Exercises for Chapter 1

8/8/2019 Exercises for Chapter 1

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CHAPTER 1 BASIC CONCEPTS

1.1 Points, Lines and Planes

Exercises

A. Answer the following as instructed.

1. Classify each statement as either true or false.

_______a. s contains J, K and H

_______b. m and p are concurrent

_______c. J, K and F are coplanar

_______d. A,K and F are collinear

_______e. C, D, A and G are coplanar and collinear

_______f. K, G and A coplanar

_______g. lines JH and AF intersect at K

_______h. plane M intersects s at K

_______i. plane M contains s

_______j. m and s are in plane M

_______k. D, E and F are collinear

_______l. J, D, E, B and C are coplanar points

_______m. A, K and G are in plane M, thus coplanar

_______n. A, B, F, and J are non-coplanar

_______o. F, E, D, and C are non-collinear

2. What is the simplest figure being studied in geometry?

3. What is the implication when two figures do not intersect?


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4. Does a plane have edges?

5. If line m contains points B and C, what is the other

possible name of the line?

6. If line m is in plane A, what is the intersection of plane A

and line m?

7. Complete the following statements using the words

intersection, contain(s), coplanar, non-coplanar, collinear,

and non-collinear.

a. Points B, C, K and J are __________.

b. Points K and J are both _________ and __________.

c. Line p ___________ the points M, D and E.

d. The ____________ of line m and plane L is line m.

e. Points B, C, G, H, K and J are ____________ and __________.

f. The _____________ of planes L and K is line m.

g. Points A, B, C, K, J, D, G and H are _____________ and

______________.

h. A point is the __________ of lines m and p.


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i. Lines m and p are ______________.

j. Points K and J are both _________ in planes L and K.

8. State whether the given points in each set is collinear or

non-collinear.

_______a. {T, O, R}

_______b. {N, C, G, R}

_______c. {A, B, U, I, Z}

_______d. {C, E, L}

_______e. {N, O}

9. Identify whether the given points in each set is coplanar or

non-coplanar.

_______a. {A, C, E, U, B}

_______b. {C, T, O, N, R, G}

_______c. {T, O, R}

_______d. {C, E, L}

_______e. {B, I, Z, C, E, T, O}

_______f. {Z, G, O, N, R, U}

_______g. {T, G, R, O, N, Z}

10. Give a pair of coplanar lines.

11. Give two pairs of non-coplanar lines.

For Nos. 8-12


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12. Using a single letter, name the plane that contains A, E,

C, L and U.

13. Is there any other plane that contains line t and r?

14. Points I and Z are on plane H. Is on plane H?

15. Is there a line that contains B and N?

B. Use the provided figure to answer the following.

1. Name a point that is the same plane as A, U, and T.

2. Name the intersection of plane TON and CNO.

3. Name a set of four points that are collinear.

4. Give one set of five points that are coplanar.

5. Complete the statement: Line UC is the intersection of plane

UCN and plane ______.

6. Complete the statement: Lines TO and NO intersect at _____.


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7. True or False: There are several planes that contain the

points T, Z, A and U. ________

8. True or False: Z, C, B, I, and N are non-coplanar

points.______

9. Name the plane which contains F. ________

10. True or False: Plane SONC contains only the following

points- S, O, N, C, B and I. __________

Sketch and label the figure as described.

11. A line containing points C, D and E which lies in plane B.

12. A plane containing non-collinear points P, Q and R where Q

and R both lie in line m.

13. Planes R and P intersect at line AB.

14. Line k intersects plane R at point B. Line AB lies in plane

R.


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15. Plane M contains the points A, B and C. A line containing

a point, D, not on the plane, intersects plane M at A, B and C.

Use the figure below to answer the remaining questions.

16. How many lines contain

_____________a) A?

_____________b) B and F?

_____________c) B, C, D and E?

17. Is there any other line that contains points B and C?

18. How many planes contain

____________a) line k?

____________b) line k and point F?

____________c) ?

____________d) points B, C, and F?

19. If line BF is on plane M, is line k also on plane M?


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20. If line k and point A are on plane X, is there any other

plane that may contain A? _________

C. Identify the postulate or theorem used in each of the

following statements.

1. Line m contains points B and C.

2. Points G and H are on plane K, then is on plane K.

3. At least one plane will contain points A, B and C.

4. Points A, B and C are non-collinear so there is one and only

one plane that contains these points.

