Identify Points, Lines and Planes

“We can redeem anyone who strives

unceasingly.” -Goethe

Identifying shapes

Undefined terms

O Point-Has no dimension. It is represented by a dot.

O Line-Has one dimension. It is represented by a line

with two arrowheads, but continues forever. (note:

through any two points there is EXACTLY ONE line,

so we can name a line using only two points)

O Plane-Has two dimensions. It is represented by a

shape that looks like a wall or floor, but this also

continues forever in every direction.

Other shapes

Other Shapes

O Line segment-called a segment because it is

a finite part of the line. Labeled by its

endpoints.

O Ray-One side has endpoints but the other

extends forever. Labeled going from

endpoint to infinity.

Ex. 1: Name all UNIQUE rays, lines, segments,

planes, and points.

A B C D

E F

G

H I J

R

S

K

Intersection

O The set of point where two shapes meet.

O The intersection of two unique lines is

always a point.

O The intersection of two unique planes is

always a line.

m

n

A B

S

R

C

Ex. 3: Sketch all interactions of a line and

plane. (there are three different ways)

Ex. 3: Name the intersections of:

1)Plane ABD and Plane AEF

2)Line AD and Line HD

3)Segment EF and Segment GF

4)Plane EFH and Plane ABC

5)Segment AD and Segment HG

A

B C

D

E

F G

H

Summary

O You should now be able to:

O Identify points, lines, segments, rays, and

planes.

O Identify points of intersection.

O Make sketches of each type of shape.

Identify Angles and Angle Measures Part 1

“People don’t ever seem to realize that

doing what’s right is no guarantee against

misfortune.” –William McFee

Parts of an angle

O Sides-line segments or rays that form the

angle.

O Vertex-the endpoint of the rays or the place

where the sides of an angle meet. (the

corner or sharp point of the angle)

O Naming angles

B

A

C

D E

F

G

O Ex. 1: Name all the angles in the figure below.

A

B

C

D

E G

F

Measuring with a Protractor

O Line up the protractor with a side and make

sure the vertex is in the center of the dot.

O Read off angle according to the type of

angle.

O Postulate 3: Protractor Postulate

O The measure of an angle is equal to the

absolute value of the difference between the

numbers for the sides.

Identify Angles and Angle Measures Part 2

“People don’t ever seem to realize that

doing what’s right is no guarantee against

misfortune.” –William McFee

Angle Addition Postulate

O Postulate 4: If P is in the interior of angle

RST then the measure of angle RST is equal

to the sum of the measures of angle RSP

and angle PST.

R

S

P

T

O Ex. 2: Given that the measure of angle RST is 80°,

find the measure of angle PST and the measure of

angle RSP.

R

S

P

T

(3x – 5)°

(2x + 10)°

Congruent and Bisectors

O Two angles are considered congruent if they have

the same measure. Congruent means that two

shapes are the same size and same shape. (like

the equal sign but for shapes instead of for

numbers). In Ex. 2, angle RSP is congruent to angle

PST.

O An angle bisector is a segment, ray, or line that

divides an angle into two smaller congruent angles.

In Ex. 2, ray SP ended up being an angle bisector of

angle RST.

O Ex. 3: Ray AB bisects angle CAD. The measure of

angle BAD is 54°, what is the measure of angle

CAD and the measure of angle CAB?

Summary

O You should know how to identify an angle.

O You should know how to measure an angle

using a protractor.

O You should know what a bisector does and

what congruent means.

Segment Length and Congruence

“Our minds are lazier than our bodies.”

–La Rochefoucauld

Ruler Postulate

O Postulate 1: The points on a line can be

matched one to one with the real numbers.

These numbers are called coordinates of the

point. The distance between points A and B,

written as AB, is the absolute value of the

difference of the coordinates of A and B.

O Ex. 1: Measure the length of these segments to the

nearest tenth of a centimeter.

Segment Addition Postulate

O Postulate 2: If B is between A and C, then

AB + BC = AC. The converse is true also. If

AB + BC = AC, then B is between A and C.

A B C

Congruence

O Two segments are congruent if they have the

same length. If AB = BC, then segment AB is

congruent to segment BC.

O Ex. 2: Use the diagram to find BC.

A B C

B A C

56

20

x

25 35

O Ex. 3: Plot J(-2, 5), K(1, 5), L(3, 4), and M(3, -1) in

a coordinate plane. Then determine whether

segment JK and segment LM are congruent.

Summary

O You should know what postulate 1 and 2

allow you do and understand that postulates

are rules that can’t be proved. These rules

are the foundations of geometry.

O You should understand the difference

between congruent and equal.

Midpoint and Distance Formulas

“This is nothing wrong with making

mistakes. Just don’t respond with encores.

–Anon.

Midpoint

O The midpoint of a segment is the point that

divides the segment into two smaller

congruent segments.

O A segment bisector is a shape that goes

through the midpoint of a segment.

A B M

C

D

O Ex. 1: M is the midpoint of segment AB. If AM is

25, what is BM and AB?

O Ex. 2: M is the midpoint of segment AB. If AM is 2x-

5 and MB is x+3, what is AB?

The Midpoint Formula

O The coordinates of the midpoint of a

segment are the averages of the coordinates

of the endpoints. If 𝐴(𝑥1, 𝑦1) and 𝐵(𝑥2, 𝑦2)

are points on a coordinate plane, then the

midpoint M of segment AB has the

coordinates 𝑀(𝑥1+𝑥2

2,

𝑦1+𝑦2

2).

