Chapter 1Tools of Geometry

The students will…

◦ identify and model points, lines, and planes.

◦ identify intersecting lines and planes.

1-1: Points, Lines and Planes

Some terms in geometry are considered undefined terms because they are only explained using examples and descriptions.

A point is a location.◦ It does not have size or shape.

A point A

A line is a set of points that extends forever in two opposite directions.◦ Has no thickness or width◦ There is exactly one line through any two points.

A

B

line m

line AB

line BA

AB

BAm

A plane is a set of points that extends forever in all directions.◦ There is exactly one plane through any three

points not on the same line.

A

B

plane K

plane ABC

plane ACB

plane BAC

plane BCA

plane CBA

plane CAB

K

C

Collinear points are points that lie on the same line.◦ Noncollinear points do not lie on the same line.

Coplanar points are points that lie in the same plane.◦ Noncoplanar points do not lie on the same plane.

Example: Use the figure to name each of the following:◦ A line containing point C◦ A plane containing point A

C A B

F

ED

Pr

s

The intersection of two or more geometric figures is the set of points they have in common.

a

b

X

Example: Draw and label a diagram for each of the following:◦ Points J(-4,2), K(3,2), and L are collinear◦ Line p lies in plane N and contains point L ◦ Line s intersects plane A at point P

Example: Use the figure below:◦ How many planes appear in the figure?◦ Name three points that are collinear.◦ Name the intersection of plane HDG with plane X.◦ At what point do LM and EF intersect?◦ Are points E,D,F, and G coplanar?◦ At what point or in what line do planes JDH, JDE,

and EDF intersect?

Pg. 8-12 #14-36 even, 53, 59

How many points determine a line? How many points determine a plane? Refer to the points below:

◦ Are these points collinear? Why or why not?◦ Are these points coplanar? Why or why not?

Review

What is the intersection of two nonparallel lines?

What is the intersection of two nonparallel planes?

Draw a figure to represent each of the following:◦ Line a lies in plane P and contains point Z.◦ Line b intersects plane Q at point Y.◦ Planes Q and R intersect at line WX.

The students will…

◦ measure segments.

◦ calculate with measures.

1-2: Linear Measure

A line segment is a portion of a line that has two endpoints.◦ The measure of AB is written as AB.

A

B

AB

BA

AB = 4.4 cm = 1¾ in

“Betweenness” of Points◦ Point M is between points P and Q if and only if P,

Q, and M are collinear and PM + MQ = PQ.

P

M

Q

Example: Find JL. Assume that the figure is not drawn to scale.

J

K

L

8.4 cm

2.3 cmJK + KL = JL

2.3 + 8.4 = JL

10.7 cm = JL

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

S

T

U

5x - 3

7x

ST + TU = SU

7x + 5x - 3 = 45

12x - 3 = 45

12x = 48

x = 4

ST = 7(4) = 28

Example: Find the value of a and XY if Y is between X and Z, XY = 3a, XZ = 5a – 4, and YZ = 14.

Segments that have the same measure are called congruent segments.

Pg. 18-21 #10-30 even, 37

A

B

Y

Z

YZAB

Draw and label a figure for each relationship:◦ FG lies in plane M and contains point H.◦ Lines r and s intersect at point W.

What is the length of CD if CE = 1.1 in, ED = 2.7 in, and E is between C and D.

Find the value of x and BC if B is between C and D: CB = 2x, BD = 3x + 5, BD = 12.

Review

The students will…

◦ find the distance between two points.

◦ find the midpoint of a segment.

1-3: Distance and Midpoint

The distance between two points is the length of the segment with those points as its endpoints.

Distance Formula (on a number line)◦ d = |x2 – x1|

Distance Formula (in a coordinate plane)◦ d =

Distance can be rational or irrational.◦ Rational – whole numbers, terminating decimals,

repeating decimals◦ Irrational – decimal that doesn’t repeat or

terminate

On your calculator:◦ When you square a negative number, put it in

parenthesis (-12)2

When you square a number, it will ALWAYS be positive.

Example: Find the distance between (-4, 1) and (3, -1)

Example: Find the distance between (-4, -6) and (5, -1).

d = √(x2 – x1)2 + (y2 – y1) 2

d = √(3 + 4)2 + (-1 – 1) 2

d = √(7)2 + (-2) 2

d = √49 + 4

d = √53 ≈ 7.28

Example: The United States Capitol is located 800 meters south and 2300 meters to the east of the White House. If the locations were placed on a coordinate grid, the White House would be at the origin. What is the distance between the Capitol and the White House?

The midpoint of a segment is the point halfway between the endpoints of the segment.◦ If X is the midpoint of AB, then AX = XB and

AX = XB.~

A

X

B

2 in

2 in

Midpoint Formula (on a number line)◦ M =

Example: The temperature on a thermometer dropped from a reading of 28o to -8o. What is the average temperature?

