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iBooks Author
Lines and Angles
1
1. Lines
2. Angles
3. More on Lines and
Angles
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iBooks Author
Foreword
The purpose of this book is to teach essential geometry, lines
and angles, in the simplest way possible through interactive diagrams. The goal is for the reader to have an intuitive feel for geometry.
If a picture is worth a thousand words, then an interactive diagram is worth thousand pictures..
Click and try the interactive diagrams to better understand the
concepts. Sometimes it takes a few seconds to load the diagram so be patient.
The review at the end of each section should be attempted, as
this will solidify the learning process.
The book is in three sections:
a) Lines
! line
! point
! line segment
! ray
! intersection point!
b) Angles
! Angles
! Congruent
! Classification of Angles
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iBooks Author
!
!
!
!
!
!
!
!
Straight line sum of angles
Sum of Angles of Circle
Vertical Angles
Angle Bisector
Line Bisector
Perpendicular Lines
Parallel Lines
Transversal Line
The goal of this book is to allow students to quickly grasp essentials that they can apply.
The book is deliberately made available only in landscape
mode as this makes the placement of interactive diagrams optimal.
Geometry can be interesting and fun!
I am grateful to my family, Sanjana, Nikhil, Trisha, and Krish for
their support.
c) Lines and Angles, Using Algebra
! using algebra to solve geometry problems!
! using ratios to solve geometry problems
Sections (a) and (b) lay down the foundation for understanding
the basics of lines and angles and should be accessible to all.
Section (c) is optional as it contains more intermediate to advanced material.
Krishan Jhunjhnuwala
This is the first volume of a five volume series on Geometry.
1. Lines and Angles
2. Triangles
3. Circles
4. Polygons
5. Cylinders, Cones, and Spheres
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iBooks Author
About the Author
Krishan Jhunjhnuwala has MS degree in Mathematics
from Rutger University. He also has a MBA from NYU
and BA in Economics from USC.
He has had a varied career, starting out in Sales and
Marketing as an entrepreneur. For the last fifteen years
he has worked in the software field mainly in Investment
Banking.
He was the youngest chess champion of Hong Kong
and is an internationally ranked chess FIDE master.
He is motived to write interactive books to make
Geometry fun and easy to understand.
He strongly believes that a strong foundation in the
essentials will greatly facilitate the learning of more
advanced materials.
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iBooks Author
Credits
1. Interactive programs and figures created by using:
geogebra, http://www.geogebra.org/ (GPL).
2. Book Title and Intro media pages designed and created by
Keval Amin and used with permission.
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iBooks Author
Section 1
A straight line usually is drawn with a ruler on a piece of paper.
In Geometry a line extends forever in both direction. If we have
two different points (A and B) on a line, we call this “Line AB.”
Lines
I NTERACTIVE 1.1
Line AB
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A point is like a dot on a piece of paper. We can label this point
with a letter or number. A point specifies one location.
A Line Segment is similar to a Line except that it has two distinct end points. The name of the Line Segment with endpoints
A and B is written as “Line Segment AB”.
I NTERACTIVE 1.2
I NTERACTIVE 1.3
Points
Line Segment
Points A, B, C, D
Line Segment AB
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A Ray is a straight line that begins at a certain point and extends forever in one direction. The point where the ray begins is
its end point. A ray with endpoint A and passing through point
B is called “Ray AB”
An end point is a point used to define a Line Segment or a Ray.
A Line Segment has two endpoints while a Ray has one endpoint. In interactive diagram 1.3, points A and B are endpoints.
In interactive diagram 1.4, A is an endpoint.
The point where lines, rays or line segment meet is called the
intersection point.
I NTERACTIVE 1.5
Intersection Point
I NTERACTIVE 1.4
Ray
Point E is the intersection point for lines AB and CD
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Ray AB
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iBooks Author
R EVIEW 1.1
Section Review
Question 1 of 4
Match the labels.
Points
Line
Segment
Ray
Ray
Segment
Points
Line
Check Answer
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iBooks Author
Section 2
Angles
An angle is formed when two lines intersect. The point of intersection is
called the Vertex.
