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FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS Course: Formal Geometry S1 (#2215) 2016-2017

Released 8/22/16

Instructional Materials for WCSD Math Common Finals

The Instructional Materials are for student and teacher use and are aligned

to the 2016-2017 Course Guides for the following course:

Formal Geometry S1 (#2215)

When used as test practice, success on the Instructional Materials does not

guarantee success on the district math common final.

Students can use these Instructional Materials to become familiar with the

format and language used on the district common finals. Familiarity with

standards and vocabulary as well as interaction with the types of problems

included in the Instructional Materials can result in less anxiety on the part

of the students. The length of the actual final exam may differ in length

from the Instructional Materials.

Teachers can use the Instructional Materials in conjunction with the course

guides to ensure that instruction and content is aligned with what will be

assessed. The Instructional Materials are not representative of the depth

or full range of learning that should occur in the classroom.

*Students will be allowed to use a

non-programmable scientific calculator

on Formal Geometry Semester 1 and

Formal Geometry Semester 2 final

exams.

Released 8/22/16

Formal Geometry Reference Sheet

Note: You may use these formulas throughout this entire test.

Linear Quadratic

Slope 𝑚 =𝑦2 − 𝑦1

𝑥2 − 𝑥1

Vertex-Form 𝑦 = 𝑎(𝑥 − h)2 + 𝑘

Midpoint 𝑀 = (𝑥1 + 𝑥2

2,𝑦1 + 𝑦2

2) Standard Form 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐

Distance 𝑑 = √(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2 Intercept Form 𝑦 = 𝑎(𝑥 − 𝑝)(𝑥 − 𝑞)

Slope-Intercept Form 𝑦 = 𝑚𝑥 + 𝑏

Exponential Probability

(h, k) Form 𝑦 = 𝑎𝑏𝑥−h + 𝑘 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵)

𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵|𝐴)

𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)

Volume and Surface Area

𝑉 = 𝜋𝑟2ℎ

𝑆𝐴 = 2(𝜋𝑟2) + ℎ(2𝜋𝑟)

𝑉 =4

3𝜋𝑟3

𝑆𝐴 = 4𝜋𝑟2

𝑉 =1

3𝜋𝑟2ℎ

𝑆𝐴 = 𝜋𝑟2 +1

2(2𝜋𝑟 ∙ 𝑙)

𝑉 =1

3𝐵ℎ

𝑆𝐴 = 𝐵 +1

2(𝑃𝑙)

Where 𝐵 =base area

and 𝑃 =base perimeter


Released 8/22/16

Multiple Choice: Identify the choice that best completes the statement or answers the question. Figures are not necessarily drawn to scale.

1. Identify which of the following is the best name for the figure formed by the coordinates:

𝐴(−1, −4), 𝐵(1, −1), 𝐶(2, −2).

A. scalene triangle C. equilateral triangle

B. isosceles triangle D. obtuse triangle

2. A pilot is flying an airplane on a straight path from Norfolk to Madison. On the trip, the

pilot stops to refuel exactly halfway in between at Columbus and decides to program the

autopilot for the rest of the trip. The pilot knows the coordinates for Norfolk are

(36.9, −76.3) and the coordinates for Columbus are (39.9, −83.0). What coordinates

should the pilot use for Madison?

A. (−1.5, −3.3) C. (33.9, −69.6)

B. (61.5, 56.6) D. (42.9, −89.7)

3. In the diagram below, 𝑅 is the midpoint of 𝐴𝐵̅̅ ̅̅ . 𝑇 is the midpoint of 𝐴𝐶̅̅ ̅̅ . 𝑆 is the midpoint

of 𝐵𝐶̅̅ ̅̅ . Find the area of ∆𝑅𝑆𝑇 and 𝐴𝐵.