5. Point H is not on line CD. There is one and only one plane

that contains H and line CD.

6. Lines a and b intersect at point K. There is no other point

where lines a and b intersect.

7. Plane M contains non-collinear points A, B and C.

8. lines a and b intersect a point K. There is a unique plane

that both contains a and b.


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9. Two intersecting lines determine a plane.

10. Three non-collinear points assure the existence of plane.

Answer the following as required.

11. State whether the statement is always, sometimes or never

true.

_________a. A plane passes through three points.

_________b. Line AB is on plane X. A and B are also on plane

X.

_________c. Two lines j and k pass through the points

G and T.

_________d. Two lines j and k are both on plane J and K.

_________e. A, B, C and D are non-coplanar points. These

points are contain in a space.

12. State whether it is possible for the figure described to

exist.

_________a. Three points all lie in each of three planes.

_________b. Two points both in each of two lines.

_________c. Four points all lie in each of five planes.

_________d. Collinear points which are not coplanar.

_________e. Two points, C and D, lie on plane A, and another

point E lie on plane B, where C, D and E are collinear.


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13. In a rough surface, which is better to use a chair with

three legs or a chair with four legs?

14. Why is it that a three-legged support is more stable than a

four-legged support?

15. Draw: Points A and B are plane K, points C and D are on

plane M. A, B, C and D are not collinear but coplanar.

D. Answer the following.

a. How many lines can be drawn from three non-collinear

points?

b. How many lines can be drawn from 8 points, where no

three of which are collinear?

c. How many lines can be drawn from

n points, where no three of which are

collinear?


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1.2 Segments and Rays

Exercises

A. Answer the following.

________1. Name the segment at the right.

________2. Name the ray at the right.

3. Name all the segments defined by the indicated points.

4. Name all pairs of opposite rays defined by the indicated

points.

5. Name the bisectors of the segment LC if

B is its midpoint.

Find the distance between the following pairs of points.

________6. A and B


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________7. B and D

________8. C and F

________9. E and G

________10. F and J

Using the preceding number line, find the following.

________11. AC + CE

________12. DE + EF

________13. EG + GH

________14. BC + CF + FH

________15. AD + DE + EG + GH

Complete the following statements.

16. M is the midpoint of segment AB. If MB = 8cm, then AM =

______.

17. bisects at R. If CR = 12cm, then RD = ______.

18. . If LM = 23in, then KJ = _____.

19. . If PR = 8, then .

20. . If CD + EF = 12, then AB + GH = _______.

B. Refer to the provided figure to answer the following.

1. If the distance between A and P is 8, what is the coordinate

of P?


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2. If the length of segment AL is 15.3, what is the coordinate

of L?

3. Assumed that A is the midpoint of . If LA = 8,

____________a. what is the length of ?

____________b. What is the coordinate of P?

____________c. coordinate of L?

4. If M is another point on the number line and its distance

from A is

units, what is its coordinate?

5. X is another point on the number line. If its distance from

A is

and its coordinate is less than As coordinate, what is

its coordinate?

6. Given: BZ = 2x; ZA = 3x-1; BA = 8x 7

Find: x, BZ, ZA and BA


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7. Given: UZ = a + 12; ZA = 3a + 12; UA = 8a + 4

Find: a, and UA

8. Given: BU = b; UZ = 3b; BZ = 5b 3

Find: b, UZ and BZ

9. Given: ; BU = 2c + 4; UZ = c + 8

Find: c, BU and BZ

10. Given: ; UZ = 2d 3; UA = 6d 16

Find: d, ZA and UA


11. If M is the midpoint of , then, by definition of

midpoint, __________________ or ________________.


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12. If , then, by definition of congruent segments,

______________.

13. If and C, M and D are collinear, then M is the

_____________ of .

14. If T is between J and K, then, by Segment Addition

Postulate, _________________.

15. intersects at its midpoint. is a _______________

of .

C. Answer the following.

1. J is between K and L, such that the coordinate of J is

greater

than the coordinate of K.. If KJ = 3x, JL = 2x+4, KL = x + 12

and the coordinate of K is -3, find

a. the value of x c. KL

b. JK d. coordinates of J and L


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2. C is between D and E. The coordinate of D is larger than

the coordinate of E. If EC = y + 4, CD = 2y + 2, ED = 5y-2,

and the coordinate of C is 3, find

a. the value of y c. CD

b. CE d. coordinates of E and D.