Angles are measures in degrees. The angle is named by indicating the
points on the lines creating the angle. In this case, the points are A, B,
and C. The notation to use for angle is ∠. So we can have ∠ABC. The
angle itself can also be named. In the diagram below, the angle is
named “a”. So we can also call this ∠a.
I NTERACTIVE 1.6
Angles
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iBooks Author
If two angles have the same measure, they are congruent. In figure 1.1
∠a and ∠b are congruent, each being 90°.
Angles are classified according to their measures in degrees.
Acute Angle is less
than 90°
Type/
Degree
< 90°
Acute
✓
90°
Right Angle = 90°
Obtuse Angle is
greater than 90°, less
than 180°
F IGURE 1.1 Congruent Angles
Straight Angle =
180°
Right
Obtuse
Straight
Reflex
90°
to
180°
180°
180°
to
360°
✓
✓
✓
✓
Reflex Angle is between 180° and 360°
I NTERACTIVE 1.7
Classification Of Angles
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The sum of two or more angles on a straight line is always 180°.
The sum of all the angles around a point is always 360°
I NTERACTIVE 1.8
I NTERACTIVE 1.9
Straight line sum of angles
Sum of Angles of Circle
Angle a + b + c + d = 360°
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When two lines intersect, the two angles on opposite sides are called
vertical angles. Vertical angles are congruent (equal)
In interactive 1.11, line L divides ∠ABC into two equal angles, a and b.
Line L bisects ∠ABC . The dotted Line L is called the Angle Bisector.
I NTERACTIVE 1.10
I NTERACTIVE 1.11
Vertical Angles
Angle Bisector
Angle a and b are congruent(equal)
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iBooks Author
I LLUSTRATION 1.1
I LLUSTRATION 1.2
In the figure below, A, C, B are all on a straight line.
What is angle b? (See Interactive 1.5)
In the figure below, A, C, B are all on a straight line.
What is angle b? (See Interactive 1.5)
Answer:
Since a straight line line is 180°,
Answer:
We have, “angle a” + “angle b” = 180°.
Since a straight line line is 180°,
But “angle a” = 30°.
We have, a + b + c = 180°.
Therefore: 30° + “angle b” = 180°
But a = 30° and c = 60°.
!
!
Therefore: 30° + b + 60° = 180°
!
!
“angle b” = 180° - 30°
!
= 150°
b = 150°
!
!
!
!
b = 90°
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iBooks Author
b”= 180° - 30° - 60°
!
= 90°
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I LLUSTRATION 1.3
I LLUSTRATION 1.4
In the figure below, the two line are straight lines .
What is angle b? (See Interactive 1.7)
Lines L, J and K intersect at B. If Line J bisects
∠ABC, What is the value of z? (See Interactive
1.8 and 1.5)
Answer:
Answer:
Since vertical angles are congruent,
Since ∠ABC + d = 180° and ∠ABC = z + u
b=a
we get, z + u + d = 180°, but d = 120°
So, z + u + 120° = 180°
b = 36°
therefore z + u = 180° - 120° = 60°
Now u = z, because ∠ABC is bisected
z + z = 60°
2z = 60°
z = 60°/2
z = 30°
z = 30°
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iBooks Author
In interactive 1.12, line L divides line segment AB into two equal segments. Line L bisects the line segment. C is the midpoint of segment AB.
Line L which cuts segment AB into two equal length is called the Line Bisector.
Two lines that intersect to form right angles(90°) are said to be perpendicular. In interactive 1.13, Line AB and Line P intersects at point C
which is a right angle. Therefore Line AB and Line P are perpendicular.
I NTERACTIVE 1.13
I NTERACTIVE 1.12
Perpendicular Lines
Line Bisector
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On the other hand, two lines that never intersect are parallel lines. Because the lines never meet, no angle is ever formed between them. In
interactive 1.14 Line P1 and Line P2 are parallel.
A transversal line is a line that intersects a pair of parallel lines. In interactive 1.15 Line 1 and Line 2 are parallel.
