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geometry 45 to79

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GEOMETRY

Congruence, Similarity,

Right Triangles, and

Trigonometry

MAFS.912.G-CO.1.1

Drag the words to the correct descriptions. Not all options will be used.

Circle

the set of points extending in one

direction in a plane

.
.

Line

.

the set of points between two endpoints

in a plane

.

the set of all points the same distance

from a given point in a plane

Angle

Line Segment

Ray

.

the set of points extending in opposite

directions in a plane

MAFS.912.G-CO.1.1

Classify each statement in the table as correct or incorrect.

Select one cell per row.

Correct

If two lines in the same plane do not intersect, the lines

must be parallel.

If two line in the space do not intersect, the lines must be

parallel.

If two lines are parallel, the lines must lie in the same

plane.

Incorrect

MAFS.912.G-CO.1.1

Points ?, ?, and ? are distinct points, and ?? = ??. Which of these

statements must be true? Select all that apply.

? ?, ?, and ? are coplanar.

? ?, ?, and ? are collinear.

?

? K is the midpoint of ??.

? ?? ? ??

? The measure of ???? is 90°.

MAFS.912.G-CO.1.2

Pentagon ABCDE is shown in the ?? -coordinate plane.

Pentagon ABCDE will be rotated 90° clockwise about the

point (1, 1) to form pentagon ABCDE.

Choose the graph that shows the correct placement of

ABCDE after the transformation.

A

B

C

D

MAFS.912.G-CO.1.2

Triangle ??? is shown in the ??-coordinate

plane. It will be rotated 90 degrees clockwise

about the origin to form triangle ???.

Select the correct orientation of ??? and

place it correctly in the coordinate plane.

MAFS.912.G-CO.1.2

A triangle is shown on the coordinate

grid.

Use the Connect Line tool to draw the

triangle after a transformation

following the rule ?, ? ? ? – 4, ? + 3

MAFS.912.G-CO.1.2

The pre-image of ?ABC and its image ?A’B’C’ are shown on the

coordinate plane.

Which rule describes the transformation

represented in the graph?

A.

1

?

2

1

2

B.

2? + 2, 2? – 3

C.

1

?

2

D.

2? – 2, 2? + 3

+ 2, ? – 3

1

2

– 2, ? + 3

MAFS.912.G-CO.1.4

A rotation about Point D maps Point ? to ?’ and Point ? to ?’.

Which statement must be true? Select

all that apply.

? ?? = ?’ ? ‘

? ?? = ?’ ?

? ????? = ?????

? ???DB’ = ?????

? ????? = ?????

? If Point ? is ?, ? , then Point ? is -?, ?

? If Point B is (?, ?), then Point ? is (?, ?)

MAFS.912.G-CO.1.4

On a coordinate plane, each point on ?? is defined by coordinates ?, ? . The segment

is rotated 90° counterclockwise to create ?’?’. Which of the following defines a point

on ?’?’ that corresponds to a point on ??.

A. ?, -?

B. -?, ?

C. ?, ?

D. -?, ?

MAFS.912.G-CO.1.3

Which figure always has exactly four lines of reflection that map the

figure onto itself?

A. square

B. rectangle

C. regular octagon

D. equilateral triangle

MAFS.912.G-CO.1.3

Select the values that correctly complete the sentence about the

symmetry of a regular pentagon.

5

A regular pentagon has _____

lines of symmetry and

1

2

5

6

72 degree

has ___________

rotational symmetry.

60 degree

72 degree

108 degree

540 degree

MAFS.912.G-CO.1.3

Trapezoid RSTU is shown.

Write the equation for the line that would map the

trapezoid onto itself.

MAFS.912.G-CO.1.5

In the diagram below, ???? ? ????.

Which sequence of transformations maps ????

onto ?????

A. a reflection over the ? -axis followed by a

translation

B. a reflection over the ? -axis followed by a

translation

C. a rotation of 180° about the origin followed

by a translation

D. a counterclockwise rotation of 90° about

the origin followed by a translation

MAFS.912.G-CO.1.5

Quadrilateral ???? is shown on the

coordinate grid.

Keisha reflects the figure across the

line ? = ? to create ?’ ? ‘ ?’ ? ‘ .

Use the Connect Line tool to draw

quadrilateral ?’ ? ‘ ?’ ? ‘ .

MAFS.912.G-CO.1.5

The right triangle in the coordinate plane is rotated 270° clockwise about the point

(2, 1) and then reflected across the ? -axis to form ? ?’ ?’ ? ‘ .

Drag and drop the appropriate orientation for triangle ??? into the correct

position on the coordinated plane.

MAFS.912.G-CO.1.5

?ABC is shown on the coordinate plane.