3. The coordinates of E and G are -15 and 3, respectively. F

is between E and G. If the length of is 2x and the length

of is 5x + 4, find the coordinate of F.

4. The coordinates of P and R are

and

, respectively. If

PQ = a + 3 and RQ = 2a + 4, find the coordinate of Q.


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5. B is between A and C, C is between B and D, and As

coordinate is greater than the Ds coordinate. If DC = x+1, CB

= 2x+2, BA = 3x+3 and DA = 4x+12, find x and CA.

6. P is between K and A, A is between K and L. The coordinate

of A is less than the coordinate of L. If LA = 2a, KP = 2a+4,

PA = 2a+2 and KL = 12a 12.

7. A-P-C-D-B. The coordinates of P and C are -2 and 11. If AP =

2x, CD = x + 3, DB = 3x+1 and PC = 3x 2, find x and AB.

8. E is between C and F. J is between F and K. If the

coordinates of K and F are 10 and 2, respectively, FJ = a+1,


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JK = 3a-1, CE = 2a+1 and EF = 5a-8, what are the values of a

and CK?

9. Complete the table below.

No. of

indicated

points

2 3 4 10 20 N

No. of

distinct

rays

2 3

10. Complete the table below.

No. of

indicated

points

2 3 4 10 20 N

No. of

distinct

segments

1 3 6

D. An ant is travelling at the edge of a meter stick. The ant

starts from a non-zero marked in the stick. When the ant is

halfway its destination, the coordinate of its position 6cm

more than three times the coordinate of its starting point.


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When it reached its destination, the coordinate of its position

is 4cm less than twice the coordinate of its position when it

is halfway. Find the coordinate of ants starting position and

its distance traveled.

1.3 Angles

Exercises

A. Name the vertex and sides of the following angles.

_____________1.

_____________2.

_____________3.

_____________4.

_____________5.

Classify the following angles as acute, right, or obtuse angle.

_____________6.

_____________7.

_____________8.

_____________9.

_____________10.

_____________11.

_____________12.

_____________13.

_____________14.


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_____________15.


16.

17.

18.

19. If bisects , then _____ = _____ or _____ _____.

20. If , then _______ bisects __________.

B. Determine the measures of the following angles. Use the

figure provided.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Name the angle(s)that are congruent to the following angles.

11.

12.

13.

14.


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15.

Use the figure at the right to complete the following

statements.

16. If and , then .

17. If and , then =_________.

18. If and , then .

19. If and , then .

20. If and , then .

21. If ray OR bisects angle QOS and

the measure of angle QOR is 34.9

degrees, then the measure of angle

ROS is ___________ and the measure

of angle QOS is ______________.

22. If , then bisects __________.

23. If , then bisects _________.

24. ________ and _________ are adjacent to .

25. R is in the ____________ of .

26. By Segment Addition Postulate, .

27. If bisects , then ____________________.

28. and are adjacent to .

29. If , then, by Definition of Congruent

Angles,________________.

30. If and , then ____________.


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C. Answer the following problems.

1. If and

, find

2. If and , find

.

3. If , and

, find .

4. If and ,

find the measure of angle 3.


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5. If and , find the

measures of angles 1, 2 and 3.

6. If and ,

find the measures of angles 1 and 3.

7. is congruent to . If and ,

find the value of 3x + 18.

8. bisects . If and , find


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9. and bisects . If and

, find the measure of angle 2.

10. and bisects . If and

, find m

D. Determine the number of distinct angles formed by non-

collinear rays that have the same endpoint.

CHAPTER TEST

Choose thebest answer from the given options.

_____1. If points A, B and C are non-collinear, which of the

following is true?

I. There is exactly one plane that contains them.

II. Three distinct lines can be drawn through these

points.


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III. The points are enough to determine a space.

IV. There is at least one plane that contains them.

a. I, II b. I, II, III c. III, IV d. II, III, IV

_____2. Which of the following is true about coplanar points?

a. There is exactly one plane that contains them.

b. There is a plane that contains them.

c. Points lie on the same line.

d. If plane K contains them, then there is no other plane that

contains them.

_____3. Select the odd one out.

a. Two distinct points determine a line.

b. If points A and B lie on line m, then there is no other line

that contains them.

c. If there is line, say m, then two points can be named from

this line, say A and B.

d. Through any two points there is exactly one line.

