Chapter 1 “Basics of Geometry”

Notes & HW Packet

Belongs to:

___________________

2017-2018

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Table of Contents:

1.1 Notes Identify Points, Lines, Planes 3

1.1 HW 5

1.2-1.3 Notes Using Segments, Congruence, and Midpoints 7

1.2-1.3 HW 12

Review 1.1-1.3 14

1.4 Notes Measure and Classify Angles 17

1.4 HW 21

1.5 Notes Special Angle Pair Relationships 23

1.5 HW 28

Chapter 1 Test Review 30

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Vocabulary:

Definition Naming Diagram

Point- thought of as a _________ that

represents a _________________ in

a plane or in space.

Line- understood to be _________

and contains an infinite number of

points (at least ____)

- extends forever in two directions

and no thickness

Collinear- points that lie on

the _______________________

Non-collinear- points that do ______

lie on the _____________________

Plane- thought of as a __________

surface that extends forever in all

_________________.

Space- a boundless, Coplanar- points that lie on the

3D set of all points _______________________

collinear:

Line Segment/Segment- consists of

_______________ and all points in

between

- parallelogram

-3 non-collinear

points OR an

upper case

script

2 endpoints

with a line, no

arrows

Date___________ Day _____

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Ray- extends indefinitely in ______

direction with exactly _____ endpoint

Opposite Rays- two rays on the same

line (__________) that extend in

____________ directions, share the

same endpoint

Intersection- the point or set of points that two or more figures have in ____________

Practice with your compass:

endpoint first

with a one

arrow line

H is the common

endpoint and is

between F and G

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In exercises 1-8, use the diagram.

1.) Give two other names for AB .

2.) Name three points that are collinear.

3.) Give another name for plane F.

4.) Name a point that is not coplanar with A, B, and C.

5.) Give another name for CD . 7.) Name three rays with endpoints B.

6.) Name a pair of opposite rays. 8.) Give another name for CD .

Sketch the figure described. 9.) Three collinear points 10.) Four coplanar points 11.) Three lines that intersect at one point. 12.) A line and a plane that intersect at one point.

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In Exercises 13-20, use the diagram.

13.) Name the intersection of KL and PQ .

14.) Name the intersection of PQ and plane KMN.

15.) Name the intersection of plane R and

plane S. Critical Thinking 16.) A four-legged table is placed on a flat surface. The table rocks from side to side. Explain why this might occur. (Try to use new vocabulary.)

17.) Simplify: (x – 2)(x + 5) 18.) Factor: x2 – 9

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Postulate (axiom): an accepted rule without a given proof.

Postulate 1 Ruler Postulate: The points on a line can be matched one to one with the real numbers. The distance between points A and B is the absolute value of the difference. When three points are collinear, you can say that one point is ____________ the other two.

Postulate 2 Segment Addition Postulate: If B is between A and C, then ________________. If AB + BC = AC, then B is ____________, A and C.

“_____________________________________________________________”

Example: Find a length. Use the diagram to find BC. Solution Use the Segment Addition Postulate to write an equation. Then solve the equation to find BC. AC = AB + BC Segment Addition Postulate ___= ___ + BC Substitute 32 for AC and 10 for AB. ___ = BC Subtract 10 from each side.

1.) Use the diagram to find BC.

(a) (b)

2.)Draw a sketch of the three collinear points,

No line

over AB

means

distance

!

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where X is between Y and Z. Then write the segment addition postulate for the points. 3.) Suppose K is between L and M. Use the segment addition postulate to solve for the variable. Then find the lengths of LK, KM, and LM. LK= 7y + 9 KM = 3y + 4 LM = 143

Congruent Segments: Line segments that have the same length.

WRITE: Describe the differences between when you use “equal” and when you use

“congruent?” Give a SPECIFIC example of each.

Construct!

Construct a segment GH that is congruent to EF

NOTICE!!!

1) Hash marks on

segments

2) Different

notation between

length and

congruent

4.) Let J be between R and N. If RJ = 3x + 16, NJ = 2x – 8, and NR = 7x – 12. Find the length of the entire segment.

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5.) In the diagram, points P, Q, R and S are collinear. PS = 46, PR = 18, and PQ = QR. Find the indicated length.

(a) PQ

(b) QR

(c) RS

(d) QS

Midpoint: point that divides a segment into two congruent segments

Definition of Midpoint: If K is the midpoint of MN , then _____________

Diagram/label: How do we know if a point is a midpoint? Segment Bisector: a ray, line, or line segment that intersects the segment at its midpoint

Definition of Segment Bisector: If line q bisects MN at point P, then __________

Diagram/label:

Example 1: Find Segment Lengths

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In the diagram, line l bisects AC at point B, and AB = 8 in. Find AC.