After rotating ?ABC 180° about the origin and

then reflecting it over the x-axis, what are the

coordinates of ?A?B?C??

A. A?(2, 6), B?(5, 4), C?(2, 1)

B. A?(6, 2), B?(4, 5), C(1, 2)

C. A?(-2, 6), B?(5, 4), C?(-2, 1)

D. A?(6, -2), B?(4, -5), C?(1, -2)

MAFS.912.G-CO.1.5

Given quadrilateral ABCD, what are the coordinates for the resulting image,

A?B?C?D?, after the two transformations listed?

First transformation: Rotate 90° clockwise about the origin.

Second transformation: Translate ?, ? ? (? + 1, ? – 2).

Enter the coordinates for the resulting image

A?B?C?D? in the boxes.

A =

B =

C =

D =

MAFS.912.G-CO.1.5

Triangle EGF is graphed below.

Triangle EGF will be rotated 90° counterclockwise

around the origin and will then be reflected

across the y-axis, producing an image triangle.

Which additional transformation will map the

image triangle back onto the original triangle?

A. rotation 270° counterclockwise around the

origin

B. rotation 180° counterclockwise around the

origin

C. reflection across the line ? = ?

D. reflection across the line ? = ?

MAFS.912.G-CO.1.5

Parallelogram ???? has midpoints ?, ?, ?, ? marked on the sides as shown.

Which rigid motion could be applied to ????

to show that ???? ? ?????

A. reflection over ??

B. reflection over ??

C. rotation 90° clockwise about the

intersection point of ?? and ??.

D. rotation 180° clockwise about the

intersection point of ?? and ??.

MAFS.912.G-CO.2.6

Triangle ??? is shown in the ??-coordinate plane. The

triangle will be rotated 180° clockwise around the point

(3, 4) to create triangle ???. Indicate whether each of

the listed features of the image will or will not be the

same as the corresponding feature in the original

triangle by selecting the appropriate box in the table.

the

coordinates

of A

will be the

same

will not be

the same

the

coordinates

of C

the

perimeter

of ??’ ?’ ?’

the area

of ??’ ?’ ?’

the

measure of

?B

the slope

of ?’ ?’

MAFS.912.G-CO.2.6

In the illustration, figure I and figure II are isosceles trapezoids.

Tran makes a conjecture that figure I is congruent to figure II.

Select each transformation or combination of transformations

that can help Tran prove his conjecture.

Select all that apply.

?

?

?

?

?

?

Rotate figure I 60° clockwise around point P.

Rotate figure I 120° clockwise around point P.

Reflect figure I across line ?, and then reflect the image across line ?.

Reflect figure I across line ?, and then reflect the image across line ?.

Rotate figure I 180° around point P, and then reflect the image across line ?.

Rotate figure I 120° counterclockwise around point P, and then reflect the

image across line ?.

MAFS.912.G-CO.2.7

All corresponding sides and angles of ? ??? and ? ??? are congruent.

Select all the statements that must be true.

? There is a reflection that maps ?? to ??.

? There is a dilation that maps ? ??? to ? ???.

? There is a translation followed by a rotation that maps ?? to ??.

? There is a sequence of rigid motions that maps ? ??? to ? ???.

? There is not necessarily a sequence of rigid motions that maps ? ??? to ? ???.

MAFS.912.G-CO.2.7

?ABC and ?DEF are plotted on the coordinate plane shown.

Which conclusions can be made about ?ABC and ?DEF if ?ABC

is mapped onto ?DEF by reflecting ?ABC over the ?-axis and

reflecting it over the ?-axis? Select all that apply.

? ?ABC ? ?DEF

? The corresponding sides are proportional:

??

??

=

??

??

=

??

??

.

? Reflecting ?ABC across the ?-axis and then the ?-axis

yields the same transformation as rotating ?ABC 90°

counterclockwise around the origin.

? ?ABC ~?DEF

? ?ABC is acute, but ?DEF is obtuse.

? All corresponding sides are congruent.

MAFS.912.G-CO.2.8

Aniyah has repeatedly stenciled the triangle

shape shown on her bedroom wall.

Her best friend, Daniela, wants to copy the exact same shape on her bedroom

wall. Which statement has sufficient information about the triangle for Aniyah to

give to Daniela to guarantee Daniela will have the exact same triangle?

A. Construct a triangle with angles whose measures are 33°, 42°, and 105°.

B. Construct a triangle with sides of measure 22 inches and 27 inches where the

included angle is 105°.

C. Construct a triangle with sides of measure 39 inches and 22 inches and a

nonincluded angle of measure 33°.

D. Construct a triangle with a 105° angle opposite from a side of length 39 inches

where the remaining two sides differ in length by 5 inches.

