Module Three: Geometry LUOA

Given that SQ bisects ∠PSR and ∠SPQ≅∠SRQ, which of the following proves that PS≅SR?

1.∠SPQ≅ ∠SRQ (Given) 2.SQ bisects ∠PSR.(Given) 3.∠PSQ≅∠QSR (Def. of ∠ bisect) 4.SQ≅SQ (Reflex. Prop. of ≅) 5.△PQS≅△RQS (AAS Steps 1, 3, 4) 6.PS≅SR (CPCTC)

Identify the side lengths of the triangle.

16; 16; 6

The measure of one of the acute angles in a right triangle is 59°. What is the measure of the other acute angle?

31°

Identify the side lengths of the triangle.

50; 50; 50

If the measures of the angles of a triangle are in the ratio of 19: 13: 4, then the expressions 19x, 13x, and 4x represent the measures of these angles. What are the measures of the angles?

95°, 65°, 20°

Given that polygon PQRS≅ polygon TUVW, identify the congruent corresponding part for QR.

UV

Find the value of b.

b = 14

Classify △LMN△LMN by its angle measures.

equiangular

Classify △PQR by its side lengths.

isosceles

△QRS≅△JKL, RS=−3m+10, and KL=2m+15. Find m and KL.

m = −1 and KL = 13

Identify m∠EFHm.

m∠EFH = 112°

Which of the following represents the diagram below?

square with side length 6m units

Find the value of x.

x = 19

Find the measures of ∠B and ∠C.

∠B = 40° and ∠C = 40°

Find the measures of ∠N and ∠P.

∠N = 25° and ∠P = 25°

Identify the equation in point-slope form for the perpendicular bisector of the segment with endpoints J(3,−4) and K(−7,−2).

y + 3 = 5(x + 2)

△KWJ has vertices K(3,3), W(1,−5), and J(−5,−1). Find the equation of the median through J in slope-intercept form.

y = −1

△DEF has vertices D(−5,−1), E(3,3), and F(1,−5). Find the equation of the altitude through F in slope-intercept form.

y = −2x − 3

△DEF has vertices D(−5,−1), E(3,3), and F(1,−5). Find the equation of the median through F in slope-intercept form.

y = −3x − 2

Identify the equation in point-slope form for the perpendicular bisector of the segment with endpoints J(−24,−4) and K(−4,6).

y − 1 = −2(x + 14)

Identify the equation in point-slope form for the perpendicular bisector of the segment with endpoints B(6,9) and C(8,−7).

y−1= 1/8(x−7)

Find the measures of ∠L and ∠H.

∠L = 20° and ∠H = 20°

Which of the following are the coordinates of a rectangle with a length of 8 units and width of 5 units in the coordinate plane?

(4, 1), (−4, 1), (4, 6), (−4, 6)

Which of the following are the coordinates of a rectangle with a length of 4 units and width of 3 units in the coordinate plane?

(−2, 1), (−2, −2), (2, 1), (2, −2)

Given that SQ bisects ∠PSR and PS≅SR, which of the following proves that △PQS≅△RQS?

1.PS ≅SR (Given) 2.SQ bisects ∠PSR.(Given) 3.m∠PSQ=m∠QSR (Def. of bisect) 4.∠PSQ≅∠QSR (Def. of ≅) 5.SQ ≅SQ (Reflex. Prop. of ≅) 6.△PQS≅△RQS (SAS Steps 1, 4, 5)

Identify the proof to show that △ABD≅△CBD, where ∠BDA≅∠BDC are right angles, D is the midpoint of AC, AB ≅BC, and BD bisects ∠B.

1.∠BDA and ∠BDC are right angles (given) 2. D is midpoint of AC (given) 3. AB ≅BC (given) 4. BD bisects ∠B (given) 5. ∠BDA≅∠BDC (Right ∠≅ Thm.) 6. AD ≅DC (definition of midpoint) 7. BD ≅DB (Reflex. Prop. of ≅) 8. ∠DBA≅∠DBC (Def. of bisector) 9. ∠A≅∠C (Third angles Thm.) 10. △ABD≅△CBD (definition of congruent triangles)

Given that ∠FAB≅∠GED and C is the midpoint of AE, which of the following proves that △ABC≅△EDC?

