Download Geometry Chapter 1

GEOMETRY
Amy Brunsting
Introduction
Why do we study geometry?
Geometry is the study of the properties of and
relationships between points, lines, surfaces and
solids. People use geometry to solve everyday
problems, such as determining how much paint to
buy to paint a room or figuring out what is the
shortest distance between two places.
Since shapes are easy to understand, the study of geometry is also used to teach logic. You will
learn how to collect facts, apply rules, and use deductive and inductive reasoning to draw
conclusions.
The “Father of Geometry”
In Alexandria, Egypt around 300 B. C., a Greek mathematician named Euclid wrote a collection of 13
books called the Elements. They contained definitions, theorems, rules, and constructions that
covered most of the mathematics that was known at that time, both geometry and algebra. The
Elements applied logic to the study of math, and it was used up until the 20th century as the main
textbook for teaching higher level mathematics.
The Master Mathematician
Leonhard Euler (pronounced “Oiler”) was a Swiss mathematician (1707 – 1783) who made
tremendous contributions to the fields of math and physics. He wrote most of the rules you
probably learned in algebra. Euler was the creator of graph theory – the study of diagrams that can
be used to model relationships between parts of a system – and he is credited with creating the
Euler diagram to represent sets and their relationships. You will use both of these techniques, as
well as the algebra you have learned, to solve problems in geometry.
Chapter 1
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Chapter1
Lesson 1: The Basics
Points
A point is a location in space. It has no size. It’s just a point. When you draw a point on paper, you
will make a dot and label it with a capital letter. The figure below represents Point Q. When you
are writing about Point Q, you can just call it Q.
·Q
Lines
For geometry, a line is straight and it goes on forever in both directions. When you draw a line on
paper, you will draw a straight line with arrows on both ends. Lines are labeled two different ways:
using one lowercase letter or two uppercase letters. The uppercase letters represent points on the
line.
l
or
A
B
When you are writing about a line, you could refer to Line l or AB. You could also call the second
line BA. The order of the points doesn’t matter since the line continues in both directions forever.
Points that are on the same line are called collinear. Points A and B on the line above are collinear.
You learned that geometry is used to teach logic. You will be asked to prove that a statement is
true. To do that, you’ll need facts. One type of fact you can use is called a postulate. A postulate is
a statement that is accepted as true without proof. The first three postulates you will learn are
about lines.
Postulate: The shortest distance between two points is a straight line.
Postulate: Any two distinct points in space have exactly one line that contains them.
Postulate: Two straight lines can intersect in only one point.
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Chapter 1
Planes
A plane is a flat surface that extends forever in all directions. When you draw a plane, it usually will
look like a parallelogram. We will label planes with three or more capital letters that represent
points in the plane.
·Y
Horizontal Plane EFG
Vertical Plane WXY
·W
·E
·F
·G
·X
Points that are in the same plane are called coplanar. Points W, X, and Y in the plane above are
coplanar.
Here are two more postulates for you. This time, they are about planes.
Postulate: Any three distinct points that are not all on the same line have
exactly one plane that contains them.
Postulate: When two planes intersect, they form a straight line.
Lesson 1 Homework
For Problems 1 – 8, use the diagram.
1. Name three points that are collinear, then name the line that contains them.
E
2. Name the intersection of the plane ABC and EG.
C
3. Are points D and F collinear? Explain
4. Are points A, C, and G coplanar? Explain.
5. Name a line that is contained in plane AFD.
B
F
A
D
G
6. How many straight lines can pass through points A and D?
7. How many planes can contain points E and B?
8. How many planes contain points E, B and C?
Chapter 1
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Lesson 2: Line Segments
A line segment is a small piece of a line between two points. It has length and can be measured.
When you draw a line segment, you’ll draw a short straight line and label the ends of the line
segment with capital letters. This is line segment RS.
R
S
When you write about line segments, you will name it by its endpoints and draw a short line above
the two letters, like this: R S. You could also call this segment S R because the direction doesn’t
matter.
Since line segments can be measured, you need a rule that allows you to measure them. (Yes, it
sounds ridiculous, but you will use the rule when you do proofs and you have to give it a name.)
Postulate: The Ruler Postulate – The points on a line correspond with real numbers on a number
line. The distance between points A and B equals the absolute value of the difference of their
coordinates.
For example, if you placed AB next to a number line. The coordinate 3 corresponds to A, and the
coordinate 9 corresponds to B. To find the length of AB, subtract the coordinates, 3 – 9 = -6. Since
distances are always positive, take the absolute value of your answer, |-6| = 6. The length of AB is 6
units long.
A
B
•
-1
•
0
•
1
•
2
•
3
•
4
•
5
•
6
•
7
•
8
•
9
•
10
•
11
The length of two segments can be equal. When two objects have exactly the same shape, they are
called congruent. The symbol for congruence looks like an equal sign with a wave on top, ≅. If RS
and PQ have the same measurement, you could write RS ≅ PQ.
If a line segment can be measured, it can have a middle, or a midpoint. A midpoint is the point that
divides the line into two equal parts. You now have another fact to learn, but this time, it isn’t a
postulates. It’s a theorem. A theorem is a statement that can be proven based on accepted facts.