MAFS.912.G-CO.3.9

Quadrilateral ???? is shown.

For what value of ? will line ?? be the

perpendicular bisector of segment ???

A. 5

B. 9

C. 14

D. 17

MAFS.912.G-CO.3.9

Mikayla is using the following information to prove

that ???? and ???? are complementary angles in the

diagram shown. Part of her proof is shown below.

Given: The ray ?? bisects ???? and the ray ??

bisects ????.

Statements

Which statements could be used

to complete Mikaylas proof?

Reasons

A.

4. 2 · m???? = 2 · m????

5. ????? = ?????

???? and ???? are

supplementary angles

TP is a line

2

????? + m???? = 180°

Definition of supplementary

angles

B.

4. 2 · m???? = 2 · m????

5. ????? + m???? = 90°

3

????? = 2 · m????

????? = 2 · m????

Property of angle bisectors

C.

4. 2 · ????S + 2 · m???? = 180°

5. ????? = ?????

D.

4. 2 · ????S + 2 · m???? = 180°

5. ????? + m???? = 90°

1

Substitution

4

Division property of equality

5

6

???? and ???? are

complementary angles

Definition of complementary

angles

MAFS.912.G-CO.3.9

In the figure shown, ?? intersects ?? and ?? at points ? and ?, respectively.

Part A

Given:???? ? ????

Prove:???? ? ????

Select from the drop-down menus to support each line

of the proof.

Statement: ???? ? ???? Reason:

Choose one…

Statement: ???? ? ???? Reason:

Choose one…

Statement: ???? ? ???? Reason:

Choose one…

MAFS.912.G-CO.3.9

In the figure shown, ?? intersects ?? and ?? at points ? and ?, respectively.

Part B

Given:m???? = ?????

Prove:m???? + ????? = 180°

Select from the drop-down menus to support each line

of the proof.

Statement: ????? = ????? Reason:

Choose one…

Statement: m???? + ????? = 180° Reason:

Choose one…

Statement: m???? + ????? = 180° Reason:

Choose one…

MAFS.912.G-CO.3.9

Given: ?? is the perpendicular bisector of ??.

Prove: J is equidistant from L and K

Statements

Reasons

?? is the perpendicular bisector of

??

Given

???? and ???? are right angles

Definition of right angle

?? ? ??

???? ? ????

???? ? ????

All right angles are congruent

?? ? ??

Definition of bisector

???? ? ????

Reflexive property of equality

Definition of right angle

???? ? ????

SAS

Definition of perpendicular

? ? ??

??

Corresponding parts of congruent

triangles are congruent

J is equidistant from L and K

Definition of equidistant

?? ? ??

Corresponding parts of congruent

triangles are congruent

MAFS.912.G-CO.3.10

Roads connecting the towns of Oceanside, River City, and Lake View form

a triangle. The distance from Oceanside to River City is 38 kilometers. The

distance from River City to Lake View is 26 kilometers. What is the

smallest possible whole number of kilometers between Lake View and

Oceanside?

Enter your answer in the box.

MAFS.912.G-CO.3.10

Which of the following sets of lengths could form the sides of a triangle?

Choose all that apply.

? 5, 9, 7

? 4, 8, 3

? 11, 11, 22

? 2, 5, 4

? 7, 9, 12

? 10, 8, 2

MAFS.912.G-CO.3.10

Complete the proof by filling in the missing

reasons from the Reasons Bank below.

Given: ?? ? ??

Reasons Bank

Prove: ????~????

AA Similarity Postulate

Statements

?? ? ??

?? ? ??

???? ? ????

????~????

Reasons

Given

If 2 parallel lines are cut by a

transversal, then the alternate

interior angles are congruent

Vertical angles are congruent

AA Similarity Postulate

SAS Similarity Theorem

SSS Similarity Theorem

Vertical angles are congruent

If 2 parallel lines are cut by a transversal, then

the alternate interior angles are congruent.

If 2 parallel lines are cut by a transversal, then

the corresponding angles are congruent

MAFS.912.G-CO.3.11

Parallelogram ABCD is shown.

What are the values of ? and ??

Enter the correct values in the boxes.

?=

?=

MAFS.912.G-CO.3.11

In quadrilateral ????, ?? and ?? intersect at point ?, so that ?? ? ?? and

?? is parallel to ??. Which additional information is needed to prove that

figure ???? is a rectangle?

A. ?? ? ??

B. ?? ? ??

C. ???? ? ????

D. ???? ? ????

MAFS.912.G-CO.3.11

Which set of statements would describe a parallelogram that can always

be classified as a rhombus?