1.∠FAB≅∠GED (Given) 2.∠BAC is the supp. of ∠FAB;∠DEC is thesupp. of ∠GED (Def. of Supp. ∠s) 3.∠BAC≅∠DEC (≅ Supp. Thm.) 4.∠ACB≅∠DCE (Vert. ∠s Thm.) 5. C is the midpoint of AE (Given) 6.AC ≅EC (Def. of mdpt.) 7.△ABC≅△EDC (by ASA Steps 3, 6, 4)

Identify the proof to show that △JKN≅△MKL, where K is the midpoint of JM and NL ∠L≅∠N, and LM≅JN.

1.∠L≅∠N (given) 2. K is midpoint of JM and NL (given) 3. JK ≅KM ;NK ≅KL (definition of midpoint) 4. LM ≅JN (given) 5. ∠JKN≅∠LKM (vertical angles theorem) 6. ∠J≅∠M (third angles theorem) 7. △JKN≅△MKL (definition of congruent triangles)

Identify the proof to show that △PQO≅△RSO, where O is the midpoint of PR and QS, ∠P≅ ∠R, and PQ ≅ RS.

1.∠P≅∠R (given) 2. O is midpoint of PR and QS (given) 3. PO ≅OR ;QO ≅OS (definition of midpoint) 4. PQ ≅ RS (given) 5. ∠POQ≅∠ROS (vertical angles theorem) 6. ∠Q ≅∠S (third angles theorem) 7. △PQO ≅△RSO (definition of congruent triangles)

Given that polygon PQRS≅ polygon TUVW, identify the congruent corresponding part for ∠Q.

∠U

Which of the following represents the diagram below?

rectangle with length 5r units and width 8r units

Which of the following represents the diagram below?

right triangle with leg lengths 4f units and 7g units

Classify △STU by its side lengths.

scalene

△PQR is the midsegment triangle of △ABC. Identify the perimeter of △PQR.

15

△LMN is the midsegment triangle of △XYZ. Identify the perimeter of △LMN.

19.5

CD is a median of △ABC, AD=3x+10, and DB=7x+2. Find AB.

AB = 32

Given that AD is the perpendicular bisector of BC, AB=2a+7, and AC=6a−21, identify AC.

AC = 21

Given that AD is the perpendicular bisector of BC, AB=15, AC=x, and BD=0.25x, identify BC.

BC = 7.5

Identify the measure of EF in the diagram.

EF = 8

If HJ is a median of △IKH, which of the following statements is always true?

IJ≅JK

Identify the value of n in the diagram.

Identify the value of nn in the diagram.

Identify the measure of PR in the diagram.

PR = 11

Given that PS is the perpendicular bisector of QR, PR=2+7a, and PQ=3a+38, identify PR.

PR = 65

Identify the measure of XZ in the diagram.

XZ = 16

Identify the value of m in the diagram.

m = 3

Given that m∠ABC=44∘, identify m∠DBC.

m∠DBC = 22°

Given that m∠IHJ=(3a+6)∘ and m∠KHJ=(5a−18)∘, identify m∠IHJ.

m∠IHJ = 42

Given that m∠IHJ=(−3y+6)∘ and m∠KHJ=(31+2y)∘, identify m∠IHK.

m∠IHK = 42°

Given that m∠IHJ=(2x+17)∘ and m∠KHJ=(5x−10)∘, identify m∠IHK.

m∠IHK = 70°

Given that m∠OMP=(88−2x)∘ and m∠NMP=(5.5x−32)∘, identify m∠NMO.

m∠NMO = 112°

Given that m∠USV=41∘, identify m∠TSU.

m∠TSU = 82°

Given that m∠TSV=32∘, identify m∠UST.

m∠UST = 64°

Given that C is the midpoint of BD and that ∠BAF≅∠DEG, which of the following triangle congruence statements can be used to prove that BA≅DE?

Angle-Angle-Side Congruence

Given that H is the midpoint of FK and that ∠F≅∠K, which of the following triangle congruence statements can be used to prove that JK≅FG?

Angle-Side-Angle Congruence

Given that AD is the perpendicular bisector of BC, AB=12, AC=m−6, and BD=0.25m+2, identify BC.

