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FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS
Course: Formal Geometry S1 (#2215)
2016-2017
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
𝑦2 − 𝑦1
𝑥2 − 𝑥1
Vertex-Form
𝑦 = 𝑎(𝑥 − h)2 + 𝑘
𝑥1 + 𝑥2 𝑦1 + 𝑦2
𝑀=(
,
)
2
2
Standard Form
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
Intercept Form
𝑦 = 𝑎(𝑥 − 𝑝)(𝑥 − 𝑞)
Slope
𝑚=
Midpoint
Distance
Slope-Intercept Form
𝑦 = 𝑚𝑥 + 𝑏
Exponential
(h, k) Form
Probability
𝑦 = 𝑎𝑏 𝑥−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 = 𝜋𝑟 2 ℎ
4
𝑉 = 𝜋𝑟 3
3
𝑆𝐴 = 2(𝜋𝑟 2 ) + ℎ(2𝜋𝑟)
𝑆𝐴 = 4𝜋𝑟 2
1
1
𝑉 = 𝜋𝑟 2 ℎ
3
1
𝑆𝐴 = 𝜋𝑟 2 + (2𝜋𝑟 ∙ 𝑙)
2
Released 8/22/16
𝑉 = 𝐵ℎ
3
1
𝑆𝐴 = 𝐵 + (𝑃𝑙)
2
Where 𝐵 =base area
and 𝑃 =base perimeter
FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS
Course: Formal Geometry S1 (#2215)
2016-2017
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)
̅̅̅̅ . 𝑇 is the midpoint of 𝐴𝐶
̅̅̅̅ . 𝑆 is the midpoint
3. In the diagram below, 𝑅 is the midpoint of 𝐴𝐵
̅̅̅̅
of 𝐵𝐶 . 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
FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS
Course: Formal Geometry S1 (#2215)
4.
2016-2017
̅̅̅̅ and 𝑋𝑌
̅̅̅̅ and determine which of the following
Given the coordinates below, compare 𝑅𝑆
statements is true:
𝑅(−2, −7)
𝑆(5, −1)
𝑋(−3, 3)
𝑌(6, −1)
A. The midpoints of ̅̅̅̅
𝑅𝑆 and ̅̅̅̅
𝑋𝑌 have the same 𝑥-coordinate.
B. The midpoints of ̅̅̅̅
𝑅𝑆 and ̅̅̅̅
𝑋𝑌 have the same 𝑦-coordinate.
̅̅̅̅ and the length of 𝑋𝑌
̅̅̅̅ are the same.
C. The length of 𝑅𝑆
D. The length of ̅̅̅̅
𝑅𝑆 is longer than the length of ̅̅̅̅
𝑋𝑌.
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
FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS
2016-2017
Course: Formal Geometry S1 (#2215)
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
⃡ ∥ 𝑄𝑅
⃡ ?
Which of the following justifies the statement 𝑃𝑆
⃡ because ∠𝑇𝑃𝑆 and ∠𝑃𝑄𝑅 are congruent corresponding angles.
A. ⃡𝑃𝑆 ∥ 𝑄𝑅
⃡ because ∠𝑇𝑃𝑆 and ∠𝑃𝑄𝑅 are congruent alternate interior angles.
B. ⃡𝑃𝑆 ∥ 𝑄𝑅
⃡ ∥ 𝑄𝑅
⃡ because 𝑃𝑆
⃡ does not intersect 𝑄𝑅
⃡ .
C. 𝑃𝑆
⃡ because a line can be drawn through point 𝑃 not on 𝑄𝑅
⃡ .
D. ⃡𝑃𝑆 ∥ 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
FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS
Course: Formal Geometry S1 (#2215)
2016-2017
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
FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS
Course: Formal Geometry S1 (#2215)
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.