Solution:

Point B is the midpoint of AC . So, AB = BC = 8 in.

Exercises:

1.) Find AC if AB = 10 cm. 2.) Point W bisects UV .

Find UV if WV = 14 inches.

3.) Find MF.

AC = AB + BC Segment Addition Postulate

= ____ + ____ Substitute 8 for AB and 8 for BC.

= ____ in. Add.

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--------------------------------------------------------------------------------------------------------------------

Construct a segment bisector:

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#1-8 Find the indicated length. 1.) Find GJ. 2.) Find KM. 3.) Find ST. 4.) Find UV.

5.) Find LM. 6.) Find YZ.

7.) Point B is between A and C on AC . Use the following information to (a) draw a diagram,

(b) write an equation in terms of x, (c) solve the equation, and then (d) find AB and BC.

AB = 2x – 5 BC = 6x AC = 27

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8.) In the diagram, points P, Q, R and S are collinear. PS = 46, PR = 18, and PQ = QR. Find the indicated length.

(e) PQ

(f) QR

(g) RS

(h) QS

Line l bisects the segment. Find the indicated length.

9.) Find AC if AB = 6 cm. 10.) Find PQ if NQ = 24.5 in.

11.) Line CD bisects AB at point C. Find AC if AB = 56 feet. (Draw the diagram.) In each diagram, M is the midpoint of the segment. Find the indicated length.

12.) Find XM. 13.) Find LN.

14.) Distances: Your house and your school are 8.4 miles apart on the same straight road. A baseball field is halfway between your house and your school, on the same road. Draw a sketch to represent this situation. Mark the locations of the house, school, and field. How far is your house from the baseball field?

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1.) Complete the sentence.

a. Collinear points are points that _________________________________________

b. Coplanar points are points that _________________________________________

c. If M is the midpoint of PQ , then _________________________________________

d. The difference between BC and CB is ________________________________________

2.) Find ST. 3.) Find UV.

4.) Find LM. 5.) Find YZ.

6.) Suppose J is between H and K. Use the Segment Addition Postulate to solve for x. Then find the length of each segment.

18

73

52

KH

xJK

xHJ

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In each diagram, M is the midpoint of the segment. Find the indicated length.

7.) Find XM. 8.) Find LN.

9.) H is the midpoint of WY . WH = 5x + 4 and HY = 2x + 28. Find x, and WH.

(Write an equation and solve. Find the desired measures.)

10.) Line CD bisects AB at point C. Find AC if AB = 56 feet. (Draw the diagram.)

11.) In the diagram of collinear points, .and IJHIGHHJGK 10,24 Find the length of

each segment.

a. HI = ______ b. JK = ______

c. IJ = ______ d. IG = ______

e. GH = ______ f. IK = ______

G KH I J

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12.) Construct a segment HI that is congruent to LO

13.) Label the endpoints of the segment “X” and “Y”. Construct a bisector of XY . Label the midpoint “M”. Complete the statements below.

a) ______ + _______ = ________

b) 2(_________) = _________

c) __________ __________

L O

H

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Vocabulary:

Angle- consists of two different ____________ with the same ______________.

(a) Label:

sides- vertex-

(b) Give three names for the angle.

Example:

1.) Name all the angles having R as the vertex.

2.) What are other names for 1?

3.) Is there an angle that can be named R?

Classifying Angles:

Protractor Postulate - use a protractor - the measure of an angle is in degrees

0 < mQRS < 180 - WORDS: the measure of QRS is equal to ____ - SYMBOL: mQRS = ______

1

Two angles are _____________

if they have the same measure.

Classifying Angles:

R

S

Q

Date___________ Day _____

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Construction: Angle congruent to a given angle

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Angle Addition Postulate If R is in the interior of PQS, then mPQR + mRQS = mPQS.

Remember…

4.) In the diagram, mABD = 90° 5.) Given that mLKN = 145°, find

4.) In the diagram, mABD = 90° 5.) Given that m LKN = 145°, find

and mDBC = 45°, find mABC. m LKM and mMKN

Angle Bisector: An angle bisector is a _____ that divides an angle into two angles that are congruent.

If QR is an angle bisector of PQS, then

___________________and ____________________

Remember…

EXAMPLES: 6.) If AT bisects CAN and mCAN=130 , find m1 and m2.

“ ____ ”

1

A

C

N

T

2

EXAMPLES:

“ ____ ”

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7.) If BD bisects ABC. Find mABC, given mABD=(2x+20) and mDBC=(4x)

-------------------------------------------------------------------------------------------------

Construction: Bisector of a given angle

1st: Sketch the diagram 2nd: Label the given info 3rd: Set up the equation 4th: Solve for x. 5th: Substitute the value of x.

