J.

4. Sample answer: Plane A contains the noncollinear points D, H, and I.

5. Sample answer: Points E and F lie in plane B. So,

EF

lies in plane B.

6. Sample answer: Planes A and B intersect at .CG

7. Sample answer:

C

D

E

BA

C

D

B

A

X HA

YGQ

S

TV

U R

Answers


A11

8. Sample answer:

9. no 10. yes 11. yes 12. no

13. converse: If two planes share a line, then the two planes intersect. inverse: If two planes do not intersect, then their intersection is not a line. contrapositive: If the intersection of two planes is not a line, then the two planes do not intersect; The converse, inverse, and contrapositive are true.

14. yes; Because three noncollinear points define a plane, two points on a line define an infinite number of planes.

15. no; The line that passes through a point on a plane does not lie in the plane unless there is another point on the line that is also in the plane.

16. no; yes; Because of the Plane Line Postulate

(Post. 2.6), EF

only lies in plane Z when it contains two points in plane Z.


1. There exists exactly one plane that contains both lines m and n.

2. Line-Point Postulate (Post. 2.2); A line contains at least two points.

3. Line Intersection Postulate (Post. 2.3); If two lines intersect, then their intersection is exactly one point.

4. Three Point Postulate (Post. 2.4); Through any three non-collinear points, there exists exactly one plane.

5. Plane-Line Postulate (Post. 2.6); If two points lie in a plane, then the line containing them lies in the plane.

6. 7.

8. 9.

10. ;XC

If two planes intersect, their intersection is a line; Plane Intersection Postulate (Post. 2.7).

11. no; Points C, D, E, and X would all be collinear, but

line and EX CX

intersect only at point X, so this is impossible.

12. no; This does not follow the Plane-Line Postulate (Post. 2.6) because D lies in plane P but not in plane Q, and B lies in plane Q but not in plane P.

2.3 Puzzle Time

BECAUSE IT WOULD TAKE THE GEESE FOREVER TO WALK

2.4 Start Thinking

Sample answer: One instance when it is necessary is when there is a quotient containing the variable and addition or subtraction in the numerator and a real number in the denominator;

420

7

x + =−

2.4 Warm Up

1. Each side of the equation was subtracted by 23 rather than added; 23 17;f − = −

23 23 17 23;f − + = − + 6f =

2. Each side of the equation was divided by 8,−

rather than 8; 8 4 1

8 4; ; 8 8 2

rr r= = =

3. The right side of the equation was multiplied by 47

rather than 74; ( ) ( )7 74 4

7 4 7 422; 22;m m= =

38.5m =

4. Each side of the equation was multiplied by 6,

rather than 6;− 3;6

n− = 6

6 3;1 6

n − • − = − •

18n = −

a b c

s

CA

B

ACX

DE

F

B

Q

P

AB

T

S

C

s

rC

D

M

Answers



1. , , ADC ADB BDC∠ ∠ ∠

2. , , EHG EHF GHF∠ ∠ ∠

2.4 Practice A

1. Equation Explanation and Reason

3 4 31 Write the equation; Given

3 27 Subtract 4 from each side; Subtraction Property of Equality

9 Divide each side by 3; Division Property of Equality

x

x

x

+ ==

=


( ) Write the equation; Given3 2 1 15

6 3 15 Multiply; Distributive Property

6 12 Subtract 3 from each side;Subtraction Property of Equality

2 Divide each side by 6; Division Property of Equality

x

x

x

x

+ =+ =

=

=


( ) ( )12

Write the equation; 16 18 2 16Given

8 4 2 32 Multiply; Distributive Property

6 36 Add 4 to each side and subtract 2 from each side; Additionand Subtraction Properties of Equality

6 Divide each sid

x x

x x

xx

x

− = +

− = +

=

= e by 6; Division Property of Equality


2 Write the equation; Given

Divide each side by 2; Division Property of Equality2Rewrite the equation; Symmetric Property of Equality2

p v

pv

pv

=

=

=


2

2

2

2

Write the equation; Given

Divide each side by ; Division Property of Equality

Rewrite the equation; Symmetric Property of Equality

V r h

V rhr

Vh

r

ππ

π

π

=

=

=


2

2 2

2

2

Write the equation; Given

Subtract from each side; Subtraction Property of Equality

Divide each side by ; Division Property of Equality

Rewrite the equation; Sym

S rs r

S r rs r

rS rs

r

S rs

r

π π

π π π

πππ

ππ

= +

− =

− =

−= metric Property of Equality

7. Multiplication Property of Equality

8. Transitive Property of Equality

9. 30 m K° + ∠ 10. GH 11. 3 ; 21x


( )