I NTERACTIVE 1.14
I NTERACTIVE 1.15
Parallel Line
Transversal Line
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If a Transversal Line cuts a pair of parallel lines that is not perpendicular
to the parallel lines (as in Figure 1.2.), the following are true:
F IGURE 1.2
1. Angles a, c, e, and g are congruent and acute. (green angles)
Transversal Angles
2. Angles b,d, f, and h are congruent and obtuse. (red angles)
3. The sum of an acute and obtuse angle is 180°. ( A pair of green and
red) e.g. angle a + b = 34° + 146° = 180°.
4. Angle c and e are called Alternate Interior Angles and are congruent
(equal). Angle f and d are also Alternate Interior Angles.
5. Angle g and a are called Alternate Exterior Angles and are congruent (equal). Angle h and b are also Alternate Interior Angles.
6. Angle f and b are called Corresponding Angles and are congruent
(equal). Angle e and a are also Corresponding Angles.
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iBooks Author
I NTERACTIVE 1.16
G ALLERY 1.1
Angle Relationship for a Transversal Line EF that
intersect a pair of parallel lines CD and AB
Transversal Angles
Angle c and e are called Alternate Interior Angles and are congruent (equal). Angle f and d are also Alternate Interior Angles
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iBooks Author
I LLUSTRATION 1.5
I LLUSTRATION 1.6
Lines L and M are parallel. What is the
value of b? (see interactive 1.13)
Lines L and M are parallel. What is the value of c
and d? (see interactive 1.13)
Answer:
Answer:
Since Lines L and M are parallel,
Since a = 125° is an obtuse angle, and b is an acute angle, we know a + b = 180°.
c = a, congruent acute angles
c = 38°
Now, c + b + d = 180° because straight line angle is 180°
So, 125° + b = 180°
So, 38° + 90° + d = 180°
b = 180° - 125°
d = 180° - 38° - 90°
b = 55°
d = 52°
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iBooks Author
R EVIEW 1.2
Section Review
Question 1 of 18
Which of the following two lines are congruent(equal in length)?
A. a and b
B. b and c
C. c and d
D. a and c
Check Answer
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iBooks Author
Section 3
Lines and Angles,
Using Algebra
This section uses basic algebra and ratios to solve geometry
problems.
A) Review the following terms and examples in order to understand the illustrations.
1. Straight Angle.
2. Vertical Angles.
3. Sum of all the angles around a point.
B) Basic algebra examples.
Example 1
What is the value of a in the equation below?
a+1=2
Answer:
=> a = 2 -1
=> a = 1
Example 2
What is the value of x in the equation below?
2x - 1 = x + 1
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iBooks Author
Answer:
=> 2x - x = 1 + 1
=> x = 2
I LLUSTRATION 1.7
Example 3
What is the value of a and b?
What is the value of a in the equation below?
2a - b = a
Answer:
=> 2a - a = b
=> a = b
Answer:
2a + 50° = 180° (Angle on a Straight Line)
2a = 180° - 50°
2a = 130°
a = 65°
a + b = 2a (Vertical Angles)
b=a
b = 65°
a = 65°
b = 65°
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iBooks Author
I LLUSTRATION 1.8
What is the value of a, b, and c?
I NTERACTIVE 1.17
Illustration 1.7 Interactive
Answer:
b = c (vertical angles) (call this: line 1)
2a = a + b (vertical angles)
a = b (call this: line 2)
a = b = c (from line 1 and line 2) (call this: line 3)
b + 2a + c + (a+b) = 360° (Sum of all the angles around a point)
a + 2a + a + (a+a) = 360° (from line 3)
6a = 360°
a = 60°
b = 60° (from line 3)
c = 60° (from line 3)
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iBooks Author
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I LLUSTRATION 1.10
I LLUSTRATION 1.9
The clock below is showing 2 o’clock. What is the
angle between the hour and minute hand?
What is the value of x?
Answer:
2x - 30° = x + 10° (vertical angles)
Answer:
2x - x = 10° + 30°
Since the hour hand is at 2, the number of degrees from 12 is:
2/12 * 360 = 60°
x = 40°
(Since each hour is 1/12 * 360 = 30°.
