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2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino

Part ½

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Chapter 1: Tools of Geometry Chapter 2: Reasoning and Proof Chapter 3: Parallel and Perpendicular Lines Chapter 4: Congruent Triangles Chapter 5: Relationships Within Triangles Chapter 6: Quadrilaterals Chapter 7: Similarity Chapter 8: Right Triangles and Trigonometry Chapter 9: Transformations Chapter 10: Area Chapter 11: Surface Area and Volume Chapter 12: Circles

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Tools of Geometry 1. Identify the ray that is the opposite of

.

• • • • 2. Which segment is NOT skew to segment

?

• • • • 3. The complement of an angle is 29°. What is the measure of the angle?

•

61°

•

29°

•

151°

•

149° 4.

bisects ∠RST. Find the measure of ∠RST if m∠QST = 45°.

5

2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino •

40°

•

100°

•

45°

•

90°

6

5. In the figure shown, m∠AED = 97. Which statement is false?

•

m∠AEB = 83

•

m∠BEC = 97

•

∠AEB and m∠DEC are congruent angles.

•

∠BEC and m∠CED are vertical angles. 6. a. If EF = 5w + 28, FG = 7w + 20, and EG = 36, find the value of w. b. Find EF and FG.

•

a. w = 2; b. EF = 38, FG = 34

•

a. w = –1; b. EF = 23, FG = 13

•

a. w = 4; b. EF = 48, FG = 48

•

a. w = 3; b. EF = 43, FG = 41 7. What is the next term in the sequence? 4, 7, 10, 13, ….

•

16

•

10

•

14

•

•

19 8. Anita is practicing to join the swim team. The first time she timed her lap it was 1 minute and 12 seconds. After one week of practice, her best lap was 1 minute and 9 seconds. After a second week of practice, her best lap was 1 minute and 6 seconds. Use inductive reasoning to predict how long her best lap will be after 3 weeks of practice.

1 minute 3 seconds


1 minute 9 seconds

•

1 minute 6 seconds

•

1 minute 5 seconds 9. The points of a line can be put into one-to-one correspondence with the real numbers so that the distance between any two points is the _____ of the _____ of the corresponding numbers.

•

product, absolute values

•

absolute value, difference

•

difference, values

•

7

sum, squares 10. Which is an incorrect way to name the line?

• •

line t

• • 11. If two lines intersect, then they intersect in exactly _____.

•

one point

•

one line

•

two points

•

one plane 12. Determine the coordinates of the midpoint of and B(2, 2).

•

midpoint = ( ,

); distance =

•

midpoint = ( ,

); distance = 5

•

midpoint = ( ,

); distance = 5

•

midpoint = ( ,

); distance =

and find the distance EB for the points E(3, 3)


8

13. If m∠DOE = 23° and m∠COD = 20°, then what is the measure of ∠COE?

•

40°

•

45°

•

48°

•

43° 14. Find AC and BD.

•

AC = 10; BD = 2

•

AC = 11; BD = 3

•

AC = 3; BD = 11

•

AC = –11; BD = –3 15. May's Internet Services designs Web sites. May noticed an increase in her customers over a period of 5 consecutive weeks. The first week she had 1 customer, the second week she had 3 customers, and the third week she had 5 customers. Use inductive reasoning to make a conjecture about the number of customers May will have in the sixth week.

•

10

•

12

•

9

•

11 16. Which drawing shows a three-dimensional figure sketched from these views?


9

•

•

•

• 17. If AB = 19 and BC = 3, then AC = _____?

•

19

•

25

•

22

•

16 18. A highway map of Ohio has a coordinate grid superimposed on top of the state. Springfield is at point (1, –4) and Columbus is at point (7, 1). The Springfield youth soccer team is going to Columbus to see a professional soccer match. The map shows a highway rest area halfway between the cities. What are the coordinates of the rest area? What is the distance between Springfield and Columbus? (one unit = 5.38 miles)

•

Rest area = (3,

•

Rest area = (–3,

); distance = 40 miles ); distance = 46 miles

2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino • •

10

Rest area = (– , 4); distance = 18 miles Rest area = (4, – ); distance = 42 miles 19. Write a formula that will give the area of the shaded region in the figure below.