2 2 Write the equation; 2 Given

2 2 2 Subtract 2 from each side; Subtraction Property of Equality

Factor ; 2 2 2Distributive Property

2 Divide each side by 2 2 ; Divis2 2

A w hhw

A h w hw h

wA h w h

A hw

hh

= ++

− = +

− = +

− = ++

ion

Property of Equality

2 Rewrite the equation;Symmetric Property2 2of Equality

3

A hw

h

w

−=+

=

13. 22 units;P = Commutative and Addition Properties of Equality

Answers


A13

2.4 Practice B


( ) Write the equation; 3 4 3 2Given


2 7 Add 9 to each side andsubtract from each side; Addition and Subtraction Properties of Equality

3.5 Divide each side b

x x

x x

xx

x

− + = −

− + = −

=

= y 2; Division Property of Equality


( ) ( )

( )

Write the 1 5 3 2 1equation; Given

Multiply; 5 3 2 1Distributive Property


10 2 Add 5 to each sideand subtract 9 from each side; Addition and Sub

x x x

x x x

x x x

xx

− + = + −

− − = + −

− − = + −

− =

traction Properties of Equality

0.2 Divide each side by 10; Division Property of Equality

x = −


2

2

2

2

2

1 Write the equation; Given2Multiply each side by 2;2Multiplication Property of Equality

2 Divide each side by ; Division Property of Equality

2 Rewrite the equation; Symmetric Proper

I mr

I mr

I rmr

Im

r

=

=

=

=ty of Equality


2

22

2

2

12

12

12

1 Write the 9.8

equation; Given21 Subtract 9.82 from each side;

Subtraction Property of Equality

Divide each side by 9.8 ; Division 9.8Property of Equality

Rewrit

9.8

E mv mh

mvE mv mh

E mvh mm

E mvh

m

= +

− =

−=

−=

e the equation; Symmetric Property of Equality

5. Multiplication and Subtraction Properties of Equality

6. Transitive and Addition Properties of Equality

7. 60° 8. 3 5 2x y x y+ = −


1 Write the equation; Given2

2 Multiply each side by 2 and divideeach side by ; Multiplication andDivision Properties of Equality

2 Rewrite the equation; Symmetric Property of Equality

8

V bh

Vb

hlh

Vb

hb m

=

=

=

=

10. Sample answer: 65m BCD∠ = ° so

65m GCF∠ = ° and 115 ,m FCD∠ = ° so

57.5m FCE m DCE∠ = ∠ = °

Answers



1.

2.

3. a. Given

b. Addition Property of Equality

c. Multiplication Property of Equality

d. Simplify.

e. Distributive Property

f. Subtraction Property of Equality

g. Distributive Property

4. a. ( )1n c r= +

b. ( )1n c r= + Given

1n

rc

= + Division Property of Equality

1n

rc

− = Subtraction Property of Equality

c. 2%r =

d. 10.249.85;

1.04c = ≈ Solve the formula

( )1 ,n c r= + for r to yield 1 .n

rc

− =

Substitute 10.24 and 0.04n r= = into the

equation to yield 10.24

1 0.04.c

− = Using the

Addition Property of Equality, add 1 to each side

to obtain 10.24

1.04.c

= Next, multiply by c and

divide by 1.04 to obtain 10.24

9.85.1.04

c = ≈

2.4 Puzzle Time

MAKE APPLESAUCE

2.5 Start Thinking

The formula for the area of a triangle is derived directly from the formula for the area of a rectangle. By drawing a diagonal, the rectangle is now split into two congruent triangles. So, each triangle is half the area of the rectangle, and the formula for the area of a triangle