A. Area of ∆𝑅𝑆𝑇 = 4; 𝐴𝐵 ≈ 4√5

B. Area of ∆𝑅𝑆𝑇 = 8; 𝐴𝐵 ≈ 4√5

C. Area of ∆𝑅𝑆𝑇 = 4; 𝐴𝐵 ≈ 8√5

D. Area of ∆𝑅𝑆𝑇 = 8; 𝐴𝐵 ≈ 8√5


Released 8/22/16

4. Given the coordinates below, compare 𝑅𝑆̅̅̅̅ and 𝑋𝑌̅̅ ̅̅ and determine which of the following

statements is true:

𝑅(−2, −7)

𝑆(5, −1)

𝑋(−3, 3)

𝑌(6, −1)

A. The midpoints of 𝑅𝑆̅̅̅̅ and 𝑋𝑌̅̅ ̅̅ have the same 𝑥-coordinate.

B. The midpoints of 𝑅𝑆̅̅̅̅ and 𝑋𝑌̅̅ ̅̅ have the same 𝑦-coordinate.

C. The length of 𝑅𝑆̅̅̅̅ and the length of 𝑋𝑌̅̅ ̅̅ are the same.

D. The length of 𝑅𝑆̅̅̅̅ is longer than the length of 𝑋𝑌̅̅ ̅̅ .

5. Given the following:

∠𝐵 is a complement of ∠𝐴

∠𝐶 is a supplement of ∠𝐵

∠𝐷 is a supplement of ∠𝐶

∠𝐸 is a complement of ∠𝐷

∠𝐹 is a complement of ∠𝐸

∠𝐺 is a supplement of ∠𝐹

Then which angle is congruent to ∠𝐺 ?

A. ∠𝐵 C. ∠𝐸

B. ∠𝐶 D. ∠𝐹

6. Which diagram below shows a correct mathematical construction using only a compass

and a straightedge to bisect an angle?

A.

C.

B.

D.


Released 8/22/16

7. A line is constructed through point P parallel to a given line 𝑚.

The following diagrams show the steps of the construction:

Step 1 Step 2 Step 3 Step 4

Which of the following justifies the statement 𝑃𝑆 ⃡ ∥ 𝑄𝑅 ⃡ ?

A. 𝑃𝑆 ⃡ ∥ 𝑄𝑅 ⃡ because ∠𝑇𝑃𝑆 and ∠𝑃𝑄𝑅 are congruent corresponding angles.

B. 𝑃𝑆 ⃡ ∥ 𝑄𝑅 ⃡ because ∠𝑇𝑃𝑆 and ∠𝑃𝑄𝑅 are congruent alternate interior angles.

C. 𝑃𝑆 ⃡ ∥ 𝑄𝑅 ⃡ because 𝑃𝑆 ⃡ does not intersect 𝑄𝑅 ⃡ .

D. 𝑃𝑆 ⃡ ∥ 𝑄𝑅 ⃡ because a line can be drawn through point 𝑃 not on 𝑄𝑅 ⃡ .

8. Find the values of x and y in the diagram below.

A. 𝑥 = 18, 𝑦 = 94

B. 𝑥 = 18, 𝑦 = 118

C. 𝑥 = 74, 𝑦 = 94

D. 𝑥 = 74, 𝑦 = 88


Released 8/22/16

9. Which of the following are logically equivalent?

A. A conditional statement and its converse

B. A conditional statement and its inverse

C. A conditional statement and its contrapositive

D. A conditional statement, its converse, its inverse and its contrapositive

10. Two lines that do NOT intersect are always parallel.

Which of the following best describes a counterexample to the assertion above?

A. coplanar lines

B. parallel lines

C. perpendicular lines

D. skew lines

11. Determine which statement follows logically from the given statements.

If I am absent on a test day, I will need to make up the test. Absent students take the test

during their lunch time or after school.

A. If I am absent, it is because I am sick.

B. If I am absent, I will take the test at lunch time or after school.

C. Some absent students take the test at lunch time.

D. If I am not absent, the test will not be taken at lunch time or after school.

12. Determine whether the conjecture is true or false. Give a counterexample if the

conjecture is false.

Given: Two angles are supplementary.

Conjecture: They are both acute angles.

A. False; either both are right or they are adjacent.

B. True

C. False; either both are right or one is obtuse.

D. False; they must be vertical angles.


Released 8/22/16

13. Write the statement in if-then form.

A counterexample invalidates a statement.

A. If it invalidates the statement, then there is a counterexample.

B. If there is a counterexample, then it invalidates the statement.

C. If it is true, then there is a counterexample.

D. If there is a counterexample, then it is true.

14. Which statement is true based on the figure?

A. 𝑎 ∥ 𝑏

65

65

60

60

110

110

120

120 e

d

c

b

a

B. 𝑏 ∥ 𝑐

C. 𝑎 ∥ 𝑐

D. 𝑑 ∥ 𝑒

15. In the diagram below, 𝑀𝑄 = 30, 𝑀𝑁 = 5, 𝑀𝑁 = 𝑁𝑂, and 𝑂𝑃 = 𝑃𝑄.

Which of the following statements is not true?

