Chapter 1: Tools of Geometry

Chapter 1: Tools of Geometry

Lesson 1: Points, Lines and Planes

Definitions

Point- represents a location Line- made up of points and has no thickness or

width, extends infinitely at both ends (cannot be measured)

Collinear- points on the same line Plane- flat surface made from points that has no

depth and extends in all directions infinitely Coplanar- points or lines on the same plane Space- boundless, 3-D set of all points that contains

lines and planes

Chapter 1 Foldable

Step 1- fold the construction paper in half both by width and length (hamburger and hotdog)

Step 2- Unfold the paper and hold width wise, fold in the ends until they meet at the center crease

Step 3- Cut the folded flaps along the crease so that there are now 4 flaps

Upper Left flap- Lesson 1.1 Points, Lines and Planes

Label the outside of the flap with the lesson number and title.

Inside the flap create a grid with 7 columns and 4 rows.

Copy the notes into the foldable, then draw and label your own examples based on the information in the chart.

Name Model Drawn Named By Facts Words/

Symbols

Examples

Point

As a dot A capitol letter

A point has neither size nor shape

point P

Line With an arrowhead

at both ends

Two letters representing

points on the line- or the script

letter

There is exactly 1

line through any two points

line n

line AB

line BA

Plane As a shaded,

slanted, 4-sided figure

A capital script letter or by any

three letters of non-

collinear points

There is exactly 1

plane through any three non-collinear points

plane S

plane XYZ

plane XZY

plane ZXY

plane ZYX

plane YXZ

plane YZX

P

AB

XYZ

S

n

A. Use the figure to name a line containing point K.

B. Use the figure to name a plane containing point L.

C. Use the figure to name the plane two different ways.

A. Name the geometric shape modeled by a 10 12 patio.

B. Name the geometric shape modeled by a water glass on a table.

C. Name the geometric shape modeled by a colored dot on a map used to mark the location of a city.

D. Name the geometric shape modeled by the ceiling of your classroom.

A. How many planes appear in this figure?

B. Name three points that are collinear.

C. Are points A, B, C, and D coplanar? Explain.

1.2 Linear Measure

Chapter 1: Tools of Geometry

Definitions

Line segment- part of a line that has two endpoints and can be measured(named by the letters marking the endpoints)

Congruent- same shape and size (segments that have the same measure)

A. Find LM.

B. Find XZ.

C. Find x and ST if T is between S and U, ST = 7x, SU = 45, and TU = 5x – 3.

Find SE.

Find a if AB = 4a + 10, BC = 3a – 5, and AC = 19.

Chapter 1: Tools of Geometry

Lesson 3: Distance and Midpoint

Definitions

Midpoint- the point on a segment that divides the segment into two congruent segments

Segment bisector- any line, segment or plane that intersects a segment at its midpoint

Distance and Midpoint

Distance Formula- used to find the length of a segment.

ex: Find the distance between A (5,1) and B (-3, -3).

*on a number line- subtract the endpoint values

Midpoint Formula- used to find the point half way down a segment

ex: Find the midpoint of JK if J(-1,2) and K(6, 1)

* on a number line- add the endpoint values and divide by 2

212

212 )()( yyxxd

2,

22121 yyxx

M

Use the number line to find the midpoint and the measure of AX.

Find the midpoint and distance between E(–4, 1) and F(3, –1).

Find the distance and midpoint of AM

Find the coordinates of R if N (8, –3) is the midpointof RS and S has coordinates (–1, 5).

Find LM. Assume that the figure is not drawn to scale.

Find the value of x and ST if T is between S and U, ST = 7x, SU = 45, and TU = 5x – 3.

Find the value of n and WX if W is between X and Y, WX = 6n – 10, XY = 17, and WY = 3n.

Chapter 1: Tools of Geometry

Lesson 4: Angle Measure

Definitions Degree- the unit of measurement for an angle Ray- a part of a line which has one endpoint and one end that

extends infinitely (name with the endpoint first and then any other point on the ray)

Opposite rays- two rays that share an endpoint and extend in opposite directions (together they make a line)

Angle- formed by two non-collinear rays that have a common endpoint

Sides of an angle- rays Vertex- the common endpoint of the rays of an angle Angle Bisector- a ray or line that divides an angle into two

congruent angles

Naming and Classifying Angles

Angle:

-B is the vertex

-ray BA and ray BC are the sides( BA and BC )

-Angle names:

ABC, CBA

B, 4

-Angle bisector : makes 2 congruent angles

A

BC

4

Name Measure Model

Right Angle

90

Acute Angle

Less than 90

(0 < x < 90)

Obtuse Angle

Between 90 and 180(90 < x < 180)

A. Name all angles that have B as a vertex.

B. Name the sides of 5.

C.

A. Measure TYV and classify it as right, acute, or obtuse.

B. Ray YT bisects angle SYU. Angle TYS = 2x-24, angle UYT = x+16.

Find x and the measure of angle SYU.

Chapter 1: Tools of Geometry

Lesson 5: Angle Relationships

Definitions Adjacent angles: two angles that lie in the same plane, have a

common vertex and a common side, but no common interior points

Vertical angles: two nonadjacent angles formed by two intersecting lines

Linear pair: a pair of adjacent angles with non-common sides that are opposite rays

Complementary angles: two angles with measures that add up to 90

Supplementary angles: two angels with measures that add up to 180

Perpendicular ( ): lines, segments or rays that form right angles

Angle Relationship examples

Adjacent angles Vertical angles

Linear pair Complementary angles

Supplementary angles Perpendicular lines

AB

C

D

L

M

N

O A

B

DC

E

RS

T

UV40 140

72 18

A. Name two adjacent angles whose sum is less than 90.

B. Name two acute vertical angles.

Find the measures of two supplementary angles if the measure of one angle is 6 less than five times the measure of the other angle.

A. Name an angle supplementary to BEC.

B. Name a linear pair whose vertex is E.

C. Name two acute vertical angles.

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

The supplement of A measures 140 degrees. What is the measure of the complement of A?

ALGEBRA Find x and y so thatKO and HM are perpendicular.