O Ex. 3: Find the coordinates of the midpoint M of

segment AB if A(1, 2) and B(5, -4).

O Ex. 4: Find the coordinates of the endpoint C of

segment CD. The midpoint of segment CD is N(1,4)

and the endpoint D(3, 5).

Distance Formula

O If 𝐴(𝑥1, 𝑦1) and 𝐵(𝑥2, 𝑦2) are points on a

coordinate plane, then the distance between

A and B is 𝐴𝐵 = (𝑥2 − 𝑥1)2+(𝑦2 − 𝑦1)2.

O Ex. 5: What is the length of segment AB with

endpoints A(1, 2) and B(-3, -1)?

Summary

O You should be able to use the midpoint

formula to find the coordinates of an

endpoint or midpoint of a segment.

O You should be able to use the distance

formula to find the length of segment.

Angle Pairs “Being entirely honest with oneself is a

good practice” –Sigmund Freud

Complementary and Supplementary Angles

O Complementary angles are two angles that

measures sum to 90°.

O Supplementary angles are two angles that

measures sum to 180°.

O Ex. 1: Identify/Name the supplementary angles, the

complementary angles and the adjacent angles.

A

B

C

D E

F

G

22°

112° 68°

O Ex. 2: Given that angle 1 and angle 2 are

complements and measure of angle 1 is 78°, find

measure of angle 2.

O Ex. 3: Given that angle 3 and angle 4 are

supplements and measure of angle 4 is 112°,

find the measure of angle 3.

O Additional question: Which angles given above are

adjacent?

Linear Pair

O Linear Pair-Two adjacent angles with their

noncommon sides forming opposite rays.

(could be though of two adjacent

supplementary angles)

1 2

Vertical Angles

O Two angles with sides forming two pairs of

opposite rays

1 2

O Ex. 4: Identify all vertical angles and linear pairs.

1

2 3

4

5

O Ex. 5: Two angles form a linear pair and one angle

is 3 times the measure of the other angle. What

are the measures of the angles?

O What can you conclude.

A B C

D E

Summary

O You should be able to identify

complementary and supplementary angles.

O You should be able to identify linear pairs

and vertical angles.

O You should be able to use the above

definitions to solve a problem.

Classify Polygons “Experience is a good teacher, but she

sends in terrific bills.”

-Minna Antrim

Polygons

O Poly-multi and gons-sides. Translates to

multisided shapes.

O Two requirements:

O It is formed of three or more segments called

sides.

O Each side intersects EXACTLY two sides, one

at each endpoint, so that no two sides with a

common endpoint are collinear.

Convex and Concave

O If you could draw a line from any two

vertices of a polygon and that line would

only be in the interior or the side of polygon,

then that polygon is convex.

O If at least one line drawn from two vertices

of a polygon lies in the exterior, then the

polygon is concave.

Classifying Polygons Number of Sides Type of Polygon

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon

11 Hendecagon

12 Dodecagon

n N-gon

100 100-gon

Special Polygons

O Equilateral - All sides are congruent (all

sides have the same length)

O Equiangular - All angles are congruent (all

angles have the same measure)

O Regular – Polygon is both equiangular and

equilateral

Ex. 1: Tell whether each shape is a polygon. If it is a

polygon, classify the polygon and state whether it is

concave or convex.

Ex. 2: Classify each polygon and tell whether the

polygon is equilateral, equiangular, regular, or none of

these. Explain.

Ex. 3: A stop sign is a regular octagon. The

expressions for two of the side lengths are (4x – 4)in.

and (3x + 6)in. What is the length of the edge of the

stop sign?

Perimeter, Circumference, and Area of Basic Shapes

“To travel hopefully is a better thing than to

arrive, and the true success is to labour.”

-Robert Louis Stevenson

Terms

O Perimeter (P) – the distance around a

polygon

O Circumference (C)– the distance around a

circle

O Area (A) – the amount of surface covered by

the figure.

Formulas

Square

𝑃 = 4𝑠

𝐴 = 𝑠2

Side length s

Rectangle

𝑃 = 2𝑙 + 2𝑤

𝐴 = 𝑙𝑤

Length l and width w

Formulas

Triangle

𝑃 = 𝑎 + 𝑏 + 𝑐

𝐴 =1

2𝑏ℎ

Side lengths a, b, and c;

base b; and height h

Circle

𝐶 = 𝜋𝑑 𝑜𝑟 𝐶 = 2𝜋𝑟

𝐴 = 𝜋𝑟2

Diameter d and radius r

Ex. 1: Find the perimeter and area of a football field if

it has a width of 50 yards and length of 120 yards.

Ex. 2: Find the area and perimeter of a triangle with

the given dimension below

3.5

2 2 1

Ex. 3: Triangle ABC has coordinates A(1, 6), B(1, 2),

and C(7, 3). Find its perimeter and area.

Ex. 4: Find the circumference and area with a

diameter of 10 in. and a center C(5, 7).

Summary

O Students should be able to find the

perimeter and area of a triangle, rectangle

and square.

O Students should be able to find the

circumference and area of a circle.

Bisect Segments and Angles

“The difference between what we do and what we are capable of doing would suffice to solve most of the world’s

problems.” -Gandhi

No Measuring Device

O Without a ruler or protractor we can

construct squares, perpendicular lines,

equilateral triangles, bisect angles perfectly,

etc.