A. 𝑁𝑃 = 𝑀𝑁 + 𝑃𝑄 C. 𝑀𝑄 = 3 ∙ 𝑃𝑄

B. 𝑀𝑃 = 𝑂𝑄 D. 𝑁𝑄 = 𝑀𝑃


Released 8/22/16

For #16-17 use the following:

Given: 𝐾𝑀 bisects ∠𝐽𝐾𝐿

Prove: 𝑚∠2 = 𝑚∠3

Statements Reasons

𝐾𝑀 bisects ∠𝐽𝐾𝐿 Given

∠1 ≅ ∠2 16.

𝑚∠1 = 𝑚∠2 Definition of Congruence

∠1 ≅ ∠3 17.


𝑚∠2 = 𝑚∠3 Substitution Property of Equality

16. Choose one of the following to complete the proof.

A. Definition of angle bisector- If a ray is an angle bisector, then it divides the angle

into two congruent angles.

B. Definition of opposite rays- If a point on the line determines two rays are collinear,

then the rays are opposite rays.

C. Definition of ray- If a line begins at an endpoint and extends infinitely, then it is ray.

D. Definition of segment bisector- If any segment, line, or plane intersects a segment at

its midpoint then it is the segment bisector.


A. Definition of complementary angles- If the angle measures add up to 90°, then

angles are supplementary

B. Supplemental Angle Theorem- If two angles are supplementary to a third angle then

the two angles are congruent

C. Definition of supplementary angles- If the angles are supplementary, then the

angle’s measures add to 180°.

D. Vertical Angle Theorem- If two angles are vertical angles, then they have congruent

angle measures.


Released 8/22/16

18. What are the coordinates of the point 𝑃 that lies along the directed segment from

𝐿(−5, 7) to 𝑀(4, −8) and partitions the segment in the ratio of 1 to 4?

A. (−3.2, 4) C. (1.8, −3)

B. (−2.5, 3) D. (2, −5)

19. An 80 mile trip is represented on a gridded map by a directed line segment from point

𝑀(3, 2) to point 𝑁(9, 14). What point represents 50 miles into the trip? Round your

answers to the nearest hundredth.

A. (2.31, 4.62) C. (5.31, 6.62)

B. (3.75, 7.50) D. (6.75, 9.50)

20. The equations of four lines are given. Identify which lines are parallel.

I. 3𝑥 + 2𝑦 = 10

II. −9𝑥 − 6𝑦 = −8

III. 𝑦 + 1 =3

2(𝑥 − 6)

IV. −5𝑦 = 7.5𝑥

A. I, II, and IV C. III and IV

B. I and II D. None of the lines are parallel

21. Which equation of the line passes through (4, 7) and is perpendicular to the graph of the

line that passes through the points(1, 3) and (−2, 9)?

A. 𝑦 = 2𝑥 − 1 C. 𝑦 =

1

2𝑥 − 5

B. 𝑦 =1

2𝑥 + 5 D. 𝑦 = −2𝑥 + 15


Released 8/22/16

22. Which equation of the line passes through (29, 8) and is perpendicular to the graph of the

line 𝑦 =1

13𝑥 + 17?

A. 𝑦 = 385𝑥 +

1

13 C. 𝑦 = −13𝑥 + 385

B. 𝑦 =1

13𝑥 + 385 D. 𝑦 = −13𝑥 − 13

23. Solve for x and y so that 𝑎 ∥ 𝑏 ∥ 𝑐 . Round your answer to the nearest tenth if

necessary.

A. 𝑥 = 17.6, 𝑦 = 3.1 C. 𝑥 = 54.3, 𝑦 = 8.5

B. 𝑥 = 17.6, 𝑦 = 5.5 D. 𝑥 = 54.3, 𝑦 = 26.9


Released 8/22/16


Given: 𝑝 ∥ 𝑞

Prove: 𝑚∠3 + 𝑚∠6 = 180

Statements Reasons

𝑝 ∥ 𝑞 Given

24. If two parallel lines are cut by a transversal, then

each pair of alternate interior angles is congruent.


∠5 and ∠6 are supplementary If two angles form a linear pair, then they are

supplementary.

