final additional practice exam 2nd sem 2018-19 - answers -...

Additional Practice Final Exam #1

Class Name : GEOMETRY RETAKE Instructor Name : Mr. Beckey

Student Name : _____________________ Instructor Note :

1. The triangles below are congruent and their corresponding parts are marked.

Name all the corresponding congruent angles and sides.Then, complete the triangle congruence statement.

(a) (b)

(c)

2. Each part below contains one triangle and one figure that can be adjusted to make a triangle. The markings indicate congruence.

=~∠A ∠ =~AB

=~∠B ∠ =~AC

=~∠C ∠ =~BC

=~BAC

A

B

C

Z

X

Y

© 2019 M cGr aw - H i l l Educat i on. Al l R i ghts R eser ved.Addi t i onal Pr ac t i ce F i nal Exam #1 / 24

2. Each part below contains one triangle and one figure that can be adjusted to make a triangle. The markings indicate congruence.

3.

4.

(a) Adjust the figure on the right to explore the different triangles that can beformed. Then answer the question. Note that the figure will not be graded.

Which of the following is true?

The two triangles must be congruent.

Reason: 'Side-Angle-Side (SAS) Congruence Property' , 'Angle-Side-Angle (ASA) Congruence Property' , 'Side-Side-Side (SSS) Congruence Property' , 'Angle-Angle-Side (AAS) Congruence Property'

The two triangles do not have to be congruent.

(b) Adjust the figure on the right to explore the different triangles that can beformed. Then answer the question. Note that the figure will not be graded.



Reason: 'Side-Angle-Side (SAS) Congruence Property' , 'Angle-Side-Angle (ASA) Congruence Property' , 'Side-Side-Side (SSS) Congruence Property' , 'Angle-Angle-Side (AAS) Congruence Property'


=~∠DFE ∠DFG

=~∠EDF ∠GDF

=~DE DG

D

E

F

G


4.

Given:

Prove:

5. Consider and in the figure below.

Use the figure above to complete the following.

6. Use the given information to prove that .

Given:

Prove:

7.

=~BC DC

=~∠BAC ∠DEC

=~ABC EDC

ABD DCA

(a) and have been separated. Fill in the missing vertex labels.ABD DCA

9

5

3

6

(b) Choose the correct statement below about and .

Then fill in the additional information as necessary.

ABD DCA

The triangles have a common side.

Common side: { , , , , }AB AD BD AC CD

The triangles have a common angle.

Common angle: { , , , , , , , , }∠1 ∠2 ∠3 ∠4 ∠5 ∠6 ∠7 ∠8 ∠9The triangles have neither a common side nor a common angle.

=~RST VUT

=~ST UT

=~RT VT

=~RST VUT

Statement Reason

1 =~ST Given

2 =~R T V T _________________

3 =~∠R T S ∠ Vertical Angles Property

4 =~R ST VUT _________________ 1 2 3

Line(s) Used

A

BC

D

E

B

DA

C

7

98

2

31

56

4

R

ST

U

V


7.

Given:

bisects

Prove:


Given: and are right triangles

Prove:

9.

Given: bisects

Prove:

10. Use the given information to prove that

=~GF DF

GD HE

=~HFG EFD

=~DBA BDC

DBA BDC

=~AD CB

=~DBA BDC

BE AD

AB DE

=~AB DE

~RST RUT

F

D E

GH

A

BC

D

C

A

B

D

E



Given: is a right angle

is a right angle

Prove:

11.

Given: is the midpoint of

Prove:

12. For each pair of triangles, determine whether the Hypotenuse-Leg (HL) Congruence Property can be used to prove that the triangles are congruent.

13. For each part below, use the figure to fill in the blank.

=~RST RUT

∠RTS

∠RTU

=~∠RST ∠RUT

=~RST RUT

Statement Reason

1 is a right angle∠R T S Given

2 is a right angle∠R TU Given

3 =~∠R T S ∠ All right angles are =~ 1 2

4 =~∠R ST ∠RUT _________________

5 =~R T R T _________________

6 =~R ST RUT _________________ 3 4 5

Line(s) Used

R TQ

ST PQ=~TRS QRP

(a) (b)

Can the HL CongruenceProperty be used?

Yes No Yes No

(c) (d)

24 mm 24 mm

40 mm 40 mm


Yes No Yes No

RS

T

U

PQ

S

R

T


13. For each part below, use the figure to fill in the blank. If necessary, you may learn what the markings on a figure indicate.

(a) Find . (b) Find .

______ ______

14. Suppose that is isosceles with base .

Suppose also that and .

