1

Midterm Review Geometry Fundamentals of Geometry

1. In the diagram of rAEB, !" is extended

to R and K, and m∠ 3 = m∠ 4 = 135. Triangle AEB must be (1) equilateral

(2) acute and isosceles

(3) obtuse and isosceles

(4) right and isosceles 2. In the diagram, m∠ BDC = 100°, m∠ A = 50°, and m∠ DBC = 30°.

Which statement if true? (1) rABD is obtuse.

(2) rABC is isosceles.

(3) m∠ ABD = 80°

(4) rABD is scalene. 3. In the diagram of quadrilateral NAVY, m∠ YNA = 30°, m∠ YAN = 38°,

m∠ AVY = 94°, and m∠ VAY = 46°.

Which segment has the shortest length? (1) $% (3) &$

(2) '% (4) &%

A

1800r i E B

90 xA EtxB l8At45 451

allsideslanglesE f f KA902sideslangles5

allangleslessthan90 135 1352sideslangles5 45 usoneanglegreaterthan90oneangteeg.ua ogo

s2sidesangles

BB

oneanglegreaterthan904 5030A D D c2sideslangles54 A DtxB iso Ntsb 1318

so so B Iso goo actioo 30 1841350 so gotoo so c 50

nosideslangles Ed

44 94 46 18044 40

402

shortestsideis oppthesmallestinthesmallertriangle

2

4. Using the diagram on the right, label each statement as TRUE or FALSE. I. (∠ 1 + (∠ 2 + (∠ 3 = 180 II. (∠ 2 = (∠ 3 III. (∠ 3 + (∠ 4 = 180 IV. (∠ 2 + (∠ 3 = (∠ 4 V. (∠ 1 + (∠ 2 = (∠ 4

5. Lines l and m are cut by transversal n. Which statement would prove l || m ? (1) m∠2 = m ∠6 (2) m∠2 = m ∠3 (3) m∠7 + m ∠8 = 180˚ (4) m∠3 + m ∠5 = 90˚

6. In this figure, transversal e intersects lines a, b, c, and d.

Which lines must be parallel? (1) a and c (3) b and d (2) b and c (4) a and d 7. In the diagram, $!)" ∥ +,-,

and ,! and ,) are drawn.

If m∠ EFG = 32° and m∠ AEG = 137°, what is m∠ EGF ?

(1) 11° (3) 75°

(2) 43° (4) 105°

TRUE3K's of a o upto 180

FALSETRUE

supplementary s upto180FALSETRUE

exteriors of a o is to 0thesum ofthe2 nonadjacentinterior s ofthe o fleett m t T

m mcorresponding 4s m t t m 180vertical 4 s t m

supplementary sm t

no 60 74110 sogou.iomotto 60 not so

alt.int 4 I corresponding4 E

I161143 32180

ios

4Gt43 32 18046 105

3

Constructions

8. The four diagrams below each illustrate a geometry construction. Label each one as either perpendicular bisector, altitude, median, or angle bisector.

9. Based on the construction shown, which conclusion is not always true? (1) $" ⊥ +- (3) AE = EB

(2) AB = CD (4) CE = DE 10. Based on the construction shown, which statement must be true?

(1) m∠ ABD = /0 m∠ CBD

(2) m∠ ABD = m∠ CBD (3) m∠ ABD = m∠ ABC

(4) m∠ CBD = /0 m∠ ABD

vertex midpt vertex 1 toopp cut half cutseg halfsidemedian altitude 4bisector 1 bisector

1 bisectorcontruction900 and cut in half

30 30

30 30 6030 60 30

3030 30 J

K bisectorcontructioncut in half

4

Construct each of the following. Show all construction marks.

