Skills needed: Label a diagram with given

information Set up and solve an equation given a

diagram Notation

GEOMETRY 22 MID-TERM EXAM REVIEW January 2017

Note to student: This packet should be used as practice for the Geometry 22 midterm exam. This should not be the only tool

that you use to prepare yourself for the exam. You must go through your notes, re-do homework problems, class work

problems, formative assessment problems, and questions from your tests and quizzes throughout the year thus far. The

sections from the book that are covered on the midterm exam are:

Intro: Geometry Tool Kit (pre-requisite skills students should know coming in to Geometry) - Pythagorean theorem (8.1) - Using a protractor (1.4) - Algebra (solving equations, factoring) - Basic Area (triangle, circle (area & circumference), rectangle/square, trapezoid) - Naming Polygons - Slope - Parallel lines (3.2) - Complementary & Supplementary (1.5) - Congruence - Graphing on a coordinate plane (points and lines) (p. 893) - Types of triangles: equilateral, isosceles, scalene - Types of angles: Acute, obtuse, right, straight angle (1.4) - Solving proportions (7.1) - Perimeter

Unit 1: Parallel Lines/Angles A.

- Describe lines (using notation) (1.2) - Describe a point (using notation) (1.2)

o Collinear (1.2) - Define intersection (1.2)

o The intersection of two lines is a point (1.2) - Define postulate (1.2) - Define rays and angles (using notation) & naming angles (1.2/1.4)

o Define vertex of an angle (1.4) o Define adjacent angles (1.5)

- Angle addition postulate (1.4) - Definition of a Linear Pair & Linear Pair Postulate

o Define supplementary & complementary - Define vertical angles and the property that vertical angles are equal (1.5)

B. - Proofs (simple either two-column or flow-chart) (2.5)

o Substitution Property

o Reflexive Property of Equality

o Transitive Property of Equality (2.5)

o Distributive Property

C. - Define Alternate Interior, Alternate Exterior, Same-side Interior, and Corresponding angles (3.2) - Parallel lines and transversals (definitions) (3.2) - Define congruence and notation - Define a theorem

o Alternate Interior Angles Theorem o Alternate Exterior Angles Theorem o Same-Side Interior Angles Theorem (3.2) o Corresponding Angles Postulate o Congruent Supplements/Complements Theorem (+) (2.6) o Vertical Angles Theorem (1.5)

- Define conditional and converses with truth values and counterexamples (2.2)

o Converse of Alternate Interior Theorem o Converse of Alternate Exterior Theorem o Converse of Same-Side Interior Theorem (3.3) o Converse of Corresponding Theorem

- Biconditional Statements/What is a definition (2.3)

Unit 2: Triangles A.

- Define segments (using notation) (1.2) o Segment addition postulate (1.3) o Define Midpoint o Define Equidistant (5.2)

B. - Define a triangle (using notation) (4.4) - Triangle inequality (5.6) - Angles in a triangle

o Triangle sum theorem (3.5) o Exterior angle of a triangle theorem (3.5) o Isosceles triangle theorem & converse (4.5) o Isosceles triangle theorem corollaries (4.5)

Define equilateral and equiangular (6.1) o Longest side is opposite the largest angle (8.1)

- Define Bisect (1.2) - Define Perpendicular (1.6)

o Perpendicular bisector (1.6) o Angle bisector (1.5) o Median (5.4) o Define Altitude (5.4)

The shortest distance between a point and a line is the perpendicular distance (5.2) Use altitude to find AREA of triangles (A = ½ bh)

- Points of concurrency (+) (5.3) (incenter and circumcenter) o Area and perimeter of triangles in the coordinate plane (6.7)

Midpoint Formula (1.7) Triangle midsegment theorem (5.1)

Unit 3: Transformations (Triangle Conguence & Similarity) A.

- Rigid transformations- define reflection, rotation, translation (9.1, 9.2, 9.3) - Perform transformations in coordinate plane (ch. 9)

o Define a sequence that would transform a figure onto another figure (9.5) B.

- Define congruence with triangles (using correct notation) & proofs with congruent triangles (two-column or flowchart)

o SSS o SAS o ASA (4.2/4.3) o AAS o HL (4.6) o CPCTC (4.4)

C.

- Non-rigid transformations- define dilations (9.6) - Perform dilation in coordinate plane (9.6/9.7)

D.

- Define similarity with triangles (using correct notation) & proofs with similar triangles (two-column or flowchart) (setting up & solving proportions)

o SSS o SAS (7.3) *Incorporate application problems throughout o AA o Side-splitting theorem (7.5) o Right triangle similarity (+) (7.4)

Practice Problems for Midterm Exam – be sure to check your answers online!!