𝑚∠5 + 𝑚∠6 = 180 25.

𝑚∠3 + 𝑚∠6 = 180 Substitution Property of Equality


A. ∠4 ≅ ∠5

B. ∠2 ≅ ∠8

C. ∠3 ≅ ∠6

D. ∠3 ≅ ∠5


A. Vertical Angle Theorem- If two angles are vertical angles, then they have congruent

angle measures

B. Supplemental Angle Theorem- If two angles are supplementary to a third angle then

they are congruent

C. Definition of supplementary angles- If two angles are supplementary, then their

angle measures add to 180°.

D. Definition of complementary angles- If two angles are a complementary, then their

angle measures add to 90°


Released 8/22/16

26. Line k is represented by the equation, 𝑦 = 2𝑥 + 3. Which equation would you use to

determine the distance between the line k and point (0, 0)?

A. 𝑦 = 2𝑥 C. 𝑦 = −

1

2𝑥 + 3

B. 𝑦 =1

2𝑥 D. 𝑦 = −

1

2𝑥

27. Which of the following is true?

A. All triangles are congruent.

B. All congruent figures have three sides.

C. If two figures are congruent, there must be some sequence of rigid transformations

that maps one to the other.

D. If two triangles are congruent, then they must be right angles.

28. Describe the transformation 𝑀: (−2, 5) → (−2, −5).

A. A reflection across the y-axis

B. A reflection across the x-axis

C. A clockwise rotation of 270° with center of rotation (0, 0)

D. A counterclockwise rotation of 90° with center of rotation (0, 0)

29. The endpoints of 𝐴𝐵̅̅ ̅̅ have coordinates 𝐴(1, −3) and 𝐵(−4, 5). After a translation 𝐴 is

mapped on to 𝐴′(−1, −7). What are the coordinates of 𝐵′ after the translation?

A. (−6, −1) C. (−6, 1)

B. (6, 1) D. (1, 6)


Released 8/22/16

30. Figure 𝐴𝐵𝐶 is rotated 90° counterclockwise about the point (−2, −3). What are the

coordinates of 𝐴′ after the rotation?

A. 𝐴′(−4, 5)

B. 𝐴′(−1, −6)

C. 𝐴′(−3, 0)

D. 𝐴′(4, − 5)

31. Point A is reflected over the line 𝐵𝐶 ⃡ . Which of the following is not true of line 𝐵𝐶 ⃡ ?

A. line 𝐵𝐶 ⃡ is perpendicular to line 𝐴𝐴′ ⃡

B. line 𝐵𝐶 ⃡ is perpendicular to line 𝐴𝐵 ⃡

C. line 𝐵𝐶 ⃡ bisects line segment 𝐴𝐵̅̅ ̅̅

D. line 𝐵𝐶 ⃡ bisects line segment 𝐴𝐴′̅̅ ̅̅ ̅

32. A graphic designer is creating a cover for a geometry textbook by reflecting a design

across line 𝑝 and then reflecting the image across line 𝑛. Describe a single

transformation that moves the design from its starting position to its final position.

A. clockwise rotation of 180° about the origin

B. clockwise rotation of 90° about the origin

C. translation along the line 𝑝 = 𝑛

D. reflection across the line 𝑝 = 𝑛


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33. What are the coordinates for the image of ∆𝐺𝐻𝐾 after a rotation 90° clockwise about the

origin and a translation of (𝑥, 𝑦) → (𝑥 + 3, 𝑦 + 2)?

A. 𝐺′′(−3, 2), 𝐻′′(−5, −1), 𝐾′′(−1, −2)

B. 𝐺′′(0, 4), 𝐻′′(−2, 1), 𝐾′′(2, 0)

C. 𝐺′′(1, 2), 𝐻′′(5, 1), 𝐾′′(2, −1)

D. 𝐺′′(6, 0), 𝐻′′(8, 3), 𝐾′′(4, 5)

34. Which composition of transformations maps ∆𝐴𝐵𝐶 into the third quadrant?

A. Reflection across the line 𝑦 = 𝑥 and then a

reflection across the y-axis.

B. Clockwise rotation about the origin by 180° and

then a reflection across the y-axis.

C. Translation of (𝑥 − 5, 𝑦) and then a

counterclockwise rotation about the origin by 90°.