11. perpendicular bisector of 1% 12. perpendicular to $" through point M

13. altitude to "+ from point A 14. median to $+ from point B

15. inscribed square 16. inscribed equilateral triangle

stepl step1morethan 2pointshalfway samedistafromX fromMandY step2step2 morethanconnect halfwaybothpointsd intersect step3

o connectbothpoinofintersec

Sleptummm ummmvertex 1 toopp vertex midpt morethan

step1 side halfwayfromA2points a andcamedistance

fromA Step2step2 connectorethan bothpoinhalfway ofintersectogetmidstep3 stepsconnect connec

midptbothpoints withBd intersectwithA

mum mumstepI 4 sections slept 3 sectionsrawdiameter measureradius from6 sectionsstep2

instruct1bisector step2stantonpointingdiameterendpts oncircleusingthatthebisector radiusmeasuretersectsthecircle steps2morepts connectevery

step's otherpointconnectall4pts

5

Transformations 17. Using the grid shown, find the coordinates

of each point after the given transformation.

a. Find A’ the image of A after r y = x

b. Find B’ the image of B after r x-axis

c. Find C ’ the image of C after R 90°

d. Find D’ the image of D after D 2 centered about the origin 18. Using the grid shown, find the coordinates

of each point after the given transformation.

a. Find A’ the image of A after r(0,0)

b. Find B’ the image of B after r y-axis

c. Find C ’ the image of C after r x = 1

d. Find D’ the image of D after T –2,3

19. On the set of axes shown, rectangle ABCD

can be proven congruent to rectangle KLMN using which transformation?

(1) rotation

(2) translation

(3) reflection over x = 0

(4) reflection over y = 0 20. Which coordinate rule represents a reflection of the x-axis?

(1) (x, y) → (y, x) (3) (x, y) → (–x, y)

(2) (x, y) → (–y, x) (4) (x, y) → (x, –y)

gXB C

reflectoverAL5 3 A 3 5 y x DreflectoverBC4 4 B'C4,4 x axisrotate900Counter4 2 C 2,4 clockwisedilatefromorigin

D 2 1 D 4,2 by2 A

Dreflect

AC5,3 A 5 3 thorough

reflect uoverBC4 4 B 4 4 y axisreflectover c4 2 C'f2 2 Atranslate BD 2,1 B 0,4 left2up3 11 1

mirrorxaxis imagetr4 0

is

6

21. On the graph shown, point A(3, 4) and "+ with coordinates B(4, 3) and C(2, 1) are graphed.

What are the coordinates of B ’ and C ’ after "+ undergoes a dilation centered at point A with a scale factor of 2?

(1) B ’(5, 2) and C ’(1, –2) (3) B ’(5, 0) and C ’(1, –2)

(2) B ’(6, 1) and C ’(0, –1) (4) B ’(5, 2) and C ’(3, 0) 22. On the set of axes below, rABC ≅ rDEF. Describe a

sequence of rigid motions that maps rABC onto rDEF.

23. Which rotation about its center will not carry a regular decagon onto itself? (1) 36° (3) 216°

(2) 180° (4) 270° 24. In regular hexagon ABCDEF shown, $-, "!, and

+) all intersect at G. When rABG is reflected over ", and then rotated 180° about point G, rABG is mapped onto

(1) rFEG (3) rCBG

(2) rAFG (4) rDEG

v B

VC

One of many possible answersB c Reflect over y axisthenit I Reflectover 9 2

mumlookexactlikeitstarted360

decagon To_36tdecade 36,72,108,144,180sides 216,252,288,324,360

texts

7

Properties of Congruent Triangles and Definitions/Theorems

25. State the most efficient method of proving each pair of triangles congruent.

a. b. c. d.

e. f. g. h. 26. Mark each set of diagrams with the given information.

Determine the most efficient method that will prove these triangles are congruent. a. Given: $+ ≅ "+

a. Given: +- ⊥ $"

a. Gien:

b. Given: "+ bisects ∠BCD

a. ∠$"+ ≅ ∠ADC

b. Given

a. Given:

c. Given: C is the midpoint of $!

. ∠"$+ ≅ ∠DEC

a. Given:

d. Given: 13 and %4 bisect each other

a. Gi

a. Given:

HL

X IX

H

ASA AAS SAS SASH

L LXL h K D

HSSS HL Sss ASA

AC

HL AASLx H X L

TL L

SASr ASAe il C g Ilei gC

8

27. State what pair of parts will be congruent based on the given information. a. 56 is an altitude

a.

a. Given:

≅

b. !7 is an angle bisector

a. Given:

≅

c. +- is a median

a. Give

a. Given: ≅

d. "- is the perpendicular bisector of $+

a. Given: ≅ .

and ≅ .