_____4. Which of the following is possible?

a. Points A and B lie on different lines.

b. Non-collinear points A, B and C contain in at least one

plane.

c. Two lines m and n are both on plane M and N.

d. A, B and C contain in plane M and plane N.

_____5. Which of the following is the possible intersection of

a line and a plane?


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I. a point II. a line III. a plane

a. I b. I and II C. II and III d. I and III

_____6. If C, M and D are collinear and , then which of

the following is a good conclusion to this?

a. CM + MD = CD.

b. M is between C and D.

c. M is the midpoint of .

d.

.

_____7. If two distinct planes intersect, then their

intersection is

a. a point.

b. a set of finite points contain in a segment.

c. a set of infinitely many points contain in a line.

d. a set of infinitely many points contain in a plane.

_____8. Which of the following is true?

a. Collinear points are coplanar points.

b. coplanar points are non-collinear.

c. coplanar lines are intersecting lines.

d. non-collinear points are non-coplanar points.

_____9. Which of the is NOT a valid conclusion: If two lines

intersect, then

a. they lie on the same plane.

b. their intersection is a point.


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c. they lie on different planes.

d. there is a plane that contains them.

_____10. Which of the following is NOT true about a segment,

say ?

a. If N is its midpoint, then there is no other midpoint than

N.

b. If E is between C and D, then CE + ED = CD.

c. In a number line, if the coordinate of C is -3, then the

coordinate of D can also take this value.

d. If CD = 4 units and another segment AB has the same length,

then .

_____11. . If EF = 4cm and XY = 2a+3 cm, then what is

the value of a?

a. 4 b. 2 c. 1 d. 1/2

_____12. , which of the following is not necessarily

true?

a. EF = FD.

b. F is the midpoint of .

c. If EF = 25cm, then FD = 25cm.

d. If EF = 10cm, then EF + FD = 20cm.

_____13. PK = 8 units and the coordinate of K is -12. If the

coordinate of K is greater than that of M which is greater than


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the coordinate of P, which of the following is the coordinate

of P?

a. -20 b. -4 c. 4 d. 20

_____14. If P-O-R, and PO = 2x+8, PR=8x+4 and OR=8, find PR.

a. 2 b. 12 c. 18 d. 20

_____15. Which of the following is NOT true about ?

I. A is its endpoint.

II. It can be written also as .

III. There is only one point on given a distance from

A.

IV. If R is between A and B, then can also be named as

.

a. I b. II c. III d. IV

_____16. Which of the following statements are supported by

Ruler Postulate?

I. There is a one-to-one correspondence between the set of

real numbers and the set of points in a line.

II. If point N has a coordinate of -1, then there is no

other point having such coordinate.

III. The distance between two points is defined by their

coordinates.

a. I b. II c. III d. all of these

_____17. Let ( ) denotes the coordinate of a point. If (A)=4,

(B)= -2, (C)= -4, and (D)=1, arrange AB, BC, CD and BD in

descending order.

a. AB, BD, CD, BC c. BC, BD, CD, AB

b. AB, CD, BD, BC d. CD, AB, BD, BC


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_____18. Determine which point is in the interior of which

angle if , and .

a. P b. B c. R d. T

_____19. If two angles are adjacent and congruent, then

________. Which of the following is a valid conclusion?

a. they are coplanar.

b. they have equal measures.

c. the angle formed by the non-common sides is bisected by the

common side of the two angles.

d. they dont have common points.

_____20. If two angles are adjacent, then

a. they are coplanar.

b. then they have no common interior points.

c. they have common vertex and common side.

d. they are coplanar, no common interior points but have common

vertex and common side.

_____21. is adjacent to and . If ,

and , what is the measure of ?

a. b. c. d.

_____22. Refer to problem 21. Which of the following is

congruent to ?

a. b. c. d. none of these


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_____23. is adjacent to and . If ,

and , find .

a. b. c. d.

_____24. Complete the statement: If K is in the interior of

, then there exists a segment, , such that J lies in

and L lies on ______ where K is ______________ J and L.

a. , collinear with c. , collinear with

b. , opposite d. , between

_____25. bisects . If and ,

find the value of .

a. 3 b. 9 c. 27 d. 81

ENRICHMENT

Try to solve the following problems.

1.

2.

3.

4.

Exercises for Chapter 1

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