BC = 13

Given that AD is the perpendicular bisector of BC, AB=7.4, AC=7.4, and BD=3.2, identify BC.

BC = 6.4

Identify the postulate that proves the triangles are congruent.

HL

Find the value of JK given that LJ=3.2x−4 and LK=5x−14.8.

JK = 15.2

Which of the following shows that △JHK≅△GHK when m=3?

KH ≅KH , ∠JHK≅∠KHG,and JH ≅GH So △JHK≅△GHK by SAS.

Given that KP is the perpendicular bisector of JL, KJ=5x+6, and KL=3x+24, identify KL.

KL = 51

Given that polygon PQRS≅ polygon LMNO, identify the congruent corresponding part for PQ.

LM

Classify △ACD by its angles and sides.

obtuse; scalene

Classify △SRP by its angles and sides.

obtuse; scalene

△JKL≅△RST, KL=4p−12, and ST=−3p+16. Find p and ST.

p = 4 and ST = 4

Which of the following represents the diagram below?

square with side length 4k units

Question 2: 20 ptsSkip to question text. △DEF has vertices D(−5,−1), E(3,3), and F(1,−5). Find the equation of the altitude through F in slope-intercept form.

y = −2x − 3

Identify the equation in point-slope form for the perpendicular bisector of the segment with endpoints E(−7,6) and G(9,−2).

y − 2 = 2(x − 1)

Find the value of JG.

JG = 34

Which of the following are the coordinates of a rectangle with a length of 9 units and width of 3 units in the coordinate plane?

(0, −4), (9, −1), (9, −4), (0, −1)

Which of the following are the coordinates of a rectangle with a length of 5 units and width of 2 units in the coordinate plane?

(1, 4), (−4, 4), (1, 6), (−4, 6)

The diagram is a map showing a zoo, a planetarium, and a museum. Find the measure of the angle formed at the planetarium.

88°

Given that rectangle PQRS with coordinates P(0,0), Q(8,0), R(8,4), S(0,4), U is the midpoint of PQ, and V is the midpoint of RS, which of the following proves that UV=QR?

By the Midpoint Formula, the coordinates of U are (4, 0) and V are (4, 4).Then UV = 4, and QR = 4.Thus UV = QR.

Find the value of DF given that DE=1.5m−9 and EF=4.5m−42.

DF = 7.5

Find the measures of ∠A and ∠R.

∠A = 63° and ∠R = 63°

△ABC is divided into two congruent triangles. Which of the following congruency statements is incorrect?

All True

Given that rectangle JKLM has coordinates J(0,0), K(10,0), L(10,8), and M(0,8), T is the midpoint of JK, and V is the midpoint of ML, which of the following proves that TV=JM?

By the Midpoint Formula, the coordinates of T are (5, 0) and the coordinates of V are (5, 8).Then TV = 8, and JM = 8.Thus TV = JM.

Given that FE is the perpendicular bisector of DG, DF=14, FG=4x, and DE=1.25x, identify DG.

DG = 8.75

Given that AC⊥CD, BD⊥CD, and BC≅AD, which of the following triangle congruence statements can be used to prove that ∠ADC≅∠BCD?

Hypotenuse-Leg Congruence

Which postulate, if any, can be used to prove that the triangles are congruent?

None

Given that polygon JKLM≅ polygon WXYZ, identify the congruent corresponding part for KL.

XY

Find the value of a.

a = 17

Given that polygon ABCD≅ polygon EFGH, identify the congruent corresponding part for ∠A.

∠E

Find the measures of ∠F and ∠H.

∠F = 72.5° and ∠H = 72.5°

Find the measures of ∠K and ∠L.

∠K = 42° and ∠L = 42°

Given that polygon PQRS≅ polygon LMNO, identify the congruent corresponding part for ∠Q.

∠M

Identify the proof to show that △PQS≅△RQS, where ∠QSP≅∠QSR are right angles, S is the midpoint of PR, PQ ≅QR and QS bisects ∠Q.