•

A = (29 – 12) • (3 – 2)

•

A = (29 – 3) • (12 – 2)

•

A = 29 • 12

•

A = (29 – 2) • (12–3) 20. Noam walks home (8, 4) from school (2, –4) by walking 8 blocks north, then 6 blocks east. How much shorter would his walk be if there were a direct path from the school to his house?

•

14 blocks

•

4 blocks

•

10 blocks

•

The distance would be the same.


11

Tools of Geometry 1. Identify the ray that is the opposite of

•

•

.

CORRECT: 2. Find the coordinates of the midpoint of the segment connecting H(6, 0) andK(–8, 14). CORRECT: (–1, 7) 3. Name an angle complementary to ∠COD.

•

CORRECT: ∠BOC 4. If AB = 19 and BC = 3, then AC = _____?

•

CORRECT: 22 5. Which point is NOT colinear with two other points?

•

CORRECT: A


12

6. Which drawing shows a three-dimensional figure sketched from these views?

•

CORRECT: 7. Anita is practicing to join the swim team. The first time she timed her lap it was 1 minute and 12 seconds. After one week of practice, her best lap was 1 minute and 9 seconds. After a second week of practice, her best lap was 1 minute and 6 seconds. Use inductive reasoning to predict how long her best lap will be after 3 weeks of practice.

•

CORRECT: 1 minute 3 seconds 8. What is the first step in constructing a congruent segment?

•

CORRECT: Draw a ray. 9. In the figure shown, name the ray that appears to bisect ∠JKL.

•

•

CORRECT: 10. If KM = 305, and KL = 3x + 2, and LM = 5x – 1, find the value of x. Then findKL and LM.

CORRECT: x = 38, KL = 116, LM = 189 11. Ralph wants to put up a fence around his rectangular garden, which measures 48 feet by 54 feet. He wants to include the path around it on two sides that is 3 feet wide. How much fencing material does he need?

•

CORRECT: 216 ft 12. Supplementary angles are two angles whose measures have sum _____. Complementary angles are two angles whose measures have sum _____.


13

•

CORRECT: 180; 90 13. The points of a line can be put into one-to-one correspondence with the real numbers so that the distance between any two points is the _____ of the _____ of the corresponding numbers.

•

CORRECT: absolute value, difference 14. Which figure is a net for a cube?

•

CORRECT: 15. Which of these points, if any, is the midpoint of

?

•

CORRECT: none of them 16. a. If EF = 5w + 28, FG = 7w + 20, and EG = 36, find the value of w. b. Find EF and FG.

•

CORRECT: a. w = –1; b. EF = 23, FG = 13 17. Which of the following describes the figure below?

•

CORRECT: 18. Find AB and BC. Are AB and BC congruent?

•

CORRECT: AB = 3; BC =2 ; no, they are not congruent 19. What is the next term in the sequence? 4, 7, 10, 13, ….

•

•

CORRECT: 16 20. If two planes intersect, then they intersect in exactly _____. CORRECT: one line

.


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Reasoning and Proof 1. Which of the following choices shows a true conditional with the hypothesis and conclusion identified correctly?

•

If yesterday was Saturday, then tomorrow is Monday. hypothesis: Yesterday was Saturday. conclusion: Tomorrow is Monday.

•

If yesterday was Saturday, then tomorrow is Monday. hypothesis: Tomorrow is Monday. conclusion: Yesterday was Saturday.

•

If yesterday was Saturday, then today is Monday. hypothesis: Yesterday was Saturday. conclusion: Tomorrow is Monday.

•

If yesterday was Saturday, then today is Monday. hypothesis: Tomorrow is Monday. conclusion: Yesterday was Saturday. 2. Select the appropriate property of equality for the statement. If a = b, then b= a.

•

Addition Property

•

Symmetric Property

•

Transitive Property

•

Substitution Property 3. Use the Law of Detachment and the Law of Syllogism to draw a conclusion from the given statements. Franz is taller than Isabel. Isabel is shorter than Ellen. Janina is taller than Franz.

•

Janina is shortest.

•

Janina is taller than Isabel.

•

Franz is the tallest.

•

Ellen is shorter than Isabel. 4. Determine if the conditional and its converse are true. If they are both true, select which biconditional correctly represents them. If either the conditional or the converse is false, select the counterexample which disproves the statement: If four points are non-coplanar, then they are non-collinear. If four points are non-collinear, then they are non-coplanar.

•

Counterexample: If four points are non-coplanar, they still may be collinear.

•

If and only if four points are non-collinear are they non-coplanar.