Theorem: A line segment can have only one midpoint.
Point M is the midpoint of AB. A midpoint divides a line into two equal parts, so you know that
mAM = mMB. Since their measures are equal, AM ≅ MB.
A
M
B
If you know the coordinates of the endpoints of a line segment, it’s easy to find the coordinate of
the midpoint. Just average the x values and average the y values. For example, if the endpoints of a
line are at (3, 5) and (7, 11), its midpoint is at (5, 8) because (3 + 7)/2 = 5 and (5 + 11)/2 = 8.
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Chapter 1
Segment Addition
If you have a 6-foot long board, you can cut it into a 2-foot piece and a 4-foot piece. It makes sense
that you can add the lengths of the boards together to get a total length. So it should make sense
that you can do the same thing with line segments. When you add line the length of line segments,
you are using the Segment Addition Postulate.
Postulate: Segment Addition Postulate– If B is between A and C, the AB + BC = AC
Since XY = 16 and YZ = 27, XY = 47.
X
16
Y
27
Z
Lesson 2 Homework
1. Use a ruler to draw lines of the following lengths: 3 in., 1 ¼ in., 5 cm, 3 cm
2. Estimate the length of each line in inches and in centimeters. Write down your estimate.
Check your estimate with a ruler. Write down the error in your estimate.
a.
b.
3. Draw lines without aid of a ruler (freehand) as nearly as possible of the following lengths: 4
in., 1 in., 10 cm, 2 cm. Measure your lines and write down the error in each.
4. Draw a line. Estimate its midpoint. Measure the line in millimeters and note your error.
5. Use a ruler to draw a line 5 ¼ inches long. Divide this line into three parts that have a ratio
of
1: 2: 3.
Find the indicated lengths.
6. Find XY.
30
X
Y 7 Z
7. M is the midpoint of AC. Find AM.
x+5
A
Chapter 1
2x
M
C
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In the diagram, P, Q, R, S, and T are collinear. PT = 54, QT = 42, QS = 31, and RS = 17. Find the
indicated lengths.
8. PQ
P
Q
R
S
T
9. QR
10. ST
11. RT
Point B is between A and C on AC. Use the given information to write an equation in terms of x.
Solve the equation. Find the lengths of AB and BC. Determine whether they are congruent.
12. AB = x + 3
BC = 2x + 1
AC = 10
13. AB = 11x – 16
BC = 8x – 1
AC = 78
Find the coordinates of the midpoints of the segments with the given endpoints.
14. A(3, -7), B(-1, 9)
15. C(2, -4), D(7, 1)
Tell whether the following statements are true or false.
16. If AB ≅ CD, the mAB = mCD.
17. If AX = XC, then X is the midpoint of AC.
18. If T is the midpoint of SR, then S, T, and R are collinear.
19. If B is the midpoint of PQ and PQ = 24, then PB = 12.
20. If AX + XR = AR, then X is the midpoint of AR.
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Chapter 1
Lesson 3: Angles and their Measurement
Rays
A ray is a half-line. It extends forever in one direction only. When you refer to a ray, you have to
write the endpoint first. The order matters because the ray extends only in one direction. The
figures below show ray MN or MN and ray YZ or YZ.
N
Z
Y
M
Opposite rays are two rays that share the same endpoint, but extend in opposite directions, forming
a straight line. AB and BC are opposite rays.
A
B
C
Angles
An angle is made up of two rays that have the same endpoint. The endpoint is called the vertex.
The rays are called the sides of the angle. Angles can be named two ways: with one letter or three
letters. The figure below shows ∠K, ∠JKL, or ∠LKJ. When you use three letters, the central letter is
always the vertex.
J
L
K
You will use a protractor to measure the size of angles. Since angles can be measured, they have
rules that are similar to the rules for line segments.
Postulate: The Protractor Postulate – The rays in a half-rotation can be numbered from 0° to 180°.
The measure of an angle equals the absolute value of the difference of their coordinates.
For example, say an angle was made up of OA and OB. The coordinate 39° corresponds to OA, and
the coordinate 87° corresponds to OB. To find the measure of ∠AOB, subtract the coordinates,
39° – 87° = -48°. Since measurements are always positive, take the absolute value of your answer,
|-48°| = 48°. The measurement of ∠AOB is 48°.
B
A
Chapter 1
O
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Angles can with the same measurements are congruent. If m∠G = 40° and m∠H = 40°, you know
that ∠G ≅ ∠H.
Theorem: An angle can have only one bisector.
“Bisect” means to cut or divide into two equal parts. An angle bisector is the line or ray that cuts an
angle in half. In the figure, XB bisects ∠WXY. Since ∠WXB and ∠BXY are equal, ∠WXB ≅ ∠BXY.
B
W
Y
X
Angle Addition
If two angles are adjacent, their measurements can be added to get the measurement of the entire
angle. This is called the Angle Addition Postulate.
Postulate: Angle Addition Postulate – If D is in the interior of ∠ABC, then the measure of ∠ABC is
equal to the sum of the measures of ∠ABD and ∠DBC.