D. Clockwise rotation about the origin by 270° and

then a translation of (𝑥 + 1, 𝑦).

35. The point 𝑃(−2, −5) is rotated 90° counterclockwise about the origin, and then the

image is reflected across the line 𝑥 = 3. What are the coordinates of the final image 𝑃′′?

A. (1, −2) C. (−2, 1)

B. (11, −2) D. (2, 11)


Released 8/22/16

36. Describe the rigid motion(s) that would map ∆𝐴𝐵𝐶 on to ∆𝑋𝑌𝐶 to satisfy the SAS

congruence criteria.

A. Rotation

B. Translation

C. Rotation and Reflection

D. Translation and Reflection

37. In the figure below, 𝐷𝐸 = 𝐸𝐻, 𝐺𝐻̅̅ ̅̅ ≅ 𝐷𝐹̅̅ ̅̅ , and ∠𝐹 ≅ ∠𝐺. Is there enough information to

conclude ∆𝐷𝐸𝐹 ≅ ∆𝐻𝐸𝐺? If so, state the congruence postulate that supports the

congruence statement.

A. Yes, by SSA Postulate

B. Yes, by SAS Postulate

C. Yes, by AAS Theorem

D. No, not enough information

38. If ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹, which of the following is true?

A. ∠𝐴 ≅ ∠𝐷, 𝐵𝐶̅̅ ̅̅ ≅ 𝐸𝐹̅̅ ̅̅ , ∠𝐶 ≅ ∠𝐹

B. ∠𝐴 ≅ ∠𝐷, 𝐴𝐵̅̅ ̅̅ ≅ 𝐷𝐹̅̅ ̅̅ ∠𝐶 ≅ ∠𝐸

C. ∠𝐴 ≅ ∠𝐹, 𝐵𝐶̅̅ ̅̅ ≅ 𝐴𝐶̅̅ ̅̅ , ∠𝐶 ≅ ∠𝐷

D. ∠𝐴 ≅ ∠𝐸, 𝐷𝐹̅̅ ̅̅ ≅ 𝐸𝐹̅̅ ̅̅ , ∠𝐶 ≅ ∠𝐹

39. In the figure ∠𝐺𝐴𝐸 ≅ ∠𝐿𝑂𝐷 and 𝐴𝐸̅̅ ̅̅ ≅ 𝐷𝑂̅̅ ̅̅ . What information is needed to prove that

∆𝐴𝐺𝐸 ≅ ∆𝑂𝐿𝐷 by SAS?

A. 𝐺𝐸̅̅ ̅̅ ≅ 𝐿𝐷̅̅ ̅̅

B. 𝐴𝐺̅̅ ̅̅ ≅ 𝑂𝐿̅̅̅̅

C. ∠𝐴𝐺𝐸 ≅ ∠𝑂𝐿𝐷

D. ∠𝐴𝐸𝐺 ≅ ∠𝑂𝐷𝐿


Released 8/22/16

40. You are given the following information about ∆𝐺𝐻𝐼 and ∆𝐸𝐹𝐷.

I. ∠𝐺 ≅ ∠𝐸

II. ∠𝐻 ≅ ∠𝐹

III. ∠𝐼 ≅ ∠𝐷

IV. 𝐺𝐻̅̅ ̅̅ ≅ 𝐸𝐹̅̅ ̅̅

V. 𝐺𝐼̅̅ ̅ ≅ 𝐸𝐷̅̅ ̅̅

Which combination cannot be used to prove that ∆𝐺𝐻𝐼 ≅ ∆𝐸𝐹𝐷?

A. V, IV, II

B. II, III, V

C. III, V, I

D. All of the above prove ∆𝐺𝐻𝐼 ≅ ∆𝐸𝐹𝐷

41. In the figure 𝐷𝐸̅̅ ̅̅ ≅ 𝐸𝐻̅̅ ̅̅ and 𝐺𝐻̅̅ ̅̅ ≅ 𝐷𝐹̅̅ ̅̅ . Which theorem can be used to conclude that

∆𝐷𝐸𝐹 ≅ ∆𝐻𝐸𝐺?