•

Four points are non-coplanar if and only if they are non-collinear.


• •

Counterexample: Four points may be non-collinear and yet lie in the same plane. 5. Is it possible to use the Law of Detachment to draw a conclusion? I will go to the cinema if and only if I can afford to buy a ticket. The movie starts at 7:30. Yes, it is possible. No, it is not possible. 6. One way to show that a statement is NOT a good definition is to find a _____.

•

biconditional

•

conditional

•

converse

•

counterexample 7. Which of the following is an example of the Reflexive Property of Equality?

•

If x = –2, then x + 4 = –2 + 4.

•

x–2=x–2

•

If y = x + 4, then x + 4 = y.

•

If x – 2 = y and y = 4, then x – 2 = 4. 8. Find the value of the variable.

•

2

•

3

•

4

•

15

5 9. Is the converse of the true statement also true? If four points are non-coplanar, then they are non-collinear.

•

Yes, it is true.

•

No, it is false.


16

10. Which biconditional is a good definition? Two angles are adjacent if and only if they share a common side.

•

A point is the midpoint of a segment if and only if it is between the endpoints of the

• segment. • •

Two angles are supplementary if and only if they form a linear pair. Two lines are parallel if and only if they never intersect. 11. Two angles whose sides are opposite rays are called _____. Two coplanar angles with a common side, a common vertex, and no common interior points are called _____.

•

adjacent angles; vertical angles

•

vertical angles; adjacent angles

•

adjacent angles; complementary angles

•

vertical angles; supplementary angles 12. Which is the converse of this conditional? If it is appropriate, then I play golf.

•

If it is appropriate, then I do not play golf.

•

If I play golf, then it is appropriate.

•

If I do not play golf, then it is appropriate.

•

If it is not appropriate, then I play golf. 13. Which is the Law of Syllogism in symbolic form?

•

If p → q and q → r are true statements, then p → r is a true statement.

•

If p → q is a true statement and p is true, then q is true.

•

If p → q and q → r are false statements, then p → r is a false statement.

•

If p → q is a false statement and p is false, then q is false. 14. Decide if the statement is reversible. If so, identify the true biconditional. Two lines that intersect at right angles are perpendicular.

•

If two lines intersect at right angles, then they are not perpendicular.

•

If two lines intersect at right angles, then they are perpendicular.

•

Two lines intersect at right angles if and only if they are perpendicular.

•

not reversible


17

15. Identify the converse of the conditional statement. Determine the truth values of the original conditional and its converse. If an angle is a right angle, then its measure is 90.

•

If the measure of an angle is 90, then it is a right angle. original: true converse: true

•

If an angle is not a right angle, then its measure is not 90. original: true converse: true

•

If an angle is not a right angle, then its measure is 90. original: true converse: false

•

If the measure of an angle is 90, then it is a right angle. original: true converse: false 16. Complete the proof. Given: ∠A and ∠B are right angles. Prove: ∠A ≈ ∠B

By the definition of a. _____, m∠A = 90 and m∠B = 90. By the b. _____ Property,m∠A = m∠B or ∠A ∠B.

•

a. complementary angles b. Substitution

•

a. right angles b. Reflexive

•

a. right angles b. Distributive

•

a. right angles b. Substitution 17. What is the conclusion of the following statement? "A number is even if the number is divisible by 6."

•

A number is even if the number is divisible by 6.

•

A number is divisible by 6 if the number is even.

•

A number is divisible by 6.

•

The number is even.

2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino 18. Find the value of the variables.

•

x = 29, y = 60

•

x = 26, y = 30

•

x = 26, y = 60

•

x = 29, y = 30 19. Find the value of the variable.

•

6

•

3

•

5

•

8 20. Supply the missing step in the following proof. Given: 4x – 6y = –8; x = 10 Prove: 8 = y

•

–6y = 32; Multiplication Property of Equality

•

–6y = –48; Subtraction Property of Equality

•

–6y =–48; Multiplication Property of Equality

•

–6y = 32; Subtraction Property of Equality

18


19

Reasoning and Proof 1. Which of the following is an example of the Reflexive Property of Equality? •

CORRECT: x – 2 = x – 2 2. Is the following statement reversible? If two angles are adjacent, then they share a common side.

•

CORRECT: No, it is not reversible. 3. Which is a counterexample to the following faulty definition? A square is a figure with four congruent sides.