65
60
110
120
14. Which statement is true based on the figure?
A. 𝑎 ∥ 𝑏
b
B. 𝑏 ∥ 𝑐
a
c
C. 𝑎 ∥ 𝑐
D. 𝑑 ∥ 𝑒
65
110
d
60
120
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
2016-2017
FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS
Course: Formal Geometry S1 (#2215)
2016-2017
For #16-17 use the following:
Given: 𝐾𝑀 bisects ∠𝐽𝐾𝐿
Prove: 𝑚∠2 = 𝑚∠3
Statements
Reasons
𝐾𝑀 bisects ∠𝐽𝐾𝐿
∠1 ≅ ∠2
Given
𝑚∠1 = 𝑚∠2
Definition of Congruence
∠1 ≅ ∠3
17.
𝑚∠1 = 𝑚∠3
Definition of Congruence
𝑚∠2 = 𝑚∠3
Substitution Property of Equality
16.
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.
17. Choose one of the following to complete the proof.
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
FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS
Course: Formal Geometry S1 (#2215)
2016-2017
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
3
III. 𝑦 + 1 = 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
B. 𝑦 =
1
𝑥+5
2
Released 8/22/16
C. 𝑦 =
1
𝑥−5
2
D. 𝑦 = −2𝑥 + 15
FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS
Course: Formal Geometry S1 (#2215)
2016-2017
22. Which equation of the line passes through (29, 8) and is perpendicular to the graph of the
1
line 𝑦 = 13 𝑥 + 17?
A. 𝑦 = 385𝑥 +
B. 𝑦 =
23.
1
13
1
𝑥 + 385
13
C. 𝑦 = −13𝑥 + 385
D. 𝑦 = −13𝑥 − 13
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
FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS
Course: Formal Geometry S1 (#2215)
2016-2017
For #24-25 use the following:
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.
𝑚∠3 = 𝑚∠5
Definition of Congruence
∠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
24. Choose one of the following to complete the proof.
A. ∠4 ≅ ∠5
B. ∠2 ≅ ∠8
C. ∠3 ≅ ∠6
D. ∠3 ≅ ∠5
25. Choose one of the following to complete the proof.
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
FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS
Course: Formal Geometry S1 (#2215)
2016-2017
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
B. 𝑦 =
1
𝑥
2
1
C. 𝑦 = − 𝑥 + 3
2
1
D. 𝑦 = − 𝑥
2
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 ̅̅̅̅
𝐴𝐵 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
FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS
Course: Formal Geometry S1 (#2215)
2016-2017
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)
⃡ . Which of the following is not true of line 𝐵𝐶
⃡ ?
31. Point A is reflected over the line 𝐵𝐶
A. line 𝐵𝐶
⃡ is perpendicular to line ⃡𝐴𝐴′
⃡ is perpendicular to line ⃡𝐴𝐵
B. line 𝐵𝐶
⃡ bisects line segment 𝐴𝐵
̅̅̅̅
C. line 𝐵𝐶
̅̅̅̅̅
⃡ bisects line segment 𝐴𝐴′
D. line 𝐵𝐶
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
Released 8/22/16
FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS
Course: Formal Geometry S1 (#2215)
2016-2017
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
FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS
Course: Formal Geometry S1 (#2215)
2016-2017
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, ̅̅̅̅
𝐺𝐻 ≅ ̅̅̅̅
𝐷𝐹 , 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?
̅̅̅̅ ≅ 𝐸𝐹
̅̅̅̅ , ∠𝐶 ≅ ∠𝐹
A. ∠𝐴 ≅ ∠𝐷, 𝐵𝐶
B. ∠𝐴 ≅ ∠𝐷, ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐷𝐹 ∠𝐶 ≅ ∠𝐸
̅̅̅̅ ≅ 𝐴𝐶
̅̅̅̅ , ∠𝐶 ≅ ∠𝐷
C. ∠𝐴 ≅ ∠𝐹, 𝐵𝐶
D. ∠𝐴 ≅ ∠𝐸, ̅̅̅̅
𝐷𝐹 ≅ ̅̅̅̅
𝐸𝐹 , ∠𝐶 ≅ ∠𝐹
̅̅̅̅. What information is needed to prove that
39. In the figure ∠𝐺𝐴𝐸 ≅ ∠𝐿𝑂𝐷 and ̅̅̅̅
𝐴𝐸 ≅ 𝐷𝑂
∆𝐴𝐺𝐸 ≅ ∆𝑂𝐿𝐷 by SAS?