Using the diagram to the right, give an example of each of the following:

1. Use the diagram at right to name the following using proper geometric notation.

a) An obtuse angle containing point K ___________

b) An acute angle containing point K ___________

c) A right angle other than angle LQK ___________

d) Any vertex ___________

e) A ray with endpoint P ___________

f) Another way to name a ray with endpoint P ___________

g) An angle supplementary to angle NQK ___________

h) A line named two different ways ___________ ___________

i) An angle adjacent to angle NQJ that is not its complement ___________

j) the intersection of line NP and line ML _______________________

2. Would it be accurate to name ∠𝑀𝑄𝑃 as ∠𝑄? Why or why not?

_______________________________________________________________________________________

3. Use the diagram in #1 above to fill in the blanks.

________________ + ________________ = 𝑚∠𝐽𝑄𝑃 and JQ + ______ = JK

4. Through any two points there is exactly one ___________________________.

5. Assume that B is between A and C. If AB = 6 units and BC = 13 units, what is AC? ___________

6. Using the diagram below, if JR = 4x -12, RT = 6x +4 and JT = 8x + 10, find x. ___________

Equation: ________________________

Reason: ___________________________

is the perpendicular bisector of . If the intersection of and is and 16, find .AB PQ AB PQ R PR PQ

bisects at point . If 4 11 and 7 25, find the following:

___________

__________

__________

AB CD E CE x DE x

x

DE

CD

bisects . If (8 20) and ( 11) , find each of the following:

__________

__________

__________

AK TAN m TAN x m KAN x

x

m NAK

m TAN

7. In the diagram below, if (5 11) , 27 , and (2 65)m DYL x m LYN m DYN x , solve for x and find

.m DYN

Equation: _____________________

x = ___________

Reason: ____________________________

.m DYN =________

8. If and are complementary, and (2 15) and 25 , find .A B m A x m B x

9. If 1 and 2 form a linear pair, and 1 (3 9) and 2 (2 24) , find .m x m x x

10.

PQ = ___________

11.

12.

13. If CD = 5x – 7, find the indicated values.

a) x = ____________ b) CE = ______________ c) CD= ________________

14. a) What is m DTB ? ___________

b) What is m BTC ?____________

A

B C

D

T

A

x143° 110°

(2x + 5)°

15. If 43om EFG , what is the measure of its supplement? ___________

16. In the figure below, 18 3m DAF x . Find the indicated measures.

a) x = _______________

b) m FAE = ______________

17. In the figure below, find the indicated measures.

a) x = _______________

b) m KLN = _______________

18. Solve for x in the following problems.

a. b. c.

19. Refer to the statement: “All altitudes form right angles.”

a) Rewrite the statement as a conditional.

b) Identify the hypothesis and conclusion of the conditional.

Hypothesis:

Conclusion:

c) Draw a Venn to illustrate the statement.

d) Write the converse of the conditional. e) If the converse is false, give a counterexample:

3x - 42 123°

tennis

team

Sports teams

20. Refer to the statement: “A polygon with exactly three sides is called a triangle.”

a) Rewrite the statement as a conditional.

b) Write the converse.

c) Write the biconditional.

d) Decide whether the statement is a definition. Explain your reasoning.

21. Given the following Venn diagram, state a conditional using the information.

_______________________________________________________

_______________________________________________________

22. Explain the similarities and differences between supplementary angles and a linear pair.

23. Determine whether the following is a translation, reflection, or rotation.

a) _______________________ b) _______________________

24. Reflect ABC over the x-axis.

25. On the same graph, reflect ABC over the y-axis.

26. On the same graph, rotate ABC 180° about the origin.

27. The dashed-line figure is a dilation image of the solid-line figure. The labeled point is the center of dilation.

Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation.

a. b. c.

28. Determine if the following scale factor would create an enlargement, reduction, or isometric figure.

a. 3.5 b. 2/5 c. 0.6 d. 1 e. 4/3 f. -5/8

29. In the figure at right, lines r and s are parallel, m2 = 40o, and m4 = 60o; find:

a) m1 ______________ b) m3 ______________

c) m5 ______________ d) m6 ______________

e) m7 ______________ f) m8 ______________

g) m9 ______________ h) m10 ______________

i) m11 ______________ j) m12 ______________

k) m13 ______________ l) m14 ______________

30. If lines k and l are parallel, m4 = (3x -10)o and m5 = (x + 70)o; find:

a) m8 ______________

b) m6 ______________

c) m3 ______________

d) m7 ______________

31. Using the diagram above (from #30), state the type of angle pair of the given angles;

a. ∠2 𝑎𝑛𝑑 ∠6 _________________________ b. ∠5 𝑎𝑛𝑑 ∠6 _________________________

c. ∠2 𝑎𝑛𝑑 ∠3 _________________________ d. ∠3 𝑎𝑛𝑑 ∠6 _________________________

32. If ∠2 ≅ ∠7 , then 𝒌 ∥ 𝒍 by what theorem? ________________________________________________

r 13 12 11

7 8

14

6

10

5

4

9

s

3

2 1

8

k

7

6 5

4

l

3

2 1

(1.5y - 12)

P'

O'

W'