•

CORRECT: A rhombus has four congruent sides. 4. Find the value of the variable.

•

CORRECT: 4 5. Which is a counterexample that proves this conditional is false? All prime numbers are odd.

•

CORRECT: 2 6. Which biconditional is a good definition?

•

CORRECT: Two angles are adjacent if and only if they share a common side. 7. Identify the converse of the conditional statement. Determine the truth values of the original conditional and its converse. If an angle is a right angle, then its measure is 90.

•

original: true converse: true CORRECT: If the measure of an angle is 90, then it is a right angle. original: true converse: true 8. Select the appropriate property of equality for the statement. If a = b, then a• c = b • c.

•

•

CORRECT: Multiplication Property


20

9. In the figure shown, m∠AED = 97. Which statement is false?

•

CORRECT: ∠BEC and m∠CED are vertical angles. 10. Select the appropriate property of equality for the statement. If a = b, then b= a.

•

CORRECT: Symmetric Property 11. Select the appropriate property of equality for the statement a = a.

•

CORRECT: Reflexive Property 12. A conditional can have a _____ of true or false.

•

CORRECT: truth value 13. Which is the converse of this conditional? If it is appropriate, then I go for a drive.

•

CORRECT: If I go for a drive, then it is appropriate. 14. Use the Law of Syllogism to draw a conclusion. If a country's population is more than 1.033 billion, then it has a higher population than India. If a country has a higher population than India, then it is has the highest population in the world.

•

CORRECT: If a country's population is more than 1.033 billion, then it is has the highest population in the world. 15. The complement of an angle is 29°. What is the measure of the angle?

•

CORRECT: 61° 16. Name an angle complementary to ∠COD.

•

CORRECT: ∠BOC 17. Decide if the statement is reversible. If so, identify the true biconditional. Two lines that intersect at right angles are perpendicular.

•

CORRECT: Two lines intersect at right angles if and only if they are perpendicular.


21

18. One way to show that a statement is NOT a good definition is to find a _____. •

CORRECT: counterexample 19. Which is the converse of this conditional? If it is a dog, then it is a pet.

•

CORRECT: If it is a pet, then it is a dog. 20. Use the Law of Syllogism to draw a conclusion. If a substance is water, then its molecular structure is H2O. If a substance has molecular structure H2O, then its boiling point at sea level is 100°C.

•

CORRECT: If a substance is water, then its boiling point at sea level is 100°C.

Parallel and Perpendicular Lines 1. If c ⊥ b and a || c, what is m∠2?

•

60°

•

90°

•

30°

•

not enough information 2. Which is an equation in slope-intercept form of the line that contains the pointsS(7, –9) and T(8, –7)?

•

y = 2x – 23

•

x + 2y = –23

•

x – 2y = 23

•

y = 2x + 23


22

3. Find the value of x for a to be parallel to b.

•

–

•

–

•

–

•

– 4. An irregular quadrilateral has exterior angle measures of 93, 85, and 89. What is the measure of the fourth interior angle?

•

273

•

267

•

93

•

87 5. How many sides does a regular polygon have if each exterior angle measures 12?

•

24

•

168

•

30

•

none of these 6. Find the slope of a line parallel to the graph of 4x – 2y = 9.

•

–

•

2

•

–2

•


23

7. Which of the following statements is true?

•

∠ABH and ∠DEF are alternate interior angles.

•

∠BED and ∠EBH are same-side interior angles.

•

∠ABC and ∠BED are alternate interior angles.

•

•

∠CBE and ∠DEF are corresponding angles. 8. Find the slope of a line parallel to the line containing the points A(2, –7) andB(–8, –9).

5

• • •

0.5 2 9. Below is a diagram of an airport runway intersection. There are two parallel runways, and a taxiway crosses both runways. If ∠8 measures 115 what is the sum of ∠1 and ∠4?

•

130

•

65

•

230

•

115


24

10. Which are corresponding angles?

•

∠6 and ∠16

•

∠1 and ∠12

•

∠8 and ∠16

•

none of these 11. Find the value of the variable. The diagram is NOT to scale.

∠SRT ∠STR ∠SRT = 30 ∠STU = 9x

•

–2

•

7

•

16

•

•

none of these 12. Construct a perpendicular from the given line segment that passes through the given point.


•

• •

•

•

none of these 13. Which drawing is a construction of the perpendicular line to

from S?