is 12

.A bh=

2.5 Warm Up

1. complement: 31 ,° supplement: 121°



4. complement: 67.4 ,° supplement: 157.4°




1. , BEC DEC∠ ∠ 2. 83°

3. 139° 4. 90°

2.5 Practice A

1. Symmetric Property of Segment Congruence

2. Reflexive Property of Angle Congruence

STATEMENTS REASONS

1. ( )2 12

nS a n d= + − 1. Given

2. ( )22 1S a n d

n• = + − 2. Multiplication

Property of Equality

3. ( )21 2

Sn d a

n− − = 3. Subtraction Property

of Equality

4. ( )1

2

n dSa

n

−− = 4. Division Property

of Equality

STATEMENTS REASONS

1. ( )213

3V h r hπ= − 1. Given

2. ( )23 3V h r hπ= − 2. Multiplication Property of Equality

3. 2

33

Vr h

hπ= − 3. Division Property

of Equality

4. 2

33

Vh r

hπ+ = 4. Addition Property

of Equality

5. 2 3

V hr

hπ+ = 5. Division Property

of Equality

Answers


A15

3.

4.

5.

6.

2.5 Practice B

1.

2.

3.

STATEMENTS REASONS

1. AB AB= 1. Reflexive Property of Equality

2. AB AB≅ 2. Definition of congruent segments

STATEMENTS REASONS

1. bisects BF AFC∠

1. Given

2. AFB BFC∠ ≅ ∠ 2. Definition of angle bisector

3. CFD BFC∠ ≅ ∠ 3. Given

4. BFC CFD∠ ≅ ∠ 4. Symmetric Property of Angle Congruence (Thm. 2.2)

5. AFB CFD∠ ≅ ∠ 5. Transitive Property of Angle Congruence

(Thm. 2.2)

STATEMENTS REASONS

1. A B∠ ≅ ∠ 1. Given

2. m A m B∠ = ∠ 2. Definition of congruent angles

3. m B m A∠ = ∠ 3. Symmetric Property of Equality

4. B A∠ ≅ ∠ 4. Definition of congruent angles

STATEMENTS REASONS

1. A B

B C

∠ ≅ ∠∠ ≅ ∠

1. Given

2. m A m B

m B m C

∠ = ∠∠ = ∠

2. Definition of congruent angles

3. m A m C∠ = ∠ 3. Transitive Property of Equality

4. A C∠ ≅ ∠ 4. Definition of congruent angles

STATEMENTS REASONS

1. bisects

bisects

bisects

AG CD

IJ CE

BH ED

1. Given

2. CE ED

CK KE

EF FD

===

2. Definition of segment bisector

3. 2

2

KE CE

FD ED

==

3. Definition of segment bisector

4. 2 2KE FD= 4. Transitive Property of Equality

5. KE FD= 5. Division Property of Equality

6. KE FD≅ 6. Definition of congruent segments

STATEMENTS REASONS

1. AB CD≅ 1. Given

2. AB CD= 2. Definition of congruent segments

3. CD AB= 3. Symmetric Property of Equality

4. CD AB≅ 4. Definition of congruent segments

STATEMENTS REASONS

1. bisects

bisects

bisects

E AI

BC AE

FH EI

1. Given

2. AE EI

AD DE

EG GI

≅

≅

≅

2. Definition of midpoint

3. AE EI

AD DE

EG GI

===

3. Definition of congruent segments

4. AD AD AE

EG EG EI

+ ++ =

4. Segment Addition Postulate (Post. 1.2)

5. 2

2

AD AE

EG EI

==

5. Properties of Addition

6. 2 2AD EG= 6. Substitution Property of Equality

7. AD EG= 7. Division Property of Equality

8. AD EG≅ 8. Definition of congruent segments

Answers


4.

5.


1. ;RT z=

2. 2 ;z RS=

STATEMENTS REASONS

1. 28m KMN∠ = ° 1. Given

2.

90

m KMN m JMK∠ + ∠= °

2. Definition of complementary angles

3. 28

90

m JMK° + ∠= °

3. Substitution Property of Equality

4. 62m JMK∠ = ° 4. Subtraction Property of Equality

5. 118m PTS∠ = ° 5. Given

6.

180

m PTS m STR∠ + ∠= °

6. Definition of supplementary angles

7. 118

180

m STR° + ∠= °


8. 62m STR∠ = ° 8. Subtraction Property of Equality

9. m JMK m STR∠ = ∠ 9. Transitive Property of Equality

10. JMK STR∠ ≅ ∠ 10. Definition of congruent angles

STATEMENTS REASONS

1. ADC BDE∠ ≅ ∠ 1. Given

2. m ADC m BDE∠ = ∠ 2. Definition of congruent angles

3.