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Classify the angle with the given measure as acute, obtuse, right, or straight.

1.) mA = 115° 2.) mA = 85°

3.) mA = 90° 4.) mA = 170°

Give another name for the angle in the diagram. Tell whether the angle appears to be acute,

obtuse, right, or straight.

5.) LKJ 6.) JLK

7.) KJL 8.) MKL

Find the indicated angle measure.

9.) mPRS = _____ 10.) mWXZ = _____

Use the given information to find the indicated angle measure.

11.) Given mADC = 135°, find mBDC. 12.) Given mNRQ = 78°, find mPRQ.

XZ

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Given that XZ bisects WXY, find the two angle measures not given in the diagram.

13.) 14.)

In each diagram, BD bisects ABC. Find mABC.

15.)

16.) Bridge In the bridge shown, the measure of FGH is 116° and bisects FGH. a)

What is the measure of FGK?

b) Use a compass and the given ray to construct

an angle that is congruent to HKJ

GK

The measure of ∠WXY is 146°

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1.5 – Special Angle Pair Relationships

Vertical Pair:

What’s my equation?

Linear Pair:

What’s my equation?

Determine if the angles are a vertical pair, a linear pair, an adjacent pair, or neither.

Think: Is there a relationship between angles 1, 2, and 3 together?

Bas

ed

on

th

e A

ngl

es’ _

____

___

____

___

___

___

__

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Practice Linear and Vertical Relationships:

1.

2.

3.

4.

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Complementary Pair: Two angles are complementary if their

________________ add up to ________.

What’s my equation?

Supplementary Pair: Two angles are supplementary if their

________________ add up to ________.

What’s my equation?

Do all complementary pairs form a right angle? Do all supplementary pairs form a straight angle? Are all linear pairs supplementary?

Based

on

the A

ngles’ ___

_______

______

___

____

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Practice Complementary and Supplementary Angles:

1. 2.

3. Find the measures of two complementary angles if one angle measures six degrees less than five times the

measure of the other.

4. LKH and GHI are supplementary. Find the value of x and the measure of each angle.

5.

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Extending beyond pairs:

Identify what the angles shown in the given figures should sum to. Then write an equation and solve for x.

1. Assume P, O, and Q are collinear. 2.

3.

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1.) Classify the relationship between angles a and b: vertical angles or linear pair.

a.) ________________ b.) ________________ c.) ________________ d.) ________________

2.) Find the measure of angle b.

a.) b.)

3.) Find the value of x.

a.) b.)

4.) Find the values of x and y.

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5.) Name a pair of complementary angles (C) and a pair of supplementary angles (S) for each image.

a.) b.)

C: ____________ S: _____________ C: ____________ S: _____________

6.) 1 and 2 are complementary angles. 7.) 1 and 2 are supplementary angles.

Given 1 89m , find 2m . Given 1 60m , find 2m .

8.) Find m DEG and m GEF . 9.) Find m DEG , m GEF , and m GEH

10.) Use the diagram below to tell whether the angles are vertical angles, a linear pair, or neither. a. 1 and 4 b. 1 and 2 c. 3 and 5 d. 2 and 3 e. 6 and 7 f. 5 and 6

11.) If RS bisects PRT such that (5 15)m PRS x

and 125m PRT find the value of x. (Draw diagram)

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*In addition to this study guide, some things to remember to look at are:

Vocabulary used in this chapter (there was a lot) Quizzes returned from smaller sections Homework problems – redo them especially if you struggled the first time

In Exercises 1–16, use the diagram.

1. Give another name for AB .

2. Name three points that are coplanar with both plane K and plane L.

3. Give another name for CG .

4. Name two pairs of opposite rays.

5. Are points A, G, and N collinear?

6. Are points A, G, and N coplanar?

7. Name the intersection of and .

Find the indicated length. 11. Find GJ.

15. Find PQ. 16. Find ST.

21. Find MQ.

AB MN

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Find the indicated angle measure.

23. mSTU = ____ 24. mYWZ = ____

25. Given mADC =118°, find ADB. 26. Given BD bisects ABC, mABD= (3x+17)°, and mCBD= (7x – 39)°. Find

mABC.

27. Draw obtuse angle HAT with angle bisector AS . Then name each angle formed and classify

them.

28. A and B are complementary angles. 29. A and B are supplementary angles.

Find mA and mB. Find mA and mB.

mA = (5x) mA = (x 11)

mB = (17x + 2) mB = (x 15)

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#30 & 31 Find mABC and mCBD.

30. 31. 32. The measure of one angle is 7 times the measure of its complement. Find the measure of each angle.

34. Find the values of x and y.