Practice A Isosceles and Equilateral Triangles

[Pages:2]Name

Date

LESSON Practice A 4-8 Isosceles and Equilateral Triangles

Name the parts of the figure that match the vocabulary words.

1. base:

8

2. legs:

and

3. base angles:

and

4. vertex angle:

:

9

Fill in the blanks in Exercises 5?8 to complete each theorem.

5. If a triangle is equilateral, then it is

.

6. If two angles of a triangle are congruent, then the sides those angles are congruent.

7. If two sides of a triangle are congruent, then the opposite those sides are congruent.

8. If a triangle is equiangular, then it is

.

9. A forest ranger in Grand Canyon National Park wants to find the minimum distance across the canyon. She finds a place in the Marble Canyon area of the park where the sides seem close together. She takes measurements and draws this figure. Find the distance AB. (Hint: The angles in an equiangular triangle measure 60.)

Find each value.

$

&

50? %

) '

6.3 (

10. mD

+ * 60?

,

11. GI

2

1 41?2 yd 0

12. mL

3 50? 4

5

14. mU

13. RQ

.

40

5T

-

,

15. t

Copyright ? by Holt, Rinehart and Winston. All rights reserved.

59

Class

" 60? 60? ! 693 ft #

Holt Geometry

LESSON Practice A 4-8 Isosceles and Equilateral Triangles

Name the par_ ts of the figure that match the vocabulary words.

1. base: _ ZY

_

8

2. legs: XY and XZ

3. base angles: Z and Y

4. vertex angle: X

:

9

Fill in the blanks in Exercises 5?8 to complete each theorem.

5. If a triangle is equilateral, then it is

equiangular .

6. If two angles of a triangle are congruent, then the sides

opposite

those angles are congruent.

7. If two sides of a triangle are congruent, then the

angles

opposite those sides are congruent.

8. If a triangle is equiangular, then it is

equilateral .

"

9. A forest ranger in Grand Canyon National Park wants to find

the minimum distance across the canyon. She finds a place

in the Marble Canyon area of the park where the sides seem close together. She takes measurements and draws this figure. Find the distance AB. (Hint: The angles in an equiangular triangle measure 60.)

60? 60? ! 693 ft #

693 ft

Find each value.

$

&

50? %

10. mD 50

+ * 60?

,

12. mL 60

) '

6.3 (

11. GI 6.3

1

2

41?2 yd

0

4 _1_ yd 13. RQ 2

3 50? 4

5

14. mU 65

.

40

5T

-

15. t

,

8

LESSON Practice B 4-8 Isosceles and Equilateral Triangles

An altitude of a triangle is a perpendicular segment from a vertex

(

to the line containing the opposite side. Write a paragraph proof

that the altitude to the base of an isosceles triangle bisects the base.

_ __ _

1. Given: HI HJ, HK IJ

_

_

)

+

*

Prove: HK bisects IJ.

_

_

Possible answer: It is given that HI is congruent to HJ, so I must be con-

gruent to J by the Isosceles Triangle Theorem. IKH and JKH are both

right angles by the definition of perpendicular lines, and_all right angles are_

congruent. Thu_ s by AAS, _HKI is congruent to HKJ. IK is congruent to KJ

by CPCTC, so HK bisects IJ by the definition of segment bisector.

2. An obelisk is a tall, thin, four-sided monument that tapers to a pyramidal top. The most well-known obelisk to Americans is the Washington Monument on the National Mall in Washington, D.C. Each face of the pyramidal top of the Washington Monument is an isosceles triangle. The height of each triangle is 55.5 feet, and the base of each triangle measures 34.4 feet. Find the length, to the nearest tenth of a foot, of one of the two equal legs of the triangle.

Find each value.

8

!

1

1

58.1 ft

9

:

3. mX

45

"

#

4. BC

2

1

N2

2

0 3N 18

' ( 28? )

+

*

5. PQ 36 or 9

6. mK

76

&

%

(30T 20)?