A. SSA

B. AAA

C. SAS

D. HL

42. In the figure, ∆𝐴𝐵𝐶 ≅ ∆𝐴𝐹𝐷. What is the 𝑚∠𝐷?

A. 𝑚∠𝐷 = 57°

B. 𝑚∠𝐷 = 42°

C. 𝑚∠𝐷 = 30°

D. 𝑚∠𝐷 = 25°


Released 8/22/16

For #44 use the following:

Given: 𝑄 is the midpoint of 𝑀𝑁̅̅ ̅̅ ̅; ∠𝑀𝑄𝑃 ≅ ∠𝑁𝑄𝑃

Prove: ∆𝑀𝑄𝑃 ≅ ∆𝑁𝑄𝑃

Statements Reasons

𝑄 is the midpoint of 𝑀𝑁̅̅ ̅̅ ̅; ∠𝑀𝑄𝑃 ≅ ∠𝑁𝑄𝑃 Given

[1] Definition of Midpoint

∠𝑀𝑄𝑃 ≅ ∠𝑁𝑄𝑃 Given

𝑄𝑃̅̅ ̅̅ ≅ 𝑄𝑃̅̅ ̅̅ Reflexive property of congruence

∆𝑀𝑄𝑃 ≅ ∆𝑁𝑄𝑃 [2]


A. [1] 𝑀𝑄̅̅ ̅̅ ̅ ≅ 𝑁𝑄̅̅ ̅̅

[2] AAS Congruence

B. [1] 𝑀𝑃̅̅̅̅̅ ≅ 𝑁𝑃̅̅ ̅̅

[2] Linear Pair Theorem

C. [1] 𝑀𝑄̅̅ ̅̅ ̅ ≅ 𝑁𝑄̅̅ ̅̅

[2] SAS Congruence

D. [[1] 𝑀𝑁̅̅ ̅̅ ̅ ≅ 𝑄𝑃̅̅ ̅̅

[2] SAS Congruence

43. Given ∆𝑀𝑁𝑃, Anna is proving 𝑚∠1 + 𝑚∠2 = 𝑚∠4. Which statement should be part of

her proof?

A. 𝑚∠1 = 𝑚∠2

B. 𝑚∠1 = 𝑚∠3

C. 𝑚∠1 + 𝑚∠3 = 180°

D. 𝑚∠3 + 𝑚∠4 = 180°


Released 8/22/16

45. In the figure, ∆𝑀𝑂𝑁 ≅ ∆𝑁𝑃𝑀. What is the value of y?

A. 𝑦 = 8

B. 𝑦 = 10

C. 𝑦 = 42

D. 𝑦 = 52

46. In the figure, 𝐴𝐶̅̅ ̅̅ ≅ 𝐴𝐵̅̅ ̅̅ . Find the value of y in terms of x.

A. 𝑦 = −3𝑥 + 160

B. 𝑦 = 6𝑥 − 140

C. 𝑦 = 6𝑥 + 40

D. 𝑦 =3𝑥 + 20

2


Released 8/22/16

For #47 use the following:

Given: 𝐴𝐵̅̅ ̅̅ ≅ 𝐴𝐶̅̅ ̅̅ and ∠1 ≅ ∠2

Prove: 𝐵𝐶 ⃡ ∥ 𝐸𝐷 ⃡

Statements Reasons

𝐴𝐵̅̅ ̅̅ ≅ 𝐴𝐶̅̅ ̅̅ Given

∠2 ≅ ∠3 47.

∠1 ≅ ∠2 Given

∠1 ≅ ∠3 Transitive property of congruence

𝐵𝐶 ⃡ ∥ 𝐸𝐷 ⃡

If two coplanar lines are but by a transversal so that a

pair of corresponding angles are congruent, then the

two lines are parallel.


A. Isosceles Triangle Symmetry Theorem- If the line contains the bisector of the vertex

angle of an isosceles triangle, then it is a symmetry line for the triangle.

B. Isosceles Triangle Coincidence Theorem- If the bisector of the vertex angle of an

isosceles triangle is also the perpendicular bisector of the base, then the median to

the base is the same line

C. Isosceles Triangle Base Angle Converse Theorem- If two angles of a triangle are

congruent, the sides opposite those angles are congruent

D. Isosceles Triangle Base Angle Theorem- If two sides of a triangle are congruent,

then the angles opposite those sides are congruent

48. Which of the following best describes the shortest distance from a vertex of a triangle to

the opposite side?

A. altitude

B. diameter

C. median

D. segment


Released 8/22/16

49. 𝐸𝐵 is the angle bisector of ∠𝐴𝐸𝐶. What is the value of x?