2

xx 21

Midpoint Formula (on a coordinate plane)◦ M =

Example: Find the coordinates of M, the midpoint of GH, for G(8, -6) and H(-14,12).

2

yy,

2

xx 2121

2

yy,

2

xxM 2121

2

126-,

2

148M

2

6,

2

6M

3,3M

Example: Find the coordinates of M, the midpoint of ST, for S(-6, 3) and T(1,0).

Example: Find the coordinates of J if K(-1, 2) is the midpoint of JL and L has coordinates (3, -5).

2

yy,

2

xxM 2121

12

3x

2

2

5-y

45y

9y

-23x

-5x (-5, 9)

Example: Find the coordinates of D if E(-6, 4) is the midpoint of DF and F has coordinates (-5, -3).

Example: Find the measure of PQ if Q is the midpoint of PR.

P

Q

R

9y - 2

14 + 5y

PQ = QR

9y – 2 = 14 + 5y

4y – 2 = 14

4y = 16

y = 4

PQ = 9(4) – 2 = 34

Example: Find the measure of YZ if Y is the midpoint of XZ and XY = 2x – 3 and YZ = 27 – 4x.

Any segment, line, or plane that intersects a segment at its midpoint is called a segment bisector.

Constructions are figures that are created using only a compass and straightedge.

Construct a segment bisector.

Pg. 31-33 #14-48 even, 57, 63

Find the distance between the points (-1, -8) and (3,4).

Find the value of a and ST is S is between R and T: RS = 7a, ST = 12a and RT = 76.

What is the midpoint of a segment that has endpoints at (3, 4) and (-15, 2)?

Review

The students will…

◦ measure and classify angles.

◦ identify and use congruent angles and the bisector of an angle.

1-4: Angle Measures

A ray is part of a line that has one endpoint and extends forever in one direction.◦ When you name a ray, list the endpoint first.

ray XY

XY

X

Y

An angle is formed by two noncollinear rays that have a common endpoint.◦ The rays are called sides of the angle.◦ The common endpoint is the vertex.

X

Y

Z

vertex X

side XZ

side XY

Naming an angle:◦ Use THREE letters (the vertex must be the

middle)◦ Use a single letter (the vertex) ONLY when there is

one angle at the vertex◦ Use a number if the angle is labeled with one

X

Y

Z

A12

ZXY

YXZ

X

1

An angle divides the plane into three distinct parts:◦ The angle◦ The interior of the angle◦ The exterior of the angle

X Y

Z

Example:◦ Name all angles that have

B as a vertex.◦ Name the sides of 5.◦ Write another name

for 6.

AB

DE

F

G

5

67

34

Angles are measured in degrees.◦ m ZXY = 78o

X

Z

Y

Angles can be classified by their measures:◦ Right – equals 90o

◦ Acute – less than 90o

◦ Obtuse – greater than 90o

Example: Classify each angle.◦ MJP◦ LJP◦ NJP

JK

L M

N

P

Angles that have the same measure are called congruent angles.

Y

X

Z

B

C

A

ABCXYZ

A ray that divides an angle into two congruent angles is called an angle bisector.

Construct an angle bisector.

Y

X

Z A

Example: KN bisects JKL. If m JKN = 8x – 13 and m NKL = 6x + 11, find m JKN.

K

L

J N

Line/Angle Art

Example: Create your own line design:

Pg. 41-43 #12-44 even, 51

60o

3

2

1

4

5

6

7

8

9

9

8

7

6

5

4

3

2

1

4.5 in sides

0.5 in

Use the figure below:◦ Name a point that is collinear with points A and B.◦ What is another name for plane P.◦ Name three points that are noncollinear.

Find the midpoint and length of a segment that has endpoints of (-2, -3) and (4, 1).

Review

C A B

F

ED

Pr

s

Use the figure below:◦ If m AXC = 8x – 7 and m AXB = 3x + 10, find

m AXC.

XA

B

C

D

E

The students will…

◦ identify and use special pairs of angles.

◦ identify perpendicular lines.

1-5: Angle Relationships

Adjacent angles are two angles that lie in the same plane and have a common vertex and a common side, but no common interior points.

1 2

A linear pair is a pair of adjacent angles with noncommon sides that are opposite rays.

12

Vertical angles are two nonadjacent angles formed by two intersecting lines.◦ Vertical angles are congruent.

3

4

21

42

31

Complementary angles are two angles with measures that have a sum of 90o.

12

A B

70o

20o

Supplementary angles are two angles with measures that have a sum of 180o.◦ The angles of a linear pair are supplementary.

1 2

A B

160o

20o

Example: Find the measures of two supplementary angles if the difference in the measures of the two angles is 32o.

m A + m B = 180o A + B = 180

m A - m B = 32o A - B = 32

2A = 212

A = 106

106 + B = 180

B = 74

The two angles are 106o and 74o.

