Creating Definitions and
Angle Relationships
Homework:
Lesson 1.3/1-24
Assignment from my web page:
View PPT “Conjectures: aka Theorems and Postulates”.
Answer the ‘Time to Ponder’ questions/1-16
Due: Wednesday, October 2
Defining Terms
This statement defines a protractor:
“A protractor is a geometry tool used to
measure angles.”
First, you classify what it is (a geometry tool),
then you say how it differs from other geometry
tools (it is the one you use to measure angles).
Define a square
“A square is a figure with four equal sides.”
Is this a good definition?
Test the definition
a. Can you sketch a counterexample?
counterexample – a figure that satisfies the definition
but is NOT a square
b. If you can produce a counter example, then you need to
write a better definition for a square.
Counterexamples
A Good Definition
“A square is a quadrilateral that has all sides
congruent and all angles measuring 90 degrees.”
(a 4-sided polygon with equal sides and
equal angles)
Steps to Creating a Good Definition
1. Classify your term.
a. What is it? What class/group does it fit
into?
2. Differentiate your term.
a. How does it differ from others in that
class/group?
3. Test your definition by looking for
counterexamples.
Parallel Lines
h
h // g
g
Notation:
Picture markings
for parallel lines
Definitions:
Parallel lines are coplanar lines that do not intersect.
Skew lines are non-coplanar lines that do not intersect
Line a is skew to line b
Perpendicular Lines
l
k
Definition:
Perpendicular lines are intersecting lines that form
90˚ angles.
7 Angle relationships to define
Right angle
Acute angle
Obtuse angle
Complementary angles
Supplementary angles
Linear pair of angles
Vertical angles
Textbook Investigation:
Lesson 1.3
http://acr.kendallhunt.com/KeyPressP
ortalV5.0/Viewer/HomeFrame.htm
Types of Angles
An Acute angle is an
angle that measures
between
0° and 90°
A Right angle is an
angle that has a
measure of 90°
An Obtuse angle is an
angle that measures
between
90° and 180°
Complementary Angles
Complementary angles are two angles whose
sum of their degree measurements equals 90 degrees.
Example:
These two angles are complementary. Their sum is 90˚.
58° + 32° = 90°
Adjacent Complementary Angles.
Those same two angles can be
"pasted" together to form a right
angle!
These 2 angles are adjacent –
they share a vertex and a side.
Supplementary Angles
Supplementary angles are two angles whose sum
of their degree measurements equals 180 degrees.
Example:
These two angles are supplementary.
The sum of their measures is 180˚
139° +41° = 180 °
Linear Pair of Angles
Adjacent Supplementary Angles
A linear pair of angles are two angles
that are adjacent and supplementary
angles.
Linear Pair of angles
A linear pair of angles are two angles that are
adjacent and supplementary angles.
Conjecture: If two angles form a linear pair,
then they are supplementary.
m1  m2  180
Vertical Angles
Vertical angles are opposite angles formed by
intersecting lines.
For any two lines that meet, as in the diagram below,
<AEB and <DEC are called vertical angles.
<BEC and <AED are also vertical angles.
Vertical Pair of Angles
Vertical angles are two angles that are created by
two intersecting lines and are opposite each other.
< 1 & <3 are a vertical pair of angles
< 2 & <4 are a vertical pair of angles
Conjecture: If two angles are vertical angles,
then they are congruent.
Angles Around a Point
Angles around a point will always add up to 360 degrees.
The angles above all add to 360°
53° + 80° + 140° + 87° = 360°
We can find an unknown angle using this sum.
Example: What is angle “C"?
C
To find the measure
of < C, find the sum
of the known angles
and subtract that
from 360° .
Sum of known angles =
110° + 75° + 50° + 63° = 298°
m< C = 360° − 298°
m < C = 62°
Review
State whether the following are acute, right, or obtuse.
1.
3.
2.
5.
4.
?
?
Complementary and Supplementary
Find the missing angle.
6. Two angles are complementary.
One measures 65 degrees.
7. Two angles are supplementary.
One measures 140 degrees.
Complementary and Supplementary
Find the missing angle. You do not need a protractor.
Use the clues in the pictures.
8.
9.
x
55°
x
165°
Vertical Angles & Linear Pair
10. Find the missing angles. You do not need a
protractor. Use the clues in the pictures.
y
58°
x
More drawings
11. Find the measure of all the angles with vertex C
E
D
F
C
G
J
H
Box in the
corner
indicates a
right angle.
Final Drawing
12. Find the measure of all the angles with vertex G.
B
C
G
A
F
D
E