Points, Planes, & Lines, Study notes of Geometry

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GEOMETRY

Unit 1

Points, Planes, &

Lines

Prepared by:

G. Alvarez-Garcia, S. George, & R. Mercadante

Geometry CC RHS Unit 1 Points, Planes, & Lines 2

1 .1 (^) Points, Lines, and Planes

Definition

The “easy” definition

What does it look like?

How is it written?

A point has no ________________.

A line extends in _______

dimension.

A plane extends in _______

dimensions.

A line segment AB consists of

____________ A and B, and all the

points on line AB that are

________________ A and B.

A ray AB consists of the initial point

____ and all the points on the line AB

that lie on _____________

__________ of A as point B.

THINK ABOUT IT!

Is there a difference between lines ̅̅̅̅ and ̅̅̅̅? Is there difference between ⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗?

Example 1: Give all the names of the figures shown below.

E

C D F

E

F

I G

J

J

O P

P

R

E

F

I G

J

O (^) P

Q M^ N

J

O (^) P

O

Example 2: Using Segment Addition Postulate

1. a) O is between M and P. Find the length of̅̅̅̅̅

if and PO = 14.

b) What would your reasoning be if asked to

explain why you set up the equation above?

2. E is be between N and T. If and

, find the length of ̅̅̅̅ and explain.

3. Y is be between X and Z. If and

find the length of ̅̅̅̅ and explain.

4. A is be between H and T. If ,

, and , solve for. Explain.

5. I is be between K and D. If ,

, and , find. Explain.

6. G is be between P and A. If ,

, and , solve for. Explain.

N

T

E

H A T

Regents Test

Basic Concepts and Definitions

Questions 1 through 9 refer to the following:

1) PR and TS determine a plane. TRUE FALSE

2) Points T, Q, and S are collinear. TRUE FALSE

3) S is between P and R. TRUE FALSE

4) Points P, Q, and R are collinear. TRUE FALSE

5) PR and TS intersect at Q. TRUE FALSE

6) Points T, Q, and S are non-collinear. TRUE^ FALSE

7) Points P, Q, and T are non-collinear. TRUE^ FALSE

8) Q is between P and R. TRUE FALSE

9) Points S, Q, and R are collinear. TRUE^ FALSE

Questions 10 through 16 refer to the following:

In the diagram below, points P, N, and Q are collinear. Indicate whether the given statement is True or False.

10) Points M, N, K and Q are coplanar. TRUE^ FALSE

11) Points P, K, and Q determine a plane. TRUE FALSE

12) Only one plane contains points P, N, and Q. TRUE FALSE

13) PQ and NK intersect at N. TRUE FALSE

16) Points P, K, N, and Q are coplanar. TRUE^ FALSE

17) If two planes intersect, then their intersection is a line. TRUE FALSE

18) PQ has no endpoints. TRUE^ FALSE

19) PQ has only one endpoint. TRUE FALSE

20) A line segment has exactly one midpoint. TRUE^ FALSE

21. Tell whether a point, a line, or a plane is illustrated by.

22. Tell whether a point, a line, or a plane is illustrated by.

23) PQ has no endpoints. TRUE^ FALSE

24) FG has only one endpoint. TRUE FALSE

25) PQ has two endpoints. TRUE FALSE

26. Tell whether a point, a line, or a plane is illustrated by the top of a desk.

1900 - 1 - Page 2

27) If two lines intersect, then their intersection is a point. TRUE^ FALSE

28) It is possible to define each geometric term by using previously defined geometric terms. TRUE^ FALSE

29) AB has no endpoints. TRUE^ FALSE

30) Another name for PQ is QP. TRUE FALSE

31. Tell whether a point, a line, or a plane is illustrated by the tip of a pen.

32. Tell whether a point, a line, or a plane is illustrated by the edge of a textbook.

33) CD has two endpoints. TRUE FALSE

34. Tell whether a point, a line or a plane is illustrated by a basketball backboard.

35) Another name for PQ is QP. TRUE^ FALSE

36) PQ has two endpoints. TRUE^ FALSE

13. Define a postulate.

14) A theorem is a statement accepted without proof. TRUE^ FALSE

15) A postulate is a statement that must be proven. TRUE^ FALSE

16) A postulate can be used in the proof of a theorem. TRUE FALSE

17) A postulate cannot be used in the proof of a theorem. TRUE^ FALSE

18) A postulate is a true statement accepted without proof. TRUE^ FALSE

19. Explain the difference between a postulate and a theorem.

20) All statements in Geometry are definitions or postulates. TRUE FALSE

Regents Test

Points, Planes, & Lines: Concepts involving

Definitions Questions 1 through 5 refer to the following:

Arrange the following terms in the order in which their definitions should be stated.