A
B
Since ∠ABD = 21° and ∠DBC = 43°,
∠ABC = 21° + 43° = 64°
21°
43°
D
C
Classifying Angles
Angles can be classified according to their size.
Acute
Less than 90°.
Straight
Equal to 180°.
Right
Equal to 90°.
Reflex
Greater than 180° but less than 360°.
Obtuse
Greater than 90° but less than 180°.
Perigon
A complete rotation; equal to 360°.
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Chapter 1
Angle Pairs
There are several important pairs of angles that you should know.
1
Adjacent angles
Adjacent angles are next to each other. They share a side.
Linear pair
Two angles that are adjacent and form a straight angle.
1
2
2
1
Complementary angles
Two angles are complements if their measurements add to 90°.
Supplementary angles
Two angles are supplements if their measures add to 180°.
1
Vertical angles
2
Two opposite angles formed by the intersection of two lines. They are
adjacent to the same angle and form linear pairs with it.
1
2
Chapter 1
2
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Lesson 3 Homework
1. Name all six angles in the drawing. Name the two opposite rays.
C
B
M
O
P
2. Use a protractor to draw the following angles. Label the vertex of each angle.
a. ∠A = 53°
b. ∠B = 100°
c. ∠C = 15°
3. Using a ruler, but not a protractor, draw angles as nearly as possible to the following angles.
Label the vertex of each angle. Measure your angles with a protractor and note the error.
a. ∠D = 30°
b. ∠E = 80°
c. ∠F = 135°
∠TPS is a straight angle. Use the given information to find the indicated angle measures.
Q
R
57°
64°
T
P
S
4. m∠QPS = ?
5. m ∠TPQ = ?
6. m∠TPR = ?
7. Given m∠ABC = 133°. Find m∠ABD.
B
C
(3x + 9)°
A
(7x + 4)°
D
8. Find the value of x.
2x°
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4x° – 12°
Chapter 1
9. How many degrees are there in:
a. half of a right angle?
b. one-third of a straight angle?
c. one-fifth of a perigon?
10. Find the complement of each angle:
a. 37°
b. 72°
c. 41.2°
11. Find the supplement of each angle:
a. 35°
b. 102°
c. 89.5°
12. If two complementary angles are equal, what is the size of each angle?
13. Angle Q and Angle P are complements. m∠Q = (3x + 2)°, m∠P = (x – 4)°. Find the measure
of ∠Q and ∠P.
14. Angle X and Angle Y are complements. Angle X is twice the size of Angle Y. What is the
measure of each angle?
15. Angle A and Angle B are supplementary. The ratio of the size of the angles is 2:3. Find the
measurement of both angles.
Tell whether the statement is always, sometimes, or never true. Explain your reasoning.
16. An obtuse angle has a complement.
17. An acute angle has a supplement.
18. The complement of an acute angle is an acute angle.
19. The supplement of a right angle is an acute angle.
Chapter 1
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Lesson 4: Constructions
Constructions are drawings made using just a straightedge and a compass. A straightedge can be
any edge that is straight, such as the edge of a notebook, but you will typically use a ruler. The ruler
is not used for measuring, just for drawing straight lines. A compass is an instrument that is used for
drawing circles and arcs. It has a sharp point on one leg and a piece of graphite on the other.
Construction #1: Bisecting a Line Segment
C
A
C
A
A
B
B
B
D
D
1. To bisect AB, push the sharp end of the compass in at A.
2. Open the compass wide enough so that the graphite end is past the midpoint of the segment.
3. Hold the compass at the top and lightly swing an arc that extends above and below the
segment. Do not adjust the compass, but pick it up, and push the sharp end in at B. Swing
another arc above and below the segment.
4. Your two arcs should intersect at points C and D. Draw a line through points C and D. This line
bisects AB.
Construction #2: Bisecting an Angle
A
A
C
B
A
C
B
D
A
C
B
D
C
B
1.
2.
3.
4.
Push the sharp end of the compass in at B.
Draw a light arc that crosses both sides of the angle.
Insert the sharp end of the compass at A and draw a light arc across the interior of the angle.
Do not adjust the compass. Insert the sharp end at C and draw a light arc across the interior of
the angle.
5. The arcs should meet at point D. To bisect the angle, draw a line through points B and D. This is
the angle bisector.
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Chapter 1
Lesson 4 Homework
For each construction, use only a compass and a straightedge. Do not use a ruler or a protractor to
do the construction. Leave your construction lines. If they are not visible, you will not get credit for
the construction.
1. Copy this line on your paper.
2. Copy this angle on your paper.
3. Use a ruler to draw a 10 cm line on your paper. Bisect the line using a compass and
straightedge.
4. Use a ruler to draw a 2 ¾ inch line on your paper.
5. Use a protractor to draw a 70° angle on your paper. Bisect the angle using a compass and
straightedge.
6. Use a protractor to draw a 130° angle on your paper. Bisect the angle.
7. Use a straightedge to draw an obtuse angle. Extend one side of your angle so that it forms a
linear pair. Use your compass and straightedge to bisect both angles.
8. Use only a compass and a straightedge to draw an interesting shape or picture.
Chapter 1
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