Find the degree measure of each angle in the triangle.

15. Use the given information to complete the proof of the following theorem.

The angles opposite the two congruent sides of an isosceles triangle are congruent.

Given:

bisects

Prove:

16. In each figure below, the segments on the sides can be moved.Create the following triangles, if possible.

First figure: A triangle with side lengths of , , and .

Second figure: A triangle with side lengths of , , and .

If there is no triangle possible with the given side lengths, click on "No triangle possible".

17. For each set of three lengths, determine if they can be the side lengths of a triangle.

HI m∠K

=HI =m∠K °

GHI GI

=m∠G +5x 18 ° =m∠ I +4x 26 °

=~FD FEFG ∠DFE

=~∠D ∠E

2 2 66 6 8

G

H

I

7

JK

L

6

6

40°

+5x 18 ° +4x 26 °

H

G I

F

D

E

G


17. For each set of three lengths, determine if they can be the side lengths of a triangle.

18. The side lengths of a triangle are , , and units.

(The figure above gives only one possible configuration for the triangle.)

Determine the possible values of .Write your answer as an inequality.Use only once in your inequality.

19. Order the side lengths , , and from least to greatest.

(Note that the figure is not drawn to scale.)

20. Order the side lengths , , , , and from least to greatest.

(Note that the figure is not drawn to scale.)

21. Compare the given measures by choosing or

LengthsCan be side lengths of

a triangleCannot be side lengths

of a triangle

, 22 , 8 3

, 15 , 2 10

, 16 , 4 17

, 8.9 , 4.9 14.5

15 16 x

x

x

FG GH HF

CD DF CE CF FE

< > =

16

15 x

87°

43°G F

H

31°

80° 20°D F

C

E


21. Compare the given measures by choosing , , or .If there is not enough information, select "Cannot be determined".

If necessary, you may learn what the markings on a figure indicate.

( / / ) Cannot be determined



22. Complete the following indirect proof (proof by contradiction).

Given: Adjacent angles and , formed by the intersection of two lines

Prove: At least one of the angles and has measure or greater Proof:First, we assume that this conclusion is false. In other words, we assume that the contrary statement "_________ {(a) none of the two angles, (b) at least one of the two angles, (c)exactly one of the two angles} has measure ________________ {(a) less than , (b) greater than , (c) or less, (d) or greater}" is ______ {(a) true, (b) false}.

This assumption can be formulated as follows(1) ___ {(a) , (b) , (c) , (d) , (e) } ____

(2) ___ {(a) , (b) , (c) , (d) , (e) } ____

Using (1) and (2) and addition properties of inequalities, we conclude that + ___ {(a) , (b) , (c) , (d) , (e) } .

On the other hand, two adjacent angles form a linear pair. Thus, the last statement contradicts the _________ {(a) Definition of Right Angles, (b) Angle Addition Property, (c) TriangleAngle-Sum Property, (d) Exterior Angles Property, (e) Linear Pair Property}, which states that for a linear pair of angles and , + ___ {(a) , (b) , (c) , (d) , (e) } .

Therefore, the assumption made is ______ {(a) true, (b) false}, and the statement "at least one of the angles and has measure or greater" is _________ {(a) true, (b) false}.

23. Find the value of .

< > =

m∠WYX < > = m∠WYZ

UV < > = DF

AD < > = CD

∠A ∠B

∠A ∠B 90°

90° 90° 90° 90°

m∠A ≥ ≤ = > < °m∠B ≥ ≤ = > < °

m∠A m∠B ≥ ≤ = > < 180°

∠A ∠B m∠A

W

X

Y

Z

19

13

9

9

T

U

V

35°

D

E

F

81°

A

B

CD

71° 70°10 10


23. Find the value of .


If opposite angles of a quadrilateral are congruent, then it is a parallelogram.

By definition, a parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.Use this definition in your proof.

Given:

Prove: is a parallelogram

25. Find the sum of the interior angle measures of a convex -gon (an eleven-sided polygon).

26. Answer the questions about the figures below.

Statement Reason

1 =m ∠A m ∠C Given

2 =m ∠B m ∠D Given

3 =+m ∠A +m ∠B +m ∠C m ∠D 360° Quadrilateral Angle-Sum Property

4 =+m ∠A +m ∠B +m ∠ m ∠ ° Substitution Property (Using 3 lines) 1 2 3

5 =2 +m ∠ m ∠ ° Simplifying 4

6 =+m ∠A m ∠B 180° Multiplication and Division Properties 5

7 ∠ and ∠ are supp. angles Definition of Supplementary Angles 6

8 AD _______________

9 =+m ∠A m ∠D ° Substitution Property (Using 2 lines) 2 6

10 ∠A and ∠D are supp. angles Definition of Supplementary Angles 9

11 AB _______________

12 ABCD is a parallelogram _______________

Line(s) Used

70°137°

x °106°

DA

B C


26. Answer the questions about the figures below.

Figure A

Figure B

Figure C

Which figures are squares?Mark all that apply.