25


•

• 14. Which algebraic expression is equivalent to 3x(x² + 2) + 2(x + 1)?

•

3x³ + 2x + 8

•

3x² + 2x + 8

•

3x³ + 8x + 2

•

3x² + 8x + 2 15. Which is a correct name for the polygon?

•

BADEC

•

ABEDC

•

EDABC

•

CDEAB

26


27

16. Using the given information, which lines, if any, can you conclude are parallel? Justify your conclusion with a theorem or postulate. m∠1 + m∠2 = 180

•

g || h by the Converse of the Alternate Interior Angles Theorem

•

g || h by the Converse of the Same-Side Interior Angles Theorem

•

j || k by the Converse of the Alternate Interior Angles Theorem

•

j || k by the Converse of the Same-Side Interior Angles Theorem 17. Classify the triangle with sides of lengths 23, 23, and 21.

•

scalene

•

isosceles

•

straight

•

equilateral 18. Find the value of the variable if m || l, m∠3 = 2x + 106, and m∠4 = 4x + 38.

•

5

•

6

•

4

•

7


28

19. Which of the equations is a line parallel to 4x – 2y = 7?

•

2y = 4x + 7

•

–4x – 2y = –7

•

–2y = 4x + 2

•

•

•

•

•

4x + 2y = 2 20. Which diagram shows the correct construction of a line parallel to line l and passing through point P?


29

Parallel and Perpendicular Lines 1. If c ⊥ b and a || c, what is m∠2?

•

CORRECT: 90° 2. Which is an equation in slope-intercept form of the line that contains the pointsS(7, –9) and T(8, –7)?

•

CORRECT: y = 2x – 23 3. Find the value of x for a to be parallel to b.

•

CORRECT: – 4. An irregular quadrilateral has exterior angle measures of 93, 85, and 89. What is the measure of the fourth interior angle?

•

CORRECT: 87 5. How many sides does a regular polygon have if each exterior angle measures 12?

•

CORRECT: 30 6. Find the slope of a line parallel to the graph of 4x – 2y = 9.

•

CORRECT: 2 7. Which of the following statements is true?


30

•

CORRECT: ∠CBE and ∠DEF are corresponding angles. 8. Find the slope of a line parallel to the line containing the points A(2, –7) andB(–8, –9).

•

CORRECT: 9. Below is a diagram of an airport runway intersection. There are two parallel runways, and a taxiway crosses both runways. If ∠8 measures 115 what is the sum of ∠1 and ∠4?

•

CORRECT: 230 10. Which are corresponding angles?

•

CORRECT: ∠8 and ∠16 11. Find the value of the variable. The diagram is NOT to scale.

∠SRT ∠STR ∠SRT = 30 ∠STU = 9x

•

CORRECT: 16 12. Construct a perpendicular from the given line segment that passes through the given point.


•

•

•

CORRECT: 13. Which drawing is a construction of the perpendicular line to

CORRECT: 14. Which algebraic expression is equivalent to 3x(x² + 2) + 2(x + 1)? CORRECT: 3x³ + 8x + 2 15. Which is a correct name for the polygon?

•

from S?

CORRECT: CDEAB

31


32

16. Using the given information, which lines, if any, can you conclude are parallel? Justify your conclusion with a theorem or postulate. m∠1 + m∠2 = 180

• •

CORRECT: j || k by the Converse of the Same-Side Interior Angles Theorem 17. Classify the triangle with sides of lengths 23, 23, and 21.

•

CORRECT: isosceles 18. Find the value of the variable if m || l, m∠3 = 2x + 106, and m∠4 = 4x + 38.

•

CORRECT: 6 19. Which of the equations is a line parallel to 4x – 2y = 7?

•

CORRECT: 2y = 4x + 7 20. Which diagram shows the correct construction of a line parallel to line l and passing through point P?

•

CORRECT:


Congruent Triangles 1. Use the information given in the diagram. Tell why each statement is true. a. b.

•

a. given; b. Reflexive Property

•

a. Reflexive Property; b. given

•

a. Reflexive Property; b. Transitive Property

•

a. Transitive Property; b. Reflexive Property 2. Is ∆ABD ∆CBD by HL? If so, state the leg(s) that allows the use of HL.

•

no

•

yes,

•

yes,

•

yes,

and

and

3. ABCD is an isosceles trapezoid, and

• • • •

∆ABY ∆BAX ∠DXB ∠DYB

. Which statement is NOT true?