180

m ADC m ADE∠ + ∠= °


4.

180

m BDC m BDE∠ + ∠= °


5. m ADC m ADE

m BDC m BDE

∠ + ∠= ∠ + ∠


6. m ADC m ADE

m BDC m ADC

∠ + ∠= ∠ + ∠


7. m ADE m BDC∠ = ∠ 7. Subtraction Property of Equality

8. ADE BDC∠ ≅ ∠ 8. Definition of congruent angles

STATEMENTS REASONS

1. T is the midpoint

of .RS

1. Given

2. RT TS≅ 2. Definition of midpoint

3. RT TS= 3. Definition of congruent segments

4. TS z= 4. Given

5. RT z= 5. Substitution Property of Equality

STATEMENTS REASONS


of .RS

1. Given



4. TS z= 4. Given

5. RT z= 5. Transitive Property

6. RT TS RS+ = 6. Segment Addition Postulate (Post. 1.2)

7. z z RS+ = 7. Substitution Property of Equality

8. 2z RS= 8. Simplify.

Answers


A17

3. ;2

zRW =

4. coordinate of point P: ;4

a b3 + coordinate of

point Q: 5 3

.8

a b+

5. a. 10, 2x y= =

b. 18, 8x y= =

6.

2.5 Puzzle Time

THEY ALL DO

2.6 Start Thinking

Sample answer: factoring polynomials

2.6 Warm Up

1. 9x = 2. 35y = 3. 5x = −

4. 9y = − 5. 7x = − 6. 7x = −

STATEMENTS REASONS


of .RS

1. Given



4. TS z= 4. Given

5. RT z= 5. Transitive Property

6. W is the midpoint

of .RT

6. Given

7. RW WT≅ 7. Definition of midpoint

8. RW WT= 8. Definition of congruent segments

9. RW WT RT+ = 9. Segment Addition Postulate (Post. 1.2)

10. RW RW RT+ = 10. Substitution Property of Equality

11. 2RW RT= 11. Simplify.

12. 2

RTRW = 12. Division Property

of Equality

13. 2

zRW = 13. Substitution Property

of Equality

STATEMENTS REASONS

1. 45m ZYQ∠ = ° 1. Given

2. 45m ZQP∠ = ° 2. Given

3. m ZYQ m ZQP∠ = ∠ 3. Substitution Property of Equality

4. ZYQ ZQP∠ ≅ ∠ 4. Definition of congruent angles

5.

180

m XYQ m ZYQ∠ + ∠= °

5. Definition of linear pair

6.

180

m ZQP m ZQR∠ + ∠= °


7. m XYQ m ZYQ

m ZQP m ZQR

∠ + ∠= ∠ + ∠


8. m XYQ m ZQP

m ZQP m ZQR

∠ + ∠= ∠ + ∠


9. m XYQ m ZQR∠ = ∠ 9. Subtraction Property of Equality

10. XYQ ZQR∠ ≅ ∠ 10. Definition of congruent angles

Straight Angle

Obtuse Angle

Right Angle

AcuteAngle

Answers



1. 2.

3. 4.

5. 6.

2.6 Practice A

1. , , , A BDC BDC EDF A EDF∠ ≅ ∠ ∠ ≅ ∠ ∠ ≅ ∠ ;CDF BDE∠ ≅ ∠ A BDC∠ ≅ ∠ by definition

because they have the same measure. BDC EDF∠ ≅ ∠ by the Vertical Angles

Congruence Theorem (Thm. 2.6). A EDF∠ ≅ ∠ by

the Transitive Property. CDF BDE∠ ≅ ∠ by the Vertical Angles Congruence Theorem (Thm. 2.6).

2. 1 4, 2 5, 3 6, 2 3, ∠ ≅ ∠ ∠ ≅ ∠ ∠ ≅ ∠ ∠ ≅ ∠

2 6, 3 5, 5 6; 1 4, ∠ ≅ ∠ ∠ ≅ ∠ ∠ ≅ ∠ ∠ ≅ ∠

2 5, and 3 6∠ ≅ ∠ ∠ ≅ ∠ by the Vertical Angles

Congruence Theorem (Thm. 2.6). 2 3∠ ≅ ∠ by

definition because they have the same measure. 2 6 and 3 5∠ ≅ ∠ ∠ ≅ ∠ by the Transitive

Property of Angle Congruence (Thm. 2.2). 5 6∠ ≅ ∠ by substitution.