- Figure A- Figure B- Figure C- None of the figures

Which figures are parallelograms?Mark all that apply.


Which figures are rectangles?Mark all that apply.


27. Consider quadrilateral below.

6 mm

4 mm

6 mm

4 mm

4 mm

6 mm

4 mm

6 mm

5 mm

5 mm

5 mm

5 mm


27. Consider quadrilateral below.

Note that has vertices , , , and .Complete the following to determine if is a parallelogram.

28. Three vertices of a parallelogram are shown in the figure below.Give the coordinates of the fourth vertex.

29. Consider parallelogram below.

(a) Find the length of and the length of .

Give exact answers (not decimal approximations).

JM KL

Length of :JM

Length of :KL

(b) Find the slope of and the slope of .JM KL

Slope of :JM

Slope of :KL

(c) From parts (a) and (b), what can we conclude?

The quadrilateral is a parallelogram because it has one pair of opposite sides that are bothcongruent and parallel.

The quadrilateral is a parallelogram because it has one pair of opposite sides that arecongruent, even though those sides are not parallel.

The quadrilateral is a parallelogram because it has one pair of opposite sides that areparallel, even though those sides are not congruent.

The quadrilateral is not a parallelogram.

It cannot be determined if the quadrilateral is a parallelogram.

xx

-1-2-3-4-5-6-7-8-9 1 2 3 4 5 6 7 8 9

yy

-1-2-3-4-5-6-7-8-9

123456789

J

K

L

M

xx

yy

, , 22 66

, , −− 11 −− 33, , 55 −− 55


29. Consider parallelogram below. Use the information given in the figure to find , , and .

30. For each of the following quadrilaterals, select all the properties that must be true.

31. Find the measure of each interior angle and each exterior angle of a regular nonagon (a nine-sided polygon).

32. For the rhombus below, find the measures of , , , and .

_____

_____

_____

_____

33. Parallelogram is shown below. Give the coordinates of .


All sides congruent Four right anglesTwo pairs of parallel

sidesOnly one pair ofparallel sides

Square

Trapezoid

Parallelogram

105°

36°

42x

A B

CD

22

4444 °°

33 44

11

xx

yy

K(0, 0) N(u, 0)

M(?, ?)L(-s, t)



If a quadrilateral is a parallelogram, then its opposite sides are congruent.

By definition, a parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.Use this definition in your proof.

Given: is a parallelogram

Prove:

35. In the rectangle below, , , and . Find and .

36. Given the information marked on the figures below, classify each quadrilateral as a "Parallelogram" or "Not necessarily a parallelogram."

Statement Reason

1 QR ST is a parallelogram Given

2 QR Definition of a Parallelogram 1

3 QT Definition of a Parallelogram 1

4 =~∠ 1 ∠ If lines , then alt. int. s ∠ =~ 2

5 =~∠ 2 ∠ If lines , then alt. int. s ∠ =~ 3

6 =~ _______________

7 =~QRT _______________

8=~QR ST

_______________=~QT R S

Line(s) Used

T4

Q1

R

2

S

3

RR SS

TTUU

VV


36. Given the information marked on the figures below, classify each quadrilateral as a "Parallelogram" or "Not necessarily a parallelogram."

Note that each figure is drawn like a parallelogram, but you should not rely on how the figure is drawn in determining your answers.

___ Parallelogram___ Not necessarily a parallelogram




37. True or False?

a. Every square is a quadrilateral.

b. Every parallelogram is a quadrilateral.

c. Every quadrilateral is a rhombus.

d. Every rectangle with four congruent sides is a square.

38. Given the information marked on each figure below, select all classifications that must be true.

Note that each figure is drawn like a rectangle, but you should not rely on the way the figure is drawn in determining your answers.

___ Quadrilateral___ Parallelogram___ Rectangle



39. The convex polygon below has sides. Find the value of

A

B D

C

R S

UT

V

K

L N

M

G H

JI

G H

I J

A

B

C

D

K

L

M

N


39. The convex polygon below has sides. Find the value of .

40. Consider parallelogram below.

Note that has vertices , , , and .Answer the following to determine if the parallelogram is a rectangle, rhombus, square, or none of these.