̅̅̅̅ ≅ 𝐿𝐷
̅̅̅̅
A. 𝐺𝐸
B. ̅̅̅̅
𝐴𝐺 ≅ ̅̅̅̅
𝑂𝐿
C. ∠𝐴𝐺𝐸 ≅ ∠𝑂𝐿𝐷
D. ∠𝐴𝐸𝐺 ≅ ∠𝑂𝐷𝐿
Released 8/22/16
FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS
Course: Formal Geometry S1 (#2215)
2016-2017
40. You are given the following information about ∆𝐺𝐻𝐼 and ∆𝐸𝐹𝐷.
I.
II.
III.
IV.
V.
∠𝐺 ≅ ∠𝐸
∠𝐻 ≅ ∠𝐹
∠𝐼 ≅ ∠𝐷
̅̅̅̅
𝐺𝐻 ≅ ̅̅̅̅
𝐸𝐹
̅̅̅
̅̅̅̅
𝐺𝐼 ≅ 𝐸𝐷
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 ̅̅̅̅
𝐷𝐸 ≅ ̅̅̅̅
𝐸𝐻 and ̅̅̅̅
𝐺𝐻 ≅ ̅̅̅̅
𝐷𝐹 . 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°
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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°
For #44 use the following:
Given: 𝑄 is the midpoint of ̅̅̅̅̅
𝑀𝑁; ∠𝑀𝑄𝑃 ≅ ∠𝑁𝑄𝑃
Prove: ∆𝑀𝑄𝑃 ≅ ∆𝑁𝑄𝑃
Statements
Reasons
̅̅̅̅̅ ; ∠𝑀𝑄𝑃 ≅ ∠𝑁𝑄𝑃
𝑄 is the midpoint of 𝑀𝑁
Given
[1]
Definition of Midpoint
∠𝑀𝑄𝑃 ≅ ∠𝑁𝑄𝑃
̅̅̅̅ ≅ 𝑄𝑃
̅̅̅̅
𝑄𝑃
Given
∆𝑀𝑄𝑃 ≅ ∆𝑁𝑄𝑃
[2]
Reflexive property of congruence
44. Choose one of the following to complete the proof.
A. [1] ̅̅̅̅̅
𝑀𝑄 ≅ ̅̅̅̅
𝑁𝑄
[2] AAS Congruence
B. [1] ̅̅̅̅̅
𝑀𝑃 ≅ ̅̅̅̅
𝑁𝑃
[2] Linear Pair Theorem
C. [1] ̅̅̅̅̅
𝑀𝑄 ≅ ̅̅̅̅
𝑁𝑄
[2] SAS Congruence
D. [[1] ̅̅̅̅̅
𝑀𝑁 ≅ ̅̅̅̅
𝑄𝑃
[2] SAS Congruence
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45. In the figure, ∆𝑀𝑂𝑁 ≅ ∆𝑁𝑃𝑀. What is the value of y?
A. 𝑦 = 8
B. 𝑦 = 10
C. 𝑦 = 42
D. 𝑦 = 52
̅̅̅̅ ≅ 𝐴𝐵
̅̅̅̅. Find the value of y in terms of x.
46. In the figure, 𝐴𝐶
A. 𝑦 = −3𝑥 + 160
B. 𝑦 = 6𝑥 − 140
C. 𝑦 = 6𝑥 + 40
D. 𝑦 =
3𝑥 + 20
2
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Course: Formal Geometry S1 (#2215)
2016-2017
For #47 use the following:
Given: ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐴𝐶 and ∠1 ≅ ∠2
⃡
⃡
Prove: 𝐵𝐶 ∥ 𝐸𝐷
Statements
̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐴𝐶
Reasons
∠2 ≅ ∠3
47.
∠1 ≅ ∠2
Given
∠1 ≅ ∠3
Transitive property of congruence
If two coplanar lines are but by a transversal so that a
pair of corresponding angles are congruent, then the
two lines are parallel.