3. 13, 8x y= = 4. 5, 19x y= =

5.

Proof: Because 1 and 2∠ ∠ are supplementary and 1 and 3∠ ∠ are supplementary, 1 2 180m m∠ + ∠ = °

and 1 3 180m m∠ + ∠ = ° by the definition of supplementary angles. By the Transitive Property of Angle Congruence (Thm. 2.2),

1 2 1 3.m m m m∠ + ∠ = ∠ + ∠ By the Subtraction Property of Equality, 2 3.m m∠ = ∠ So,

2 3∠ ≅ ∠ by the definition of congruent angles.

2.6 Practice B

1. , , , D B DAC ACB BAC ACD∠ ≅ ∠ ∠ ≅ ∠ ∠ ≅ ∠ , , ,BAD BCD D BAC B BAD∠ ≅ ∠ ∠ ≅ ∠ ∠ ≅ ∠

, and ;D BCD B BCD∠ ≅ ∠ ∠ ≅ ∠ D B∠ ≅ ∠ by the Right Angles Congruence Theorem (Thm. 2.3). and DAC ACB BAC ACD∠ ≅ ∠ ∠ ≅ ∠ by definition because they have the same measures. Because 90m DAC m BAC∠ + ∠ = ° and by the Angle Addition Postulate (Post. 1.4),

DAC BAC DAB∠ + ∠ ≅ ∠ and 90 .m DAB∠ = ° By the same reasoning, 90 .m BCD∠ = ° So,

BAD BCD∠ ≅ ∠ by the Right Angles Congruence Theorem (Thm. 2.3). ,D BAC∠ ≅ ∠

, , and B BAD D BCD B BCD∠ ≅ ∠ ∠ ≅ ∠ ∠ ≅ ∠ by the Transitive Property.

P

FQ

L

M

N

M

P

N

Q

STATEMENTS REASONS

1. 1 and 2∠ ∠ are supplementary.

1 and 3∠ ∠ are supplementary.

1. Given

2. 1 2 180

1 3 180

m m

m m

∠ + ∠ = °∠ + ∠ = °


3. 1 2

1 3

m m

m m

∠ + ∠= ∠ + ∠

3. Transitive Property of Angle Congruence (Thm. 2.2)

4. 2 3m m∠ = ∠ 4. Subtraction Property of Equality

5. 2 3∠ ≅ ∠ 5. Definition of congruent angles

N EDC

D

C

Answers


A19

2. 1 3, 2 4, 1 5, 2 6, ∠ ≅ ∠ ∠ ≅ ∠ ∠ ≅ ∠ ∠ ≅ ∠

4 6; 1 3, and 2 4∠ ≅ ∠ ∠ ≅ ∠ ∠ ≅ ∠ by the

Vertical Angles Congruence Theorem (Thm. 2.6). 1 5∠ ≅ ∠ by definition because they have the same

angle measure. Because 1 and 2∠ ∠ form a linear pair and 5 and 6∠ ∠ form a linear pair,

1 2 180 and 5 6 180m m m m∠ + ∠ = ° ∠ + ∠ = ° by the Linear Pair Postulate (Post. 2.8). So, by the Congruent Supplements Theorem (Thm. 2.4),

2 6.∠ ≅ ∠ Because 1 and 4∠ ∠ form a linear pair and 5 and 6∠ ∠ form a linear pair,

1 4 180m m∠ + ∠ = ° and 5 6 180m m∠ + ∠ = ° by the Linear Pair Postulate (Post. 2.8). So, by the Congruent Supplements Theorem (Thm. 2.4),

4 6.∠ ≅ ∠

3. 8, 186x y= = 4. 4, 184x y= =

5.

Because 1∠ is a right angle and 1 3∠ ≅ ∠ by the Vertical Angles Congruence Theorem (Thm. 2.6),

1 3∠ ≅ ∠ by the Right Angle Congruence Theorem (Thm. 2.3). Because 5∠ is a right angle,

5 90 .m∠ = ° 5 and 8∠ ∠ are supplementary angles

so, 5 8 180m m∠ + ∠ = ° by the definition of supplementary angles. By the Subtraction Property of Equality, 8 90 .m∠ = ° So, 8∠ is a right angle by the definition of a right angle. Because

3 and 8∠ ∠ are right angles, 3 8∠ ≅ ∠ by the Right Angles Congruence Theorem (Thm. 2.3).