(a) Find the slope of and the slope of a side adjacent to .JK JK

Slope of :JK

Slope of side adjacent to :JK

(b) Find the length of and the length of a side adjacent to .


JK JK

Length of :JK

Length of side adjacent to :JK

(c) From parts (a) and (b), what can we conclude about parallelogram

? Check all that apply.GHJK

is a rectangle.GHJK

is a rhombus.GHJK

is a square.GHJK

is none of these.GHJK

82°

x °

61°

147°

129°

xx

-2-4-6-8 2 4 6 8

yy

-2

-4

-6

-8

2

4

6

8

G

H

J

K


Additional Practice Final Exam #1 Answers for class GEOMETRY RETAKE

1.

(a) (b)

(c)

2.

3.

4.

(a) Adjust the figure on the right to explore the different triangles that can beformed. Then answer the question. Note that the figure will not be graded.



Reason: Side-Side-Side (SSS) Congruence Property


(b) Adjust the figure on the right to explore the different triangles that can beformed. Then answer the question. Note that the figure will not be graded.



Reason: Choose one


Statement Reason Line(s) Used

1 =~∠ D F E ∠ D F G Given

2 =~∠ E D F ∠ G D F Given

3 =~D F D F Reflexive Property

4 =~D E F D G F ASA Congruence Property 1 2 3

5 =~D E D G CPCTC Property 4


4.

5.

6.

7.

8.

9.


1 =~B C D C Given

2 =~∠ B A C ∠ D E C Given

3 =~∠ A C B ∠ E C D Vertical Angles Property

4 =~A B C E D C AAS Congruence Property 1 2 3

(a) and have been separated. Fill in the missing vertex labels.

9

5

3

6

(b) Choose the correct statement below about and .Then fill in the additional information as necessary.

The triangles have a common side.

Common side: ▼(Choose one)

The triangles have a common angle.

Common angle:

The triangles have neither a common side nor a common angle.

Statement Reason

1 =~ST UT Given

2 =~R T V T _________________

3 g i v en Vertical Angles Property

4 =~R ST VUT _________________ 1 2 3

Line(s) Used


1 =~G F D F Given

2 bisectsG D H E Given

3 =~H F E F Definition of Segment Bisector 2

4 =~∠ H F G ∠ E F D Vertical Angles Property

5 =~H F G E F D SAS Congruence Property 1 3 4


1 Undefined UndefinedD B A B D C Given

2 =~A D C B Given

3 =~B D B D Reflexive Property

4 =~D B A B D C HL Congruence Property 1 2 3


9.

10.

11.

12.

13. (a) (b)

14.


1 bisectsB E A D Given

2 =~A C D C Definition of Segment Bisector 1

3 A B D E Given

4 =~∠ A B C ∠ D E C If lines , then alt. int. s ∠ =~ 3

5 =~∠ B C A ∠ E C D Vertical Angles Property

6 =~A B C D E C AAS Congruence Property 2 4 5

7 =~A B D E CPCTC Property 6

Statement Reason

1 is a right angle∠R T S Given

2 is a right angle∠R TU Given

3 =~∠R T S ∠R TU All right angles are =~ 1 2

4 =~∠R ST ∠RUT _________________

5 =~R T R T _________________

6 =~R ST RUT _________________ 3 4 5

Line(s) Used


1 is the midpoint ofR T Q Given

2 =~T R Q R Definition of Midpoint 1

3 S T P Q Given

4 =~∠ S T R ∠ P Q R If lines , then alt. int. s ∠ =~ 3

5 =~∠ T R S ∠ Q R P Vertical Angles Property

6 =~T R S Q R P ASA Congruence Property 2 4 5

(a) (b)


Yes No Yes No

(c) (d)

24 mm 24 mm

40 mm 40 mm


Yes No Yes No


14.

15.

16.

17.

18.

19.

20.

21.


1 =~F D F E Given

2 bisectsF G ∠ D F E Given

3 =~∠ D F G ∠ E F G Definition of Angle Bisector 2

4 =~F G F G Reflexive Property

5 =~F D G F E G SAS Congruence Property 1 3 4

6 =~∠ D ∠ E CPCTC Property 5

LengthsCan be side lengths of

a triangleCannot be side lengths

of a triangle

, 22 , 8 3

, 15 , 2 10

, 16 , 4 17

, 8.9 , 4.9 14.5

No triangle possible

8

6 6


21.