A. 𝑥 = 35

B. 𝑥 = 51.5

C. 𝑥 = 70.5

D. 𝑥 = 142

50. In ∆𝐷𝑂𝐺, line 𝑚 is drawn such that it is perpendicular to 𝐷𝑂̅̅ ̅̅ at point 𝑋 and 𝐷𝑋̅̅ ̅̅ ≅ 𝑂𝑋̅̅ ̅̅ .

Which of the following best describes line 𝑚?

A. altitude C. angle bisector

B. median D. perpendicular bisector

51. Reflect point H across the line 𝐹𝐺 ⃡ to form point 𝐻′, which of the following is true?

A. 𝐻𝐹̅̅ ̅̅ ≅ 𝐹𝐺̅̅ ̅̅

B. 𝐻𝐹̅̅ ̅̅ ≅ 𝐻′𝐺̅̅ ̅̅ ̅

C. 𝐻𝐺̅̅ ̅̅ ≅ 𝐻′𝐺̅̅ ̅̅ ̅

D. 𝐹𝐺̅̅ ̅̅ ≅ 𝐻′𝐺̅̅ ̅̅ ̅

52. The vertices of ∆𝐽𝐾𝐿 are located at 𝐽(−5, −3), 𝐾(3, 9), and 𝐿(7, 2). If 𝐿𝑀̅̅ ̅̅ is an altitude

of ∆𝐽𝐾𝐿, what are the coordinates of 𝑀?

A. 𝑀(7, −3) C. 𝑀(−1, 3)

B. 𝑀(1, 6) D. 𝑀(−2, 2)


Released 8/22/16

53. On the graph below, ∠𝑃𝑄𝑅 is reflected over 𝑄𝑅 ⃡ so that 𝑄𝑅 ⃡ is an angle bisector of

∠𝑃𝑄𝑃′. What are the coordinates of 𝑃′?

A. 𝑃′(5−, 3)

B. 𝑃′(1, 5)

C. 𝑃′(−9, 1)

D. 𝑃′(1, 7)

54. A segment has endpoints 𝑇(−4, 5) and 𝑈(6, 1). Find the equation of the perpendicular

bisector of 𝑇𝑈̅̅ ̅̅ .

A. 𝑥 = 1 C. 𝑦 =

5

2𝑥 +

1

2

B. 𝑦 = −2

5𝑥 + 4 D. 𝑦 =

5

2𝑥 −

21

2


Released 8/22/16


Given: 𝐺𝐹̅̅ ̅̅ is a median of isosceles ∆𝐺𝐼𝐽 with

base 𝐼�̅�

Prove: ∆𝐽𝐺𝐹 ≅ ∆𝐼𝐺𝐹

Statements Reasons

𝐺𝐹̅̅ ̅̅ is a median Given

𝐹 is a midpoint of 𝐼�̅� 55.

𝐹𝐼̅̅ ̅ ≅ 𝐹𝐽̅̅ ̅ Definition of midpoint

56. Definition of isosceles triangle

𝐹𝐺̅̅ ̅̅ ≅ 𝐹𝐺̅̅ ̅̅ Reflexive property of congruence

∆𝐽𝐺𝐹 ≅ ∆𝐼𝐺𝐹 SSS Congruence


A. Definition of angle bisector- If a ray divides an angle into two congruent angles, then

it is an angle bisector.

B. Definition of segment bisector- If any segment, line, or plane intersects a segment at

its midpoint, then it is a segment bisector.

C. Definition of isosceles triangle- If a triangle has at least two congruent sides, then it

is an isosceles triangle.

D. Definition of median- If a segment is a median, then it has endpoints at the vertex of

a triangle and the midpoint of the opposite side.


A. 𝐺𝐼̅̅ ̅ ≅ 𝐺𝐻̅̅ ̅̅

B. 𝐺𝐼̅̅ ̅ ≅ 𝐺𝐽̅̅ ̅

C. 𝐾𝐺̅̅ ̅̅ ≅ 𝐻𝐺̅̅ ̅̅

D. 𝐾𝐼̅̅ ̅ ≅ 𝐻𝐽̅̅̅̅


Released 8/22/16

57. Which of the following indirect proofs is correct given the following?

Given: ∆𝑨𝑩𝑪

Prove: ∆𝑨𝑩𝑪 has no more than one right angle

Assume: ∆𝑨𝑩𝑪 has more than one right angle

A. Assume that ∠𝐴 and ∠𝐵 are both obtuse angles. So by definition of an obtuse angle,

𝑚∠𝐴 = 120° and 𝑚∠𝐵 = 120°. According to the Triangle Angle-Sum Theorem,

𝑚∠𝐴 + 𝑚∠𝐵 + 𝑚∠𝐶 = 180°. By substitution, 120° + 120° + 𝑚∠𝐶 = 180°. Combining like terms give the equation 240° + 𝑚∠𝐶 = 180°. Subtracting 240°