28. As shown in the diagram, $+ bisects ∠BAD and ∠B ≅ ∠D.

Which method could be used to prove rABC ≅ rADC? (1) SSS (3) SAS

(2) AAA (4) AAS 29. In the diagram of quadrilateral ABCD, $" ∥ +-,

∠ABC ≅ ∠CDA, and diagonal $+ is drawn. Which method could be used to prove rABC ≅ rCDA? (1) AAS (3) SAS

(2) SSA (4) SSS

1 rights cutinhalf

7

4KLI KKLJ 4GEH 4FEH

midpoint1 rights cutinhalf

ABBA KBDC

I irmidpt AD ODAD BD l I

b t

I e

tT l

9

Congruent Triangle Proofs 30. Given: $- ⊥ -!

a. Given. : +! ⊥ -!

2. Given: $+ bisects -! at B.

Prove: rADB ≅ rCEB 31. Given: 83 bisects ∠PSR

a. Given: 93 ≅ 34

2. Given: Prove: ∠P ≅ ∠R

Statements ReasonsAD DE CE DE GivenKDE K E linesform rightanglesACbisectsDEat B GivenDBEEB abisectorcuts a segmentinto 2 parts

4DBA E EBC intersecting linesform E vertical s

OADB E oCEB ASA

tT 1

Statements Reasonsasbisects PSB GivenKPSQ E KRSQ abisectorcuts an 4 into 2 partsPs ESR GivenQs EQS Reflexive

OPQS ORQS SASKP E KR CPCTC

10

Similar Triangles

32. In the diagram, rABC ~ rDEF. If AB = 36, BC = 30,

and EF = 15, find -!.

33. Given right triangle ABC with a right angle at C, m∠B = 61°.

Given right triangle RST with a right angle at T, m∠R = 29°.

34.

EEE 15 X

3011 540DE 18

ToX 18

01 0229 61

I 2 I I2 I 2 2I 1 I 22 2 1 2

36F 2.445Ts 2.5

11

35. What additional piece of information would prove that rIJK ~ rLMN?

(1) NM = 18 (3) LM = 18

(2) NM = 15 (4) LM = 10 36. 37. In the diagram, 38 and 94 intersect at T, 98 is drawn, and 93 ∥ 84.

Prove rPST ~ rRQT.

D LMNto

811 14411 18

A

C BD

F E

w

eKH

R

Statements ReasonsPS QR GivenPST I Bat linescutbya transversal form alt int s

4SPT I QRTKSTP I QTR intersecting linesform E vertical s

OPST ORQT AAA

12

38. In the diagram of rABC, D is the midpoint of $", E is the midpoint of "+,

and F is the midpoint of $+.

If AB = 20, BC = 12, and AC = 16, what is the perimeter of parallelogram FEBD?

39. As shown in the diagram, $" and +-, and intersect at E, and $+ ∥ "-.

If DE = 6, BE = 8, AE = 12.8, BD = 5, and AC = 15, what is the length of !+?

40. In the diagram of rABC, points D and E are on $" and +" ,

respectively, such that $+ ∥ -!.

If AD = 24, DB = 12, and DE = 4, what is the length of $+?

8

10 10 6 6 320 66 8

midse.g.gg20

ftp.E 2g8 8 X X EC_9.66

521 4811 9.6

Acta TheyEffect

x x i

4 I 12I n 11 12

AD

X 36 y 12E B

C B

13

41. In the diagram of rABC, -! ∥ "+.

If AB = 10, AD = 8, and AE = 12, what is the length of !+?

42. In the diagram of rABC, ∠ABC is a right angle, AC = 12, AD = 8,

and altitude "- is drawn. 43. In right triangle PRT, m∠P = 90°, altitude 98 is drawn to hypotenuse 4;,

RT = 17, and PR = 15.

Determine and state, to the nearest tenth, the length of 48.

A8 12

D

DADETABC 7 a exec3

I CC

811112 10 12 A8 96 120 to xH2811 24IT TX 3 B C

4 L4 L SS H

X xA s 2.485

11548ntryrs

4 LL S

L A 17 1515 X

s sixH 1,7 22,5T

71 13.2352941RQ 13.2