Cannot be determined



22. Proof:First, we assume that this conclusion is false. In other words, we assume that the contrary statement "none of the two angles has measure 90° or greater" is true.

This assumption is equivalent to the following two statements:(1) and(2)

Using (1) and (2) and addition properties of inequalities, we conclude that + .

On the other hand, two adjacent angles form a linear pair. Thus, the last statement contradicts the Linear Pair Property, which states that for a linear pair of angles and , + .

Therefore, the assumption made is false, and the statement "at least one of the angles and has measure or greater" is true.

23.

24.

W

X

Y

Z

19

13

9

9

T

U

V

35°

D

E

F

81°

A

B

CD

71° 70°10 10


24.

25. Sum of the interior angle measures

26.

Which figures are squares?Mark all that apply.

- Figure C

Which figures are parallelograms?Mark all that apply.

- Figures A - Figure B - Figure C

Which figures are rectangles?Mark all that apply.

- Figure B - Figure C

27.

28.

29.

Statement Reason

1 =m ∠A m ∠C Given

2 =m ∠B m ∠D Given

3 =+m ∠A +m ∠B +m ∠C m ∠D 360° Quadrilateral Angle-Sum Property

4 =+m ∠A +m ∠B +m ∠A m ∠B 360° Substitution Property (Using 3 lines) 1 2 3

5 =2 +m ∠A m ∠B 360° Simplifying 4

6 =+m ∠A m ∠B 180° Multiplication and Division Properties 5

7 ∠A and ∠B are supp. angles Definition of Supplementary Angles 6

8 AD BC If cons. int. s are supp., then lines ∠ 7

9 =+m ∠A m ∠D 180° Substitution Property (Using 2 lines) 2 6

10 ∠A and ∠D are supp. angles Definition of Supplementary Angles 9

11 AB CD If cons. int. s are supp., then lines ∠ 10

12 ABCD is a parallelogram Definition of a Parallelogram 8 11

Line(s) Used

(a) Find the length of and the length of .


JM KL

Length of :JM 13

Length of :KL 13

(b) Find the slope of and the slope of .JM KL

Slope of :JM −2

3

Slope of :KL −2

3

(c) From parts (a) and (b), what can we conclude?

The quadrilateral is a parallelogram because it has one pair of opposite sides that are bothcongruent and parallel.

The quadrilateral is a parallelogram because it has one pair of opposite sides that arecongruent, even though those sides are not parallel.

The quadrilateral is a parallelogram because it has one pair of opposite sides that areparallel, even though those sides are not congruent.

The quadrilateral is not a parallelogram.

It cannot be determined if the quadrilateral is a parallelogram.


29.

30.

31. Measure of each interior angleMeasure of each exterior angle

32.

33.

34.

35.

36.

All sides congruent Four right anglesTwo pairs of parallel

sidesOnly one pair ofparallel sides

Square

Trapezoid

Parallelogram

Statement Reason

1 QR ST is a parallelogram Given

2 QR ST Definition of a Parallelogram 1

3 QT R S Definition of a Parallelogram 1

4 =~∠ 1 ∠ 4 If lines , then alt. int. s ∠ =~ 2

5 =~∠ 2 ∠ 3 If lines , then alt. int. s ∠ =~ 3

6 =~TR TR Reflexive Property

7 =~QRT STR ASA Congruence Property 4 5 6

8=~QR ST

CPCTC Property 7

=~QT R S

Line(s) Used


36.

___ Parallelogram

___ Not necessarily a parallelogram

___ Not necessarily a parallelogram

___ Parallelogram

37.

a. Every square is a quadrilateral.True

b. Every parallelogram is a quadrilateral.True

c. Every quadrilateral is a rhombus.False

d. Every rectangle with four congruent sides is a square.True

38.

39.

40.

(a)G H

I J

(b)

A

B

C

D (c)

K

L

M

N

Quadrilateral Quadrilateral Quadrilateral

Parallelogram Parallelogram Parallelogram

Rectangle Rectangle Rectangle

A

B D

C

R S

UT

V

K

L N

M

G H

JI


40.

(a) Find the slope of and the slope of a side adjacent to .JK JK

Slope of :JK −7

3

Slope of side adjacent to :JK1

5

(b) Find the length of and the length of a side adjacent to .


JK JK

Length of :JK 58

Length of side adjacent to :JK 26

(c) From parts (a) and (b), what can we conclude about parallelogram

? Check all that apply.GHJK

is a rectangle.GHJK

is a rhombus.GHJK

is a square.GHJK

is none of these.GHJK


final additional practice exam 2nd sem 2018-19 - answers -...

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