1. false 2. true 3. false

4. false 5. true 6. true

7.

is a right angle.

Given

Vertical Angles Congruence Theorem (Thm. 2.6)

Definition of a right angle

are supplementary.

Given Definition of supplementary angles

Right Angle Congruence Theorem (Thm. 2.3)

is a right angle.

is a right angle.

Given Subtraction Property of Equality

Right AngleCongruence Theorem (Thm. 2.3)

is a right angle.

Definition of a right angle

STATEMENTS REASONS

1. AB BD

ED BD

⊥

⊥

1. Given

2. ABD∠ is a right angle.

EDB∠ is a right angle.

2. Definition of perpendicular lines

3. 90

90

m ABD

m EDB

∠ = °∠ = °

3. Definition of right angle

4.

90

90

m ABC m CBD

m EDC m CDB

∠ + ∠= °

∠ + ∠= °

4. Angle Addition Postulate (Post. 1.4)

5. ABC EDC∠ ≅ ∠ 5. Given

6. m ABC m EDC∠ = ∠ 6. Definition of congruent angles

7. m ABC m CBD

m EDC m CDB

∠ + ∠= ∠ + ∠


8. m EDC m CBD

m EDC m CDB

∠ + ∠= ∠ + ∠


9. m CBD m CDB∠ = ∠ 9. Subtraction Property of Equality

10. CBD CDB∠ ≅ ∠ 10. Definition of congruent angles

Answers


8.

9.

2.6 Puzzle Time

MAKE SURE ONE OF THEM IS A MATCH

Cumulative Review

1. Each additional figure has an additional inscribed equilateral triangle in the previous triangle.

2. Each additional figure has a bottom row containing one more solid circle than the previous figure.

3. Each additional figure is the same rectangle with one more equal division.

STATEMENTS REASONS

1.

45

m WYZ m TWZ∠ = ∠= °

1. Given

2. and TWZ SWZ∠ ∠

are a linear pair.

and WYZ XYW∠ ∠ are a linear pair.


3. and TWZ SWZ∠ ∠

are supplementary.

and WYZ XYW∠ ∠ are supplementary.

3. Linear Pair Postulate (Post. 2.8)

4.

180

180

m TWZ m SWZ

m WYZ m XYW

∠ + ∠= °

∠ + ∠= °


5. m TWZ m SWZ

m WYZ m XYW

∠ + ∠= ∠ + ∠


6. 45

45

m SWZ

m XYW

° + ∠= ° + ∠


7. m SWZ m XYW∠ = ∠ 7. Subtraction Property of Equality

8. SWZ XYW∠ ≅ ∠ 8. Definition of congruent angles

STATEMENTS REASONS

1. The hexagon is regular.

1. Given

2. All interior angles of the hexagon are congruent.

2. Definition of regular hexagon

3. 1∠ is congruent to an interior angle of the hexagon.

3. Vertical Angles Congruence Theorem (Thm. 2.6)

4. 2∠ is supplementary to an interior angle of the hexagon.

4. Linear Pair Postulate (Post. 2.8)

5. 2∠ is supplementary to 1.∠

5. Substitution

6. 1 2 180m m∠ + ∠ = ° 6. Definition of supplementary angles

Answers


A21

4. Add 2 starting at 2; 10, 12




8. Multiply by 32

starting at 2; 81 2438 16

,

9. Multiply by 34

starting at 3; 243 729256 1024

,

10. Increase what you add by 1 each term. Start at 1 and start by adding 2; 15, 21

11. Increase what you add by 1 each term. Start at 2 and start by adding 3; 20, 27

12. a. 88 ft b. 2468 ft

13. a. 54 in. b. 216 in.