So, 1 o’clock is 30°, 2 o’clock is 60°, 3 o’clock is 90°,
4 o’clock is 120°, and so on.)
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iBooks Author
C) Ratios examples.
Example 1
I NTERACTIVE 1.18
Given this ratio a:b:c = 1:2:3, what is the value of b and c when
a = 2?
Clock angle
Answer:
if a = 2, then b = 4 and c = 6 because
b = 2/1 * 2 = 2 and
c = 3/1 * 2 = 6
Example 2
Given this ratio a:b:c = 6:1:2, what is the value of b and c when
a = 3?
Answer:
if a = 3, then b = 0.5 and c = 1 because
b = 1/6 * 3 = 0.5 and
c = 2/6 * 3 = 1
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iBooks Author
I LLUSTRATION 1.11
I NTERACTIVE 1.19
In the figure below, the ratio a:b:c = 6:1:2 and a + b
+ c = 180°. What is the value of c? ( Hint: 6b + b +
2b = 180°)
Angle relationship given the ratios among the angle a,b,c
on a straight line.
Answer:
Use the ratio in terms of b.
Since a:b:c = 6:1:2, in terms of b means that
6b + b + 2b = 180°
9b = 180°
b = 20°
But c = 2b
c = 2 * 20°
c = 40°
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iBooks Author
R EVIEW 1.3 End Of Section Review
Question 1 of 6
All the lines in the figure are straight lines. What is the value of a? (Hint: b = 3a and 2b+b+6a = 180°)
A. 12°
B. 36°
C. 24°
D. 18°
Check Answer
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iBooks Author
Acute Angle
Acute Angle is less than 90°. Angle a is acute because it is 45° (less than 90°).
Related Glossary Terms
Angles
Index
Find Term
Chapter 1 - Angles
Chapter 1 - Angles
Chapter 1 - Angles
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iBooks Author
Alternate Exterior Angles
Angle g and a are called Alternate Exterior Angles and are congruent(equal). Angle h
and b are also Alternate Interior Angles. Note: Only true when a transversal line, EF,
cuts two parallel lines, CD and AB )
Related Glossary Terms
Alternate Interior Angles, Transversal line
Index
Find Term
Chapter 1 - Angles
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iBooks Author
Alternate Interior Angles
Angle c and e are called Alternate Interior Angles and are congruent. Angle f and d are
also Alternate Interior Angles. Note: Only true when a transversal line, EF, cuts two
parallel lines, CD and AB )
Related Glossary Terms
Transversal line
Index
Find Term
Chapter 1 - Angles
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iBooks Author
Angle Bisector
An line that cuts an angle into two equal parts is called an Angle Bisector. Line L cuts
angle ABC into two equal angle a and b because a = b = 27°.
Related Glossary Terms
Angles
Index
Find Term
Chapter 1 - Angles
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iBooks Author
Angles
Angles are measures in degrees. One can name the angle by three points A,B,C as
shown in the interactive diagram. We can also name the angle. This this case we
name is a.
The notation to use for angle is ∠.
So we can have ∠ABC or ∠a.
Related Glossary Terms
Vertex
Index
Find Term
Chapter 1 - Angles
Chapter 1 - Angles
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iBooks Author
Congruent
- equilateral, equal, exactly the same (size, shape, etc.) Segment AB and CD are congruent because they are the same size. Angle a and b are both 45° and so are congruent.
Related Glossary Terms
Angles, Line Segment
Index
Find Term
Chapter 1 - Angles
Chapter 1 - Angles
Chapter 1 - Angles
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iBooks Author
Corresponding Angles
Angle f and b are called Corresponding Angles and are congruent(equal). Angle e and
a are also Corresponding Angles. Note: Only true when a transversal line, EF, cuts
two parallel lines, CD and AB )
Related Glossary Terms
Alternate Exterior Angles, Alternate Interior Angles, Angles, Transversal line
Index
Find Term
Chapter 1 - Angles
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iBooks Author
End points
Point A and B are end points for line segment AB
Related Glossary Terms
Line Segment, Point
Index
Find Term
Chapter 1 - Lines
Chapter 1 - Lines
Chapter 1 - Lines
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iBooks Author
Intersection point
The point where lines, rays or line segment meet is called the intersection point. Point
E is the intersection point.