31. If m∡HAG = (7x + 4)o, and m∡CAB = (9x – 10)o, find the following:

a) x =______________

b) m∡GAH = ______________

c) m∡CAB =______________d) m∡G = ______________

32. In the diagram below F, E, and D are midpoints.

AC = ____________ If AB = 55, then FE = __________

If AB = 55 and FD = 20, find the perimeter of ABC ________

33. x = _________________

mN =_________________

34. Solve for y, then find the value of the ∠𝑃 in the figure below.

35. Can the following groups of sides be the sides of a triangle? Explain.

a. 15, 10, 5

b. 20, 21, 3

c. 7, 4, 15

36. In the figure below, 𝑚∠𝑆 = 24 and 𝑚∠𝑂 = 130. Which side of ⊿𝑆𝑂𝑋 is the shortest side? Why?

M

N

(3x + 1)o

L

(2x - 5)o

37. Determine m∡ A, m∡ B, and m∡C if A is supplementary to B and complementary to C.

38. Write the conditional and converse of the statement, and determine if the converse is true. If it is not, write

a counterexample.

If an angle measure is 32 degrees, then it is an acute angle.

39. Find the value of x and y.

a. b.

c. d.

40. Solve for the given variable and find the angle measures.

a. b. c.

d. e. f.

41. Use the diagram to the right.

a. What type of triangle is ABD ? ___________________________

b. What type of triangle is BCD ? ___________________________

c. Find m ABC ___________________________

42. Use the congruency statement to fill in the corresponding congruent parts.

EFI HGI ∠𝐸 ≅ ∠______ FE _____ ∠𝐸𝐹𝐼 ≅ ∠______

FI _____ ∠𝐹𝐼𝐸 ≅ ∠______ IE _____

43. For the following, name which triangle congruence theorem or postulate you would use to prove the

triangles congruent.

a. b. c.

44. Mark any additional information you can FIRST (ex: vertical angles or reflexive property). Then, label and

state what ADDITIONAL information is required in order to know that the triangles are congruent for the

reason given.

a. b. c.

extra part: ________ ________ extra part: ________ ________ extra part: ________ ________

DUT Δ________ LMK Δ________ UWV Δ _______

d. e. f.

f.

extra part: ________ ________ extra part: ________ ________ extra part: ________ ________

RQS Δ________ JIH Δ________ BAC Δ _______

45. For which value(s) of x are the triangles congruent?

a. x = ______________ b. x = _______________ c. x = _______________

46. Describe what is being constructed in the figure at right.

47. Refer to the figure at the right.

a.) 𝐸𝐵̅̅ ̅̅ is a ______________________ of ∆ABC

b.) _______________ is an altitude of ∆ABC.

48. Find the value of x if 𝐴𝐷̅̅ ̅̅ is an altitude of ∆ABC.

49. Solve for 𝒙.

a. b. c.

A

D

1

m 3 = x2

m 4 = 7x - 10

B

E C

2 3 4

4x + 8 7x - 4

A

B R

C

W

S R

Z

T

x2 + 2x x2 + 24

50. In GHI , R, S, and T are midpoints. If 75m G and 63m HSR . Find the m H .

51. PM = 4x + 7 and PN = 12x – 5 71. Solve for 𝑥.

Find PL.

52. a) According to the diagram, what are the lengths of PQ and PS ?

b.) How is PR related to SPQ?

c.) Find the value of n.

d.) Find mSPR and mQPR.

53. Find the value of the missing variables in the problems below.

a. b.

54. Find the range of possible measures for 𝑿𝒀̅̅ ̅̅ in ΔXYZ.

a. XZ = 6 and YZ = 6 b. XZ = 9 and YZ = 5 c. XZ = 11 and YZ = 6

55. Draw the angle bisectors of the triangle at right.

56. Draw the perpendicular bisectors of the triangle at right.

57. Draw the medians of the triangle at right.

58. Draw the altitudes of the triangle at right.

59. If HJ = 26, then KL = ______

60. If HJ = 3x – 1 and KL = x + 1, then HJ = ______

61. Solve for x.

62. Use the distance formula to find the length of 𝐴𝐵̅̅ ̅̅ , using the given coordinates of point A and point B. Then use the

midpoint formula to find the coordinates of the midpoint of 𝐴𝐵̅̅ ̅̅ ;

a. A( -2, 4 ) , B( 6, - 8) b. A( 21, 40) , B( - 13, 6) c. A( -5, -12) , B( -3, 4)

21 84x

Q

S R

P T

63.

64.

65. Given: , RT RU TS US

Prove: TRS URS

Given: 88 , 49

Prove: 43

m SRT m Q

m P

Given: is the midpoint of ,

Prove:

Q PR P QRT

SQP TQR

Given: 1 4

Prove: is isoscelesABC

Given:

is the midpoint of

Prove:

VX VW

Y WX

VYX VYW

A

B C

66.

67.

68. Given: , 80 , (3 1)

Prove: 27

AB AC m C m B x

x

69.

70. Given: 1 and 3 are supplementary

Prove: m n

Given: and are right angles,

Prove:

TRS TRL RTS RTL

RS RL

3

2

1

p

n

m

T

S R L