33


34

4. Complete the proof. Given:

and

.

Prove: ∆FAC ∆EDB.

•

def. of congruent triangles

•

SSS

•

SAS

•

none of these 5. The two triangles are congruent. Find the missing side lengths and the missing angle measures.

•

d = 5 m; e = 32°; f = 90°; g = 58°; h = 5 m

•

d = 8 m; e = 32°; f = 90°; g = 58°; h = 9.4 m

•

d = 5 m; e = 32°; f = 90°; g = 58°; h = 9.4 m

•

d = 5 m; e = 32°; f = 58°; g = 90°; h = 5 m

2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino 6. Complete the proof. Given: bisects ∠URS and

bisects ∠UTS.

Prove: ∆URT ∆SRT.

•

Reflexive property

•

definition of angle bisector

•

HL Theorem

•

CPCTC 7. Complete the proof. Given:

bisects ∠EBC and

Prove:∆EBD ∆CBD.

bisects ∠ECC.

35


Same-Side Interior Angles Theorem

•

given

•

SSS postulate

•

Triangle Inequality Theorem

36

8. Given that ∠EAC ∠ECA, which other information would NOT help you prove that

?

∠B ∠D

• •

•

•

9. Given ∆QRS ∆TUV, find the measure of the given angles and the lengths of the given sides. a. QS = 3v + 1, TV = 5v – 9 b. m∠R = 4n + 4, m∠U = 5n – 4

•

a. 16; b. 32

•

a. 5; b. 8

•

a. 16; b. 36

•

a. 8; b. 32 10. Given

and

, use the coordinates of A, B, F, and E to prove∠B ∠E.

By adding DC to both AD and FC, which are equal,

by (a)_____, ∆ABC ∆FED, and then ∠B ∠E by (b)_____.

•

a. SAS; b. CPCTC

•

a. SSS; b. SAS

. Because AB =

=FE,

. So,

2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino • •

a. ASA; b. HL a. SSS; b. CPCTC 11. From the information in the diagram, can you prove that ∆FDG and ∆FDE are congruent? Explain.

•

yes; AAA

•

yes; ASA

•

yes; SSS

•

no 12. What is the measure of each base angle of an isosceles triangle if its third angle measures 32 degrees and its 2 congruent sides measure 15 units?

•

32°

•

74°

•

148°

•

58° 13. Why is ∆ADB ∆CDB?

•

37

SAS


SSS

•

neither of these

•

They are not congruent. 14. Use information in the figure below to find m∠D.

•

11.75°

•

47°

•

94°

•

23.5° 15. What else must you know to prove the triangles congruent for the reason shown? a. ASA b. SAS

•

a. ∠ADC ∠ABC; b.

•

a. ∠ACD ∠CAB; b.

•

a. ∠ACD ∠CAB; b.

•

a.

16. Given ∆DEF, if

•

90

•

75

•

60

•

30

; b. ∠ACD ∠CAB

and m∠E = 30, which is the measure of ∠F?

38


17. Complete the proof. Given: , ∠BAD ∠CAD. Prove: ⊥ and bisects .

•

a. ASA; b. CPCTC

•

a. SSS; b. Reflexive Property

•

a. SAS; b. Reflexive Property

•

a. SAS; b. CPCTC 18. Is ∆ABD ∆CBD by HL? If so, state the leg that allows the use of HL.

•

yes,

•

no

•

yes,

and

•

yes,

and

39

2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino 19. In which triangles could you efficiently prove ∆1 ∆2 using the HL Theorem?

•

II only

•

III only

•

II and III

•

I only 20. ∆ABD ∆CBD. Name the theorem or postulate that justifies the congruence.

•

ASA

•

SAS

•

AAS

•

none of these

Congruent Triangles 1. Use the information given in the diagram. Tell why each statement is true. a. b.

•

CORRECT: a. Reflexive Property; b. given

40


2. Is ∆ABD ∆CBD by HL? If so, state the leg(s) that allows the use of HL.

•

CORRECT: yes,

•

3. ABCD is an isosceles trapezoid, and

41

. Which statement is NOT true?

CORRECT: ∠DXB ∠DYB

•

4. Complete the proof. Given:

and

.

Prove: ∆FAC ∆EDB.