1.∠QSP and ∠QSR are right angles (given) 2. S is midpoint of PR (given) 3. PQ ≅QR (given) 4. QS bisects ∠Q (given) 5. ∠QSP≅∠QSR (Right ∠≅ Thm.) 6. PS ≅SR (definition of midpoint) 7. QS ≅SQ(Reflex. Prop. of ≅) 8. ∠SQP≅∠SQR (Def. of bisector) 9. ∠P≅∠R (Third angles Thm.) 10. △PQS≅△RQS (definition of congruent triangles)

Identify m∠Am and m∠Em.

25°, 25°

Identify the side lengths of the triangle.

36; 36; 32

Identify m∠U and m∠X.

36°, 36°

If the measures of the angles of a triangle are in the ratio of 4: 6: 8, then the expressions 4x, 6x, and 8x represent the measures of these angles. What are the measures of the angles?

40°, 60°, 80°

The measures of the angles of a triangle are in the ratio of 5:6:7. What are the measures of the angles?

50°, 60°, 70°

Identify m∠U and m∠X.

56°, 56°

Identify the postulate that proves the triangles are congruent.

AAS

It is given that JA ∥ DR, identify the postulate that proves the triangles are congruent.

AAS or ASA

Identify the true statement. Explain why the statement is true.

All equilateral triangles are acute triangles. All angles in an equilateral triangle measure 60°.

Identify the true statement.

An equilateral triangle has three congruent sides.

Given that AD is the perpendicular bisector of BC, AB=21.3, AC=21.3, and DC=9.1, identify BC.

BC = 18.2

Find the value of BC.

BC = 3

Given that AD is the perpendicular bisector of BC, AC=28, AB=28, and DC=16.5, identify BC.∠B = 40° and ∠C = 40°

BC = 33

Given that rectangle PQRS has coordinates P(−8,2), Q(2,2), R(2,8), and S(−8,8), T is the midpoint of QR, and V is the midpoint of PS, which of the following proves that QT=PV?

By the Midpoint Formula, the coordinates of T are (2, 5) and the coordinates of V are (−8, 5).Then QT = 3, and PV = 3.Thus QT = PV.

Identify the true statement.

If two angles of a triangle are congruent, then the sides opposite the angles are congruent.

Given that PS is the perpendicular bisector of QR, PQ=3.5m+18, and PR=6m+3, identify PQ.

PQ = 39

Find the value of RT.

RT = 116.2

Which postulate, if any, can be used to prove that the triangles are congruent?

SAS

Which postulate, if any, can be used to prove that the triangles are congruent?

SSS

Identify the postulate that proves that △CAB and △CAD are congruent.

SSS, SAS, HL

Given that BE ≅CE and AE ≅DE, which of the following triangle congruence statements can be used to prove that BA≅CD?

Side-Angle-Side Congruence

Given that SQ bisects ∠PSR and PS ≅SR, which of the following triangle congruence statements can be used to prove that ∠P≅∠R?

Side-Angle-Side Congruence

Which of the following uses the given set of points to prove that ∠O≅∠R? M(−1,2), N(−2,−2), O(4,−2), P(−1,−2), Q(−2,2), R(4,2)

Since MN=PQ=17‾‾‾√, NO=QR=6,and OM=RP=41‾‾‾√, ΔMNO≅ΔPQR by SSS. Therefore, ∠O≅∠R by CPCTC.

Classify △JKL by its angle measures.

acute

Classify △BGF by its angles and sides.

acute; equilateral

Identify m∠LMN.

m∠LMN = 60°

Classify △SVU△SVU by its angles and sides.

obtuse; scalene

△ABC≅△LMN, AB=−3y+10, and LM=2y+15. Find y and AB.

y = -1 and AB = 13

△PQR≅△DEF, PR=19z−9, and DF=7z+15. Find z and PR.

z = 2 and PR = 29

Two congruent triangles are given. Which of the following congruency statements is incorrect?

ΔQPR ≅ ΔZXY

Which of the following shows that △ABD≅△CBD, when y=5?

∠ADB≅∠CDB, BD ≅BD, and AD ≅CD. So △ABD≅△CBD by SAS.

Find the measures of ∠B and ∠C given that m∠B=(2.5x+6)∘ and m∠C=(7x−12)∘.

∠B = 16° and ∠C = 16°

Find the measures of ∠B and ∠C.

∠B = 72° and ∠C = 72°

Find the measures of ∠B and ∠C.

∠B = 78° and ∠C = 78°