1. ray, angle, endpoint of a ray

2. isosceles triangle, triangle, base of an isosceles triangle

3. congruent angles, angle bisector, angle

4. angle, acute triangle, acute angle

5. midpoint of a segment, segment, congruent segments

6) Point is a defined term. TRUE FALSE 7) Point is an undefined term. TRUE^ FALSE Questions 8 through 11 refer to the following:

Explain why the following isnot a good definition.

8. A line segment is a geometric figure.

9. Coplanar points are points in the same plane.

10. An isosceles triangle is when a triangle has two sides the same length.

11. A square is something not round.

12) Plane is a defined term. TRUE^ FALSE 13) Line is a defined term. TRUE FALSE

14. Define AB written out with words.

1.2 Congruent Segments

Congruent vs.̅̅̅̅

Amy’sHeight = Tim’s Height Amy ____ Tim

Segments Lengths are

Equal

Segments are

Congruent

MARKING UP CONGRUENT SEGMENTS

1. Mark the following segments congruent: ̅̅̅̅ ̅ , ̅̅̅̅ ̅̅̅̅, and̅̅̅ ̅

2. Then IJ = ____, ML = _____, and HM = _____

Midpoint: (Equal parts)

How to say it: ____ is the midpoint of

_____

Bisect: (Cuts the second one in half)

How to say it: _____ bisects _____ at point ___

Example 1: Draw a diagram and write the word form.

1. O is the midpoint of ̅̅̅̅ 2. ̅̅̅̅̅ bisects ̅̅̅̅ at O. 3. A is between F and T.

4. ̅̅̅̅̅ bisects ̅̅̅̅ at U. 5. U is between B and M. 6. E is the midpoint of̅̅̅̅

LIL = LIL LIL^ = LIL

A B

C D

Tim Amy

M D S

K P D

G

Example 2 : Finding segment lengths

1. is the midpoint of ̅̅̅̅, If AG =13, find TA.

Explain.

2. ̅̅̅̅ bisects ̅̅̅̅ at. If , find.

Explain.

3. ̅̅̅̅ bisects ̅̅̅̅ at. If and ,

find and.

Explain.

4. is the midpoint of ̅̅̅̅, If PT=26, find PF.

Explain.

3. Point S is the midpoint of ̅̅̅̅. If

and , solve for and explain.

4. A is between J and M. If ,

, and. Solve for and

explain.

E

F

M N

J

O

P

O

T A G

T

B

A

D

R

R

T

S

Regents Test

Points, Planes, & Lines: Lines and Line Segments

Questions 1 through 17 refer to the following:

Use the figure below to name a segment, ray, or point that

best completes the given statement.

1. the length of NK is G

2. G is the midpoint of MT

3) NK = NO + G

4) EA = ER - G

5) TW = TK - G

6) EA + AR = OK + G

7) WK C G

8. the ray opposite TK is G

9. G is the midpoint of NK

10. T is the midpoint of G

11) ER C^ G

12) AR = AW + G

13. the length of MW is G

14. the ray opposite AM is G

15. another name for ET is G

16. R is the midpoint of G

17. another name for AR is G

18. Find PR, given that Q is the midpoint of PR and QR = 8.

19. Find PQ, given that Q is the midpoint of PR and PR = 20.

20. Find the value ofx if B is the midpoint of AC, AB =x - 3, and AC = 6x - 38.

Regents Test

Points, Planes, & Lines: Separation of Points and Collinear Points

Questions 1 through 8 refer to the following:

Find the value ofx if A, B, and C are collinear points and

B is between A and C.

1) AB = 6x, BC =x - 5, AC = 23

2) AB = 5, BC = 3x + 7, AC = 5x - 2

3) AB = 12, BC = 5x - 2, AC = 3x + 20

4) AB =x, BC =x + 2, AC = 14

5) AB =x + 6, BC = 3x - 5, AC = 36 - x

6) AB = 2x, BC =x - 2, AC = 28

7) AB = 3x, BC = 2x - 7, AC = 2x + 35

8) AB = 5x - 1, BC = 14, AC = 25 - x

Questions 9 through 20 refer to the following:

9. Find the distance between S and F.

10. Find the distance between S and H.

11. Find the distance between A and R.

12. Find the distance between E and M.

13. Find the distance between S and A.

14. Find the distance between S and R.

15. Find the distance between M and N.

16. Find the distance between R and A.

17. Find the distance between H and A.

18. Find the distance between N and F.

19. Find the distance between E and F.

20. Find the distance between E and N.

Questions 21 through 28 refer to the following:

Find the required distance if A, B and C are collinear points and point B is between points A and C.

21) AB = 2, AC = 18, BC = G