•

CORRECT: SAS 5. The two triangles are congruent. Find the missing side lengths and the missing angle measures.

•

CORRECT: d = 5 m; e = 32°; f = 90°; g = 58°; h = 9.4 m 6. Complete the proof. Given: bisects ∠URS and bisects ∠UTS. Prove: ∆URT ∆SRT.


•

42

CORRECT: definition of angle bisector 7. Complete the proof. Given:

bisects ∠EBC and

bisects ∠ECC.

Prove:∆EBD ∆CBD.

•

CORRECT: given 8. Given that ∠EAC ∠ECA, which other information would NOT help you prove that

•

CORRECT:

?


43

9. Given ∆QRS ∆TUV, find the measure of the given angles and the lengths of the given sides. a. QS = 3v + 1, TV = 5v – 9 b. m∠R = 4n + 4, m∠U = 5n – 4 •

CORRECT: a. 16; b. 36 10. Given

and

, use the coordinates of A, B, F, and E to prove∠B ∠E.

By adding DC to both AD and FC, which are equal,

. Because AB =

=FE,

. So,

by (a)_____, ∆ABC ∆FED, and then ∠B ∠E by (b)_____. •

CORRECT: a. SSS; b. CPCTC 11. From the information in the diagram, can you prove that ∆FDG and ∆FDE are congruent? Explain.

•

CORRECT: yes; ASA 12. What is the measure of each base angle of an isosceles triangle if its third angle measures 32 degrees and its 2 congruent sides measure 15 units?

•

CORRECT: 74°

2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino 13. Why is ∆ADB ∆CDB?

•

CORRECT: SAS 14. Use information in the figure below to find m∠D.

•

•

CORRECT: 23.5° 15. What else must you know to prove the triangles congruent for the reason shown? a. ASA b. SAS

CORRECT: a. ∠ACD ∠CAB; b. 16. Given ∆DEF, if

•

and m∠E = 30, which is the measure of ∠F?

CORRECT: 75 17. Complete the proof. Given: , ∠BAD ∠CAD. Prove: ⊥ and bisects .

44


•

CORRECT: a. SAS; b. CPCTC 18. Is ∆ABD ∆CBD by HL? If so, state the leg that allows the use of HL.

•

CORRECT: no 19. In which triangles could you efficiently prove ∆1 ∆2 using the HL Theorem?

•

CORRECT: III only 20. ∆ABD ∆CBD. Name the theorem or postulate that justifies the congruence.

45


46

CORRECT: SAS

Relationships Within Triangles 1. Which lengths could be the sides of a triangle?

•

8 cm, 17 cm, 9 cm

•

22 cm, 12 cm, 9 cm

•

17 cm, 8 cm, 10 cm

•

12 cm, 22 cm, 8 cm 2. Points B, D, and F are midpoints of the sides of ∆ACE. EC = 40 and DF = 18. a. Find AC. b. Find FB.

•

a. 80; b. 9

•

a. 36; b. 20

•

a. 20; b. 18

•

a. 20; b. 36 3. List the sides of ∆ABC in order from shortest to longest, when m∠A = 10x – 7,m∠B = 7x – 10, and m∠C = 47 – 2x.

•

;

;

•

;

;

•

;

;

•

;

;


47

4. Write the conditional statement illustrated by the Venn diagram.

•

If an animal is not a cow, then it is not a mammal.

•

If an animal is a mammal, then it is a cow.

•

If an animal is a cow, then it is a mammal.

•

If an animal is not a mammal, then it is not a cow. 5. Write the side lengths from least to greatest.

•

,

•

,

• •

,

, , ,

, , 6. Which pair of statements has the same truth value?

•

the inverse and converse of a conditional statement

•

the converse and the contrapositive of a conditional statement

•

a conditional statement and its inverse

•

a conditional statement and its converse 7. Find the inverse of the statement "If she is not tall, she will not make the basketball team."

•

If she is tall, she will not make the basketball team.

•

If she will not make the basketball team, she is tall.

•

If she is tall, she will make the basketball team.

•

If she is not tall, she will make the basketball team.

2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino 8. and a. Find DC. b. Find DB. c. Find AE.

48

are perpendicular bisectors of each other.

•

a. 13; b. 24; c. 5

•

a. 13; b. 24; c. 10

•

a. 13; b. 12; c. 5

•

a. 13; b. 12; c. 10 9. Find FG.