+

Lines, segments, or rays that form right angles are perpendicular.

A

B

C

D

BDAC

Example: Find x and y so that KN and HM are perpendicular.

IJ

K LM

N

H

9xo

(3x + 6)o

(3y + 6)o

KJH = 90o

9x + 3x + 6 = 90

12x + 6 = 90

12x = 84

x = 7

MJN = 90o

3y + 6 = 90

3y = 84

y = 28

In geometry, there are some things that you cannot assume based on a picture.◦ Perpendicular lines◦ Congruent angles or segments◦ Look at Pg. 49 for a more detailed list.

Pg. 51-53 #8-22, 29, 30, 49

Find the measure of each angle in the figure below:

What is the distance between the points (2, 1) and (-3, 4)?

Draw a diagram to represent each of the following:◦ Two adjacent angles◦ Vertical angles◦ Supplementary nonadjacent angles.

Review

7x + 173x - 20

The students will…

◦ identify and name polygons.

◦ find perimeter, circumference, and area of 2D figures.

1-6: Two-Dimensional Figures

A polygon is a closed figure formed by a finite number of coplanar segments.

Polygons Not Polygons

Each point is a vertex of the polygon. Each segment is a side of the polygon. A polygon is named using the letters of the

vertices.

A

B

C

DE

vertex C

side AE Polygon ABCDE

Polygons can be convex or concave.◦ If you draw lines along each side of the polygon, if

any of the lines pass through the polygon, then it’s concave. Otherwise, it’s convex.

convex concave

Classification of polygons:

Number of Sides

Name 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

An equilateral polygon is a polygon in which all sides are congruent.

An equiangular polygon is a polygon in which all angles are congruent.

A convex polygon that is both equilateral and equiangular is a regular polygon.◦ If a polygon is not regular, then it is irregular.

Example: Name each polygon by its number of sides and classify it as convex or concave and regular or irregular.

The perimeter of a polygon is the sum of the lengths of its sides.

The circumference of a circle is the distance around the circle.

The area of a figure is the number of square units needed to cover a surface.

Review of formulas for common polygons and circles (pg. 58 in your book):

Perimeter and circumference are measured in units and area is measured in square units.

Example: Find the perimeter or circumference and the area of each figure:

3 in 2.5 cm

5.6 cm

2.5 in 2.5 in

4 in

1.5 in

Example: Teri has 19 feet of tape to mark an area in the classroom where the students may read. Which of these shapes has a perimeter or circumference that would use most or all of the tape?◦ Square with side lengths of 5 feet◦ Circle with radius of 3 feet◦ Right triangle with each leg length of 6 feet◦ Rectangle with a length of 8 feet and a width of 3

feet

Example: Find the perimeter and area of ∆PQR with vertices P(-1, 3), Q(-3, -1), and R(4, -1).

Pg. 61-63 #11-22, 24, 25, 33, 44

Name each polygon and classify it as convex or concave and regular or irregular.

Find the value of each variable:

The intersection of two planes is a ____. What is the midpoint of a segment that has

endpoints at (9,1) and (-8,5).

Review

5xx - 6

12x + 7

14x - 3

The students will…

◦ identify and name 3D figures.

◦ find surface area and volume.

1-7: Three-Dimensional Figures

A solid with all flat surfaces that enclose a region of space is called a polyhedron.◦ Each flat surface, or face, is a polygon.◦ The line segments where the faces intersect are

called edges.◦ The point where three or more edges intersect is

called a vertex.

Types of Solids:◦ Polyhedrons

A prism is a polyhedron with two parallel congruent faces called bases connected by parallelogram faces.

A pyramid is a polyhedron that has a polygonal base and three or more triangular faces that meet at a common vertex.

◦ Not Polyhedrons A cylinder is a solid with congruent parallel circular

bases connected by a curved surface.

A cone is a solid with a circular base connected by a curved surface to a single vertex.

A sphere is a set of points in space that are the same distance from a given point.

Polyhedra are named by the shape of their bases.

Triangular Prism

Rectangular Prism

Pentagonal Prism

Triangular Pyramid

Rectangular Pyramid

Pentagonal Pyramid

A polyhedron is a regular polyhedron if all of its faces are regular congruent polygons and all of the edges are congruent.◦ There are five types of regular polyhedrons, called

Platonic solids.

Surface area is a 2D measurement of the surface of a solid figure.

Volume is the measure of the amount of space enclosed by a solid.◦ Surface area and volume formulas (Pg. 69)

Example: Find the surface area and volume of each solid to the nearest tenth.

Example: The diameter of the pool Joe purchased is 8 feet. The height of the pool is 20 inches. ◦ What is the surface area of the pool?◦ What is the volume of water needed to fill the

pool to a depth of 16 inches?

Pg. 71-73 #6-13, 18-24, 37