from both sides of the equation gives 𝑚∠𝐶 = −60°. This contradicts the fact that

an angle in a triangle has to be more than 0°. Therefore, the assumption ∆𝐴𝐵𝐶 has

more than one right angle is false. The statement ∆𝐴𝐵𝐶 has no more than one right

angle is true.

B. Assume that ∠𝐴 and ∠𝐵 are both right angles. So by definition of a right angle,


𝑚∠𝐴 + 𝑚∠𝐵 + 𝑚∠𝐶 = 180°. By substitution, 180° + 180° + 𝑚∠𝐶 = 180°. Combining like terms give the equation 360° + 𝑚∠𝐶 = 180°. Subtracting 360°

from both sides of the equation gives 𝑚∠𝐶 = −180°. This contradicts the fact that

an angle in a triangle has to be more than 0°. Therefore, the assumption ∆𝐴𝐵𝐶 has


angle is true.

C. Assume that ∠𝐴 and ∠𝐵 are both right angles. So by definition of a right angle,


𝑚∠𝐴 + 𝑚∠𝐵 + 𝑚∠𝐶 = 180°. By substitution, 90° + 90° + 𝑚∠𝐶 = 180°.

Combining like terms give the equation 180° + 𝑚∠𝐶 = 180°. Subtracting 180°

from both sides of the equation gives 𝑚∠𝐶 = 0°. This contradicts the fact that an

angle in a triangle has to be more than 0°. Therefore, the assumption ∆𝐴𝐵𝐶 has


angle is true.

D. Assume that ∠𝐴 and ∠𝐵 are both acute angles. So by definition of an acute angle,


𝑚∠𝐴 + 𝑚∠𝐵 + 𝑚∠𝐶 = 180°. By substitution, 60° + 60° + 𝑚∠𝐶 = 180°.

Combining like terms give the equation 120° + 𝑚∠𝐶 = 180°. Subtracting 120°

from both sides of the equation gives 𝑚∠𝐶 = 60°. This contradicts the fact that an

angle in a triangle has to be 90°. Therefore, the assumption ∆𝐴𝐵𝐶 has more than

one right angle is true.


Released 8/22/16

58. If a triangle has two sides with lengths of 8 𝑐𝑚 and 14 𝑐𝑚. Which length below could

not represent the length of the third side?

A. 7 𝑐𝑚 C. 15 𝑐𝑚

B. 13 𝑐𝑚 D. 22 𝑐𝑚

59. Find the range of values containing x.

A. 2 < 𝑥 < 5

B. 𝑥 < 5

C. 0 < 𝑥 < 9

D. 𝑥 > 0

60. The captain of a boat is planning to travel to three islands in a triangular pattern. What is

the possible range for the number of miles round trip the boat will travel?

A. between 32 and 75 𝑚𝑖𝑙𝑒𝑠

B. between 43 and 107 𝑚𝑖𝑙𝑒𝑠

C. between 139 and 182 𝑚𝑖𝑙𝑒𝑠

D. between 150 and 214 𝑚𝑖𝑙𝑒𝑠


Released 8/22/16

Formal Geometry Semester 1 Instructional Materials 2016-2017

Answers

1. B 11. B 21. B 31. C 41. D 51. C

2. D 12. C 22. C 32. A 42. B 52. B

3. B 13. B 23. B 33. B 43. D 53. D

4. A 14. D 24. D 34. C 44. C 54. C

5. B 15. D 25. C 35. A 45. B 55. D

6. C 16. A 26. D 36. C 46. B 56. B

7. A 17. D 27. C 37. D 47. D 57. C

8. A 18. A 28. B 38. A 48. A 58. D

9. C 19. D 29. C 39. B 49. A 59. A

10. D 20. A 30. B 40. A 50. D 60. D

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