$

7. t

_4_ 3

5 11N?

3 3.5N?

4

8. n

10

"

!

$

#

9. mA

30

1 3

X

0

4

2

10. x

89

Copyright ? by Holt, Rinehart and Winston. All rights reserved.

59

Holt Geometry

Copyright ? by Holt, Rinehart and Winston. All rights reserved.

60

Holt Geometry

LESSON Practice C 4-8 Isosceles and Equilateral Triangles

Draw a figure, name the coordinates, and write a coordinate proof.

_

1. Given: ABC is an isosceles triangle. D is the midpoint of the base BC.

__

Y

Prove: AD BC

!B

Possible answer: ABC is an isosceles

triangle with vertices A (0, b), B(a, 0),

_

X

and C(a, 0). D is the midpoint of BC,

#A $ "A

so

D

has

coordinates

(0,0).

The

slope

of

_

AD

is

b_____0_ 00

_b_ 0

,

so

the

slope

is

undefined. A line with an undefined slope is a vertical line. The slope

of

_

BC

is

__0____0___ _ a (a)

_0__ 2a

0. A

_

line

with

a

zero slope is

__

a

horizontal

line.

Because AD is vertical and BC is horizontal, AD BC.

A soccer ball is covered by a pattern of regular pentagons and regular

hexagons. On a flat surface (like this page), the edges of the shapes do

not meet exactly, but on a sphere (like a ball), they cover the entire shape

without any gaps. Use the figure for Exercises 2 and 3.

2. A regular pentagon has five congruent sides and five 108 angles. Use the figure and your knowledge of isosceles triangles to find the values of x, y, and z.

x 36; y 72; z 36

3. A regular hexagon has six congruent sides and six 120 angles. If a regular hexagon has side lengths of , use the figur_ e and your knowledge of isosceles triangles to find the length of diagonal FC.

2

Z? 108?

X? Y?

!"

&

#

% $

Name the coordinates of vertex Y. Write a coordinate proof.

_

4. Given: XYZ is an equilate_ ral triangle. A is the midp_ oint of XY.

B is the midpoint of YZ. C is the midpoint of XZ.

Y 9

!

"

Prove: ABC, XAC, YAB, and CBZ are congruent equilateral triangles.

8(0, 0)

X # :(2A, 0)

Possible answer: The coordinates of Y are (a, 3a). The Midpoint Formula

shows that the

B

_3_ 2

a,

__3_ 2

a

,

midpoints and C (a,

of the sides of XYZ are 0). The Distance Formula

A

_1_ 2

a,

__3_ 2

gives these

a, distances:

AX AY AC AB BC XC BY BZ CZ a. Thus by SSS,

ABC XAC YAB CBZ, and the triangles are equilateral.

Copyright ? by Holt, Rinehart and Winston. All rights reserved.

61

Holt Geometry

LESSON Review for Mastery 4-8 Isosceles and Equilateral Triangles

Theorem Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite the sides are congruent.

Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Examples

R

T

S

_ _

If RT RS, then T S.

L

N

M

_ _

If N M, then LN LM.

You can use these theorems to find angle measures in isosceles triangles.

Find mE in DEF. mD mE

Isosc. Thm.

D 5x ?

E (3x 14)?

5x ? (3x + 14)?

Substitute the given values.

2x 14

Subtract 3x from both sides.

F

x 7

Divide both sides by 2.

Thus mE 3(7) 14 35?.

Find each angle measure.

B 78?

A

C

1. mC

51?

Q

86?

P

R

2. mQ

47?

H 8x ?

G

(6x 18)?

3. mH

J

72?

L

2x ? N

4. mM

(x 30)? M

60?

Copyright ? by Holt, Rinehart and Winston. All rights reserved.

62

Holt Geometry

Copyright ? by Holt, Rinehart and Winston. All rights reserved.

8 1 001-082_Go08an_CRF_c04.indd 62

Holt Geometry 4/12/07 12:45:28 PM

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