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Pre-AP Geometry Worksheet 1.1

Pre-AP Geometry Name: ___________________ Worksheet 1.1: Points, Lines, and Planes Date: _______ Period: ____

1. Which of the following statements cannot be assumed from the diagram?

a. Points A, B, C, and E are coplanar. b. Points F, B, and G are collinear c. !" ⊥ !" d. !" intersects plane M at point C

Use the diagram to determine if the statement is true or false.

2. Planes W and X intersect at !"

3. Points Q, J, and M are collinear

4. Points K, L, M, and R are coplanar

5. !" and !" intersect

6. !" ⊥ plane W

7. !" lies in plane X

Problem 5

Using Postulate 1-4

Use the figure at the right.

A What plane contains points N, P, and Q? Shade the plane.

The plane on the bottom of the figurecontains points N, P, and Q.

MJ

N P

Q

L

R K

B What plane contains points J, M, and Q? Shade the plane.

The plane that passes at a slant throughthe figure contains points J, M, and Q.

MJ

N P

Q

L

K

continuedProblem 4

The planes intersect in <BC

>.

Focus on plane ADC andplane BFG to see wherethey intersect.

A

E F

GB

CD

H

You can see that bothplanes contain point Band point C.

A

E F

GB

CD

H

PRACTICE and APPLICATION EXERCISES

ONLINE

HO M E W O R

K

Use the figure at the right for Exercises 1–3.

1. What are two other ways to name plane C?

2. Name three collinear points.

3. Name four coplanar points.

Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Use the figure at the right for Exercises 4 and 5.

4. Name the segments in the figure.

5. Name the rays in the figure.

For additional support whencompleting your homework, go to PearsonTEXAS.com.

m

n FBE

GC

R ST W

Scan page for a Virtual Nerd™ tutorial video.

How can you find the plane?Try to draw all the lines that contain two of the three given points. You will begin to see a plane form.

Is the intersection a segment?No. The intersection of the sides of the box is a segment, but planes continue without end. The intersection is a line.

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MJ

N P

QR K

L

Use the figure at the right for Exercises 6–15.

Name the intersection of each pair of planes.

6. planes QRS and RSW 7. planes UXV and WVS

Name two planes that intersect in the given line.

8. <QU

> 9.

<TS> 10.

<XT

> 11.

<VW

>

Copy the figure. Shade the plane that contains the given points.

12. R, V, W 13. U, V, W 14. U, X, S 15. T, U, V

Postulate 1-4 states that any three noncollinear points lie in exactly one plane. Find the plane that contains the first three points listed. Then determine whether the fourth point is in that plane. Write coplanar or noncoplanar to describe the points.

16. Z, S, Y, C 17. S, V, C, Y

18. Display Mathematical Ideas (1)(G) Draw a figure with points B, C, D, E, F, and G that shows

<CD

>, <BG

>, and

<EF>, with one of the points on all

three lines.

19. Your friend drew the diagram at the right to prove to you that two planes can intersect in exactly one point. Describe your friend’s error.

20. Which term below is undefined and which term has a definition? Write the definition of the defined term.

line opposite rays

21. Analyze Mathematical Relationships (1)(F) If one ray contains another ray, are they the same ray? Explain.

For Exercises 22–25, determine whether each statement is always, sometimes, or never true.

22. <TQ

> and

<QT

> are the same line. 23. Two intersecting lines are coplanar.

24. JK> and JL

> are the same ray. 25. Four points are coplanar.

26. Use the diagram at the right. How many planes contain both the line and the point?

a. <FG

> and point P b.

<EP

> and point G

c. Explain Mathematical Ideas (1)(G) What do you think is true of a line and a point not on the line? Explain.

8 Lesson 1-1 Points, Lines, and Planes

X W

VU

Q R

ST

Z

SV X

U

C

Y

P Q

G

FEH

27. Apply Mathematics (1)(A) A cell phone tower at point A receives a cell phone signal from the southeast. A cell phone tower at point B receives a signal from the same cell phone from due west. Trace the diagram at the right and find the location of the cell phone. Describe how Postulates 1-1 and 1-2 help you locate the phone.

28. You can represent the hands on a clock at 6:00 as opposite rays. Estimate the other 11 times on a clock that you can represent as opposite rays.

29. Apply Mathematics (1)(A) What are some basic words in English that are difficult to define?

30. a. Explain Mathematical Ideas (1)(G) Suppose two points are in plane P. Explain why the line containing the points is also in plane P.

b. Use Representations to Communicate Mathematical Ideas (1)(E) Suppose two lines intersect. How many planes do you think contain both lines? Use the diagram at the right and your answer to part (a) to explain your answer.

Display Mathematical Ideas (1)(G) Graph the points and state whether they are collinear.

31. (1, 1), (4, 4), (-3, -3) 32. (2, 4), (4, 6), (0, 2) 33. (0, 0), (-5, 1), (6, -2)

Use a Problem-Solving Model (1)(B) Suppose you pick points at random from A, B, C, and D shown at the right. Find the probability that the number of points given meets the condition stated.

34. 2 points, collinear 35. 3 points, collinear

STEM

TEXAS Test Practice

36. Which geometric term is undefined?

A. segment B. collinear C. ray D. plane

37. You want to cut a block of cheese into four pieces. What is the least number of cuts you need to make?

F. 2 G. 3 H. 4 J. 5

38. The figure at the right is called a tetrahedron.

a. Name all the planes that form the surfaces of the tetrahedron.

b. Name all the lines that intersect at D.

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A

SE

WB

A CB

A

D

CB

A

B

C

D