14. 79 15. 4 16. 29− 17. 5−

18. 11 19. 160− 20. 22− 21. 29

22. 44 23. 75− 24. 14 25. 60−

26. 24 27. 5 28. 11

29. 7 16s− + 30. 27 19c − 31. 7 9g +

32. 16 7x− + 33. 5 5m + 34. 10 3r +

35. 6 j 36. 26 42a − 37. 7 23f −

38. 6 10y − 39. 2 2b− − 40. 8 5k −

41. 2w − 42. 3 6g + 43. 3p +

44. 5b = 45. 7m = 46. 19k =

47. 15p = 48. 4a = − 49. 3x =

50. 22r = − 51. 9h = 52. 42w =

53. 22t = − 54. 3c = − 55. 7e = −

56. a. 5 lunches b. 10 lunches c. $27.75

d. $9.25 e. $37

57. a. 7 push-ups b. 9 push-ups c. day 8 d. day 11

58. 64 59. 64− 60. 32 61. 25

62. 9− 63. 9 64. 16 65. 16−

66. 16 67. 100 68. 1,000,000

69. 32− 70. 3 5 71. 2 6 72. 5 5

73. 2 7 74. 4 3 75. 4 2 76. 5 2

77. 5 6 78. 4 10 79. 10 2 80. 8 3

81. 6 2 82. 6 3 83. 5 3 84. 7 3

85. 2 5 86. 23

87. 37

88. 23

89. 56

90. 2 5

5 91.

3

3 92.

4 7

7

93. 6

2 94. 7x = 95. 3x = −

96. 9x = 97. 6x = − 98. 2x =

99. 5x = 100. 6x = − 101. 32x =

102. 14x = 103. 2x = − 104. 3x =

105. 3x = 106. 5x = − 107. 2x =

108. 3x = 109. 7x = 110. 1x = −

111. 7x =

112. a. The chair is not wood.

b. The rug is brown.

113. a. The photograph is in color.

b. Your homework is not finished.

114. a. It is not cold outside.

b. The bicycle is green.

115. 2 8 16x x+ + 116. 2 4 4x x− +

117. 2 6 9x x− + 118. 2 2 1x x− +

119. 2 18 81x x+ + 120. 2 26 169x x− +

121. 24 16 16x x+ + 122. 29 6 1x x− +

123. 225 60 36x x+ + 124. 225 10 1x x− +

125. 29 48 64x x+ + 126. 24 16 16x x− +

Answers


127. ( )( )2 7x x+ − 128. ( )( )12 11x x− +

129. ( )( )7 4x x− + 130. ( )( )5 3x x− −

131. ( )( )2 3x x− − 132. ( )( )4 9x x+ −

133. ( )( )1 1x x+ − 134. ( )( )3 3x x+ −

135. ( )( )5 5x x+ − 136. ( )( )2 3 4x x− +

137. ( )( )3 5 7x x− − 138. ( )( )5 2 4x x+ +

139. 8 and 3x x= − = −

140. 6 and 2x x= − =

141. 12 and 1x x= − =

142. 9 and 8x x= − = −

143. 5 and 4x x= − =

144. 10 and 7x x= − = −

145. 3 and 1x x= − =

146. 5 and 2x x= − =

147. 11 and 1x x= − =

148. 32

and 5x x= − =

149. 45

and 3x x= − =

150. 713 2 and x x= − =

151. a. 7 words per min

b. 87.5 words

c. 105 words

d. 17.5 words

Chapter 3 3.1 Start Thinking

right triangle; no; no; Because points B and C connect perpendicular lines, you cannot plot either point to make a perpendicular segment or a parallel segment.

3.1 Warm Up

1. Sample answer: BC

2. GE

3. CG

4. , ,AB BC BD

5. Sample answer: andFE FG

6. Sample answer: D


1. ( )4, 11K 2. ( )27, 18J − − 3. ( )21, 2K −

3.1 Practice A

1. andAB CD

2. andAC CD

3. no; AB CD

and by the Parallel Postulate (Post.

3.1), there is exactly one line parallel to AB

through point C.

4. no; They are intersecting lines.

5. 2∠ and 8,∠ 3∠ and 5∠

6. 1∠ and 7,∠ 4∠ and 6∠

7. 1∠ and 5,∠ 2∠ and 6,∠ 3∠ and 7,∠ 4∠ and 8∠

8. 2∠ and 5,∠ 3∠ and 8∠

9. no; By definition, skew lines are not coplaner.

10. 2 pairs; 4 pairs; ( )2 2n − pairs

11. a. AB

and ,CD

AC

and BD

b. AC

and ,CD

BD

and CD

c. 2∠ and 5,∠ 3∠ and 8∠

CA

B

geometry rbc a answers -...

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