I. Diagonals are perpendicular bisectors of each other.

II. Diagonals bisect the angles from which they are drawn.

III. Diagonals form four congruent isosceles right triangles.

A. I and II

B. I and III

C. II and III

D. I, II, and III

MAFS.912.G-CO.3.11

Given: Quadrilateral ???? is a parallelogram.

Prove: ?? = ?? and ?? = ??

Statements

Reasons

Quadrilateral ???? is

a parallelogram

Given

S.S.S

?? ? ??

?? ? ?? and ?? ? ??

Definition of a parallelogram

S.A.S

?? ? ??

A.S.A

?? ? ??

???? ? ????

???? ? ????

When two parallel lines are cut by a transversal,

alternate interior angles are congruent

?? ? ??

Opposite sides of a parallelogram

are congruent

When two parallel lines are cut by a transversal,

corresponding angles are congruent

???? ? ????

A.S.A

?? ? ??

?? ? ??

Corresponding parts of congruent

triangles are congruent

When two parallel lines are cut by a transversal,

same side interior angles are congruent

?? = ?? and ?? = ??

Definition of congruent segments

When two parallel lines are cut by a transversal,

alternate interior angles are congruent

MAFS.912.G-CO.3.11

Given: ABCD is a parallelogram

Prove: ?1 and ?2 are supplementary.

Proof:

It is given that ABCD is a parallelogram with angles

named, in consecutive order, ?1, ?2, ?3, and ?4.

Opposite angles of a parallelogram are congruent. Thus

m?1 = m?3 and __________

??2 = ??4

definition of congruence

__________

by the ______________.

The sum of the interior angles of any quadrilateral is

360°, so m?1 + ?2 + ?3 + ?4 = 360° .

substitution property

??1 + ??1 + ??2 + ??2 = 360°

Using the _______________,

_______________.

Therefore, 2 m?1 + ?2 = 360°

??1 + ??2 = 180°

division property

By the _______________,

_______________.

Therefore, ?1 and ?2 are supplementary.

m?1 = ??2

m?2 = ??4

m?1 = ??3

m?2 = ??3

m?1 + ??2 = 180°

m?1 + ??1 + ??2 + ??2 = 360°

m?3 + ??4 + ??3 + ??4 = 360°

Definition of congruence

Definition of equality

Division property of equality

Substitution property of equality

Transitive property of equality

MAFS.912.G-CO.3.11

The figure shows parallelogram ???? with ?? = 16.

Let ?? = ? 2 – 48 and let ?? = 2?. What are the

lengths of ?? and ???

?? =

?? =

Parallelogram ABCD is a __________________

because the ____________________.

MAFS.912.G-CO.3.11

Three vertices of parallelogram PQRS are shown:

? 8, 5 , ? 5,1 , ? 2,5

Place the statements and reasons in the table to complete the proof that shows

that parallelogram PQRS is a rhombus.

Statements

Reasons

?? = 5

Pythagorean Theorem

?? = 5

Pythagorean Theorem

?? = ??

Substitution

?? = ??

Definition of congruent lines

?? = ??

Property of a parallelogram

?? ? ??

Property of a parallelogram

Parallelogram PQRS

is a rhombus

Definition of a rhombus

?? = 5

?? = 7

?? = 5

?? = 5

?? = 7

?? = 7

???? = 90°

?? ? ??

Pythagorean

Theorem

Definition of

perpendicular lines

Property of a

parallelogram

Definition of parallel

lines

MAFS.912.G-CO.4.12

Which of the following best describes the construction?

A. ?? < ??
B. ? is the midpoint of ?.
C. ?? ? ??
D. ? is the midpoint of ?.
MAFS.912.G-CO.4.12
A student is working on a geometric construction.
If ?? is drawn, what geometric construction is shown?
A. angle bisector
B. copying an angle
C. perpendicular bisector
D. measuring an angle
MAFS.912.G-CO.4.12
The figure above shows the construction of the angle bisector of angle AOB
using a compass. Which of the following statements must always be true in the
construction of the angle bisector?
Classify each statement in the table as correct
or incorrect.
Select one cell per row.
Correct
?? = ??
?? = ??
?? = ??
?? = ??
?? = ??
Incorrect
MAFS.912.G-CO.4.12
The figure shows line ?, points ? and ? on line ?, and point ?
not on line ?. Also shown is ray ??.
Part A
Consider the partial construction of a
line parallel to r through point Q. What
would be the final step in the
construction?
A.
B.
C.
D.
draw
draw
draw
draw
a
a
a
a
line
line
line
line
through
through
through
through
P and S
Q and S
T and S
Wand S
Part B
Once the construction is complete,
which of the reasons listed contribute
to proving the validity of the
construction?
A. When two lines are cut by a
transversal and the corresponding
angles are congruent, the lines are
parallel.
B. When two lines are cut by a
transversal and the verti ...
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