Related Glossary Terms
Line , Point
Index
Find Term
Chapter 1 - Foreword
Chapter 1 - Lines
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iBooks Author
Line"
A line extends forever in both direction. A and B are points on a line.
Related Glossary Terms
Line Segment
Index
Find Term
Chapter 1 - Lines
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iBooks Author
Line Bisector
Line L which cuts segment AB into two equal length is called the Line Bisector.
Related Glossary Terms
Line , Point
Index
Find Term
Chapter 1 - Angles
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iBooks Author
Line Segment
A Line Segment is similar to a Line except that it has two distinct end points. The name
of the Line Segment with endpoints A and B is written as “line segment AB”
Related Glossary Terms
Line
Index
Find Term
Chapter 1 - Lines
Chapter 1 - Lines
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iBooks Author
Obtuse Angle
Obtuse Angle greater than 90° less than 180°. Angle a is Obtuse.
Related Glossary Terms
Drag related terms here
Index
Find Term
Chapter 1 - Angles
Chapter 1 - Angles
Chapter 1 - Angles
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iBooks Author
Parallel
Two lines that never intersect are parallel lines. Because the lines never meet, no angle is ever formed between them. Line P1 and Line P2 are parallel.
Related Glossary Terms
Line
Index
Find Term
Chapter 1 - Angles
Chapter 1 - Angles
Chapter 1 - Angles
Chapter 1 - Angles
Chapter 1 - Angles
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iBooks Author
Perpendicular
Two lines that intersect to form right angles(90°) are said to be perpendicular. Line AB
and Line P intersects at point C which is a right angle. Therefore Line AB and Line P
are perpendicular.
Related Glossary Terms
Angles, Point
Index
Find Term
Chapter 1 - Angles
Chapter 1 - Angles
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iBooks Author
Point
A point specifies a location. A, B, C, D are all points.
Related Glossary Terms
Drag related terms here
Index
Find Term
Chapter 1 - Lines
Chapter 1 - Lines
Chapter 1 - Angles
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iBooks Author
Ray
A Ray is a straight line that begins at a certain point and extends forever in one direction. The point where the ray begins is its endpoint. A ray with endpoint A and passing
through point B is called “Ray AB”
Related Glossary Terms
End points, Point
Index
Find Term
Chapter 1 - Lines
Chapter 1 - Lines
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iBooks Author
Reflex Angle
Reflex Angle between 180° and 360°. Angle a is a reflex angle.
Related Glossary Terms
Angles
Index
Find Term
Chapter 1 - Angles
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iBooks Author
Right Angle
Right Angle is 90°. Angle a is a right angle.
Related Glossary Terms
Angles
Index
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Chapter 1 - Angles
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iBooks Author
Straight Angle
Straight Angle = 180°. An angle on a straight line is 180° as shown below for angle a.
Related Glossary Terms
Drag related terms here
Index
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Chapter 1 - Angles
Chapter 1 - Lines and Angles, Using Algebra
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iBooks Author
Transversal line
A transversal line is a line that intersect a pair of parallel lines. Line EF is the transversal line. Lines CD and AB are parallel.
Related Glossary Terms
Parallel
Index
Find Term
Chapter 1 - Angles
Chapter 1 - Angles
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iBooks Author
Vertex
An angle is formed when two lines intersect. The point of intersection is called the Vertex. Point B is a Vertex.
Related Glossary Terms
Drag related terms here
Index
Find Term
Chapter 1 - Angles
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iBooks Author
Vertical angles
When two lines intersect, the two angles of opposite angles are called vertical angles.
Vertical angles are congruent (equal). Angle a and b are vertical angles.
Related Glossary Terms
Angles, Congruent
Index
Find Term
Chapter 1 - Angles
Chapter 1 - Lines and Angles, Using Algebra
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iBooks Author