⃡ ∥ ⃡𝐸𝐷
𝐵𝐶
Given
47. Choose one of the following to complete the proof.
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
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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 ̅̅̅̅
𝐷𝑂 at point 𝑋 and ̅̅̅̅
𝐷𝑋 ≅ ̅̅̅̅
𝑂𝑋.
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. ̅̅̅̅
𝐻𝐹 ≅ ̅̅̅̅
𝐹𝐺
B. ̅̅̅̅
𝐻𝐹 ≅ ̅̅̅̅̅
𝐻′𝐺
C. ̅̅̅̅
𝐻𝐺 ≅ ̅̅̅̅̅
𝐻′𝐺
̅̅̅̅ ≅ ̅̅̅̅̅
D. 𝐹𝐺
𝐻′𝐺
52.
̅̅̅̅ is an altitude
The vertices of ∆𝐽𝐾𝐿 are located at 𝐽(−5, −3), 𝐾(3, 9), and 𝐿(7, 2). If 𝐿𝑀
of ∆𝐽𝐾𝐿, what are the coordinates of 𝑀?
A. 𝑀(7, −3)
C. 𝑀(−1, 3)
B. 𝑀(1, 6)
D. 𝑀(−2, 2)
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53.
2016-2017
⃡ so that 𝑄𝑅
⃡ is an angle bisector of
On the graph below, ∠𝑃𝑄𝑅 is reflected over 𝑄𝑅
∠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 ̅̅̅̅
𝑇𝑈.
5
1
A. 𝑥 = 1
C. 𝑦 = 𝑥 +
2
2
2
B. 𝑦 = − 𝑥 + 4
5
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D. 𝑦 =
5
21
𝑥−
2
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FORMAL GEOMETRY SEMESTER 1 INSTRUCTIONAL MATERIALS
Course: Formal Geometry S1 (#2215)
2016-2017
For #55-56 use the following:
Given: ̅̅̅̅
𝐺𝐹 is a median of isosceles ∆𝐺𝐼𝐽 with
̅
base 𝐼𝐽
Prove: ∆𝐽𝐺𝐹 ≅ ∆𝐼𝐺𝐹
Statements
̅̅̅̅
𝐺𝐹 is a median
Reasons
̅
𝐹 is a midpoint of 𝐼𝐽
̅̅̅ ≅ 𝐹𝐽
̅̅̅
𝐹𝐼
55.
56.
̅̅̅̅
̅̅̅̅
𝐹𝐺 ≅ 𝐹𝐺
Definition of isosceles triangle
∆𝐽𝐺𝐹 ≅ ∆𝐼𝐺𝐹
SSS Congruence
Given
Definition of midpoint
Reflexive property of congruence
55. Choose one of the following to complete the proof.
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.
56. Choose one of the following to complete the proof.
A. ̅̅̅
𝐺𝐼 ≅ ̅̅̅̅
𝐺𝐻
̅̅̅ ≅ 𝐺𝐽
̅̅̅
B. 𝐺𝐼
C. ̅̅̅̅
𝐾𝐺 ≅ ̅̅̅̅
𝐻𝐺
̅̅̅ ≅ 𝐻𝐽
̅̅̅̅
D. 𝐾𝐼
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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 = 180° and 𝑚∠𝐵 = 180°. According to the Triangle Angle-Sum Theorem,
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
more than one right angle is false. The statement ∆𝐴𝐵𝐶 has no more than one right
angle is true.
C. Assume that ∠𝐴 and ∠𝐵 are both right angles. So by definition of a right angle,
𝑚∠𝐴 = 90° and 𝑚∠𝐵 = 90°. According to the Triangle Angle-Sum Theorem,
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
more than one right angle is false. The statement ∆𝐴𝐵𝐶 has no more than one right
angle is true.
D. Assume that ∠𝐴 and ∠𝐵 are both acute angles. So by definition of an acute angle,
𝑚∠𝐴 = 60° and 𝑚∠𝐵 = 60°. According to the Triangle Angle-Sum Theorem,
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.
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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
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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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