•

12

•

4

•

19

•

15 10. List the sides of ∆ABC in order from shortest to longest, when m∠A = 11x – 4,m∠B = 4x – 11, and m∠C = 51 – 3x.

•

;

;

•

;

;

•

;

;

•

;

;

2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino 11. Given:

is the perpendicular bisector of

49

. What can you NOT conclude?

•

∠JMP, ∠KMP, ∠KMO, and ∠JMO are rt ∠s.

•

M is the midpoint of

.

•

M is the midpoint of

.

•

12. Name the second largest angle in the figure if the side between ∠1 and ∠2 is 14 cm, the side between ∠2 and ∠3 is 15 cm, and the side between ∠3 and ∠1 is 18 cm. (not drawn to scale)

•

∠2

•

∠1

•

∠3 ∠4

• 13. Given:

•


⊥

•

E is the midpoint of

•

∠HDJ is a right angle.

•

EJ = DJ

.

. Which statement is true?

2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino 14. Find the value of x.

•

14

•

7

•

11.5

•

9 15. Determine which three lengths can be measures of the sides of a triangle.

•

10 cm, 4 cm, 13 cm

•

3 cm, 10 cm, 13 cm

•

15 cm, 3 cm, 19 cm

•

2 cm, 15 cm, 19 cm 16. Find m∠QST.

•

7.5

•

75

•

15

•

30

50

2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino 17. B is the midpoint of

•

7

•

14

•

49

•

21 18. Given:

, D is the midpoint of

is the ⊥ bisector of

, and BD = 7. Find AE.


•

L is the midpoint of

•

∠CLM, ∠MLD, ∠DLK, and ∠KLC are rt ∠s. ⊥

• •

.

19. Find the length of

•

40

•

10

•

20

given that

is a median of the triangle and BC is 20.

51


52

21 20. Identify the segment interior to the triangle with respect to the whole figure.

•

midsegment

•

perpendicular bisector

•

altitude

•

angle bisector

Relationships Within Triangles 1. Which lengths could be the sides of a triangle? •

•

CORRECT: 17 cm, 8 cm, 10 cm 2. Points B, D, and F are midpoints of the sides of ∆ACE. EC = 40 and DF = 18. a. Find AC. b. Find FB.

CORRECT: a. 36; b. 20 3. List the sides of ∆ABC in order from shortest to longest, when m∠A = 10x – 7,m∠B = 7x – 10, and m∠C = 47 – 2x.

•

CORRECT:

;

;


53

4. Write the conditional statement illustrated by the Venn diagram.

•

CORRECT: If an animal is a cow, then it is a mammal. 5. Write the side lengths from least to greatest.

•

CORRECT: , , 6. Which pair of statements has the same truth value?

•

CORRECT: the inverse and converse of a conditional statement 7. Find the inverse of the statement "If she is not tall, she will not make the basketball team."

•

CORRECT: If she is tall, she will make the basketball team. 8. and are perpendicular bisectors of each other. a. Find DC. b. Find DB. c. Find AE.

•

CORRECT: a. 13; b. 24; c. 5 9. Find FG.


54

CORRECT: 15 10. List the sides of ∆ABC in order from shortest to longest, when m∠A = 11x – 4,m∠B = 4x – 11, and m∠C = 51 – 3x.

•

CORRECT: 11. Given:

•

;

;


CORRECT: M is the midpoint of


.

12. Name the second largest angle in the figure if the side between ∠1 and ∠2 is 14 cm, the side between ∠2 and ∠3 is 15 cm, and the side between ∠3 and ∠1 is 18 cm. (not drawn to scale)

• •

CORRECT: ∠2 13. Given:

•

•


CORRECT: ⊥ 14. Find the value of x.

CORRECT: 7

. Which statement is true?

2012 UPCAT, ACET, DLSUET, USTET @ www.fb.com/Mathalino 15. Determine which three lengths can be measures of the sides of a triangle. •

CORRECT: 10 cm, 4 cm, 13 cm 16. Find m∠QST.

•

•

•

•

CORRECT: 15 17. B is the midpoint of

CORRECT: 14 19. Find the length of

, D is the midpoint of

given that

, and BD = 7. Find AE.

is a median of the triangle and BC is 20.

CORRECT: 20 20. Identify the segment interior to the triangle with respect to the whole figure.

CORRECT: angle bisector

55


Part 2/2

56

geom1

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