Creating Definitions and Angle Relationships
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Transcript of Creating Definitions and Angle Relationships
Creating Definitions and Angle Relationships
Assignment from my web page:
View PPT “Conjectures: aka Theorems and Postulates”.
Answer the ‘Time to Ponder’ questions/1-16
Due: Wednesday, October 2
Homework: Lesson 1.3/1-24
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 definitiona. 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 4-sided polygon with equal sides and equal angles)
“A square is a quadrilateral that has all sides congruent and all angles measuring 90 degrees.”
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.
Notation: h // g
Definitions:Parallel lines are coplanar lines that do not intersect.
Skew lines are non-coplanar lines that do not intersect
Parallel Lines
Picture markings for parallel lines
Line a is skew to line b
k
Definition:
Perpendicular lines are intersecting lines that form 90˚ angles.
Perpendicular Linesl
7 Angle relationships to defineRight angle
Acute angle
Obtuse angle
Complementary angles
Supplementary angles
Linear pair of angles
Vertical angles
http://acr.kendallhunt.com/KeyPressPortalV5.0/Viewer/HomeFrame.htm
Textbook Investigation: Lesson 1.3
Types of AnglesAn 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 AnglesComplementary 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°
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.
Adjacent Complementary Angles.
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 AnglesAdjacent Supplementary Angles
A linear pair of angles are two angles that are adjacent and
supplementary angles.
Linear Pair of anglesA linear pair of angles are two angles that are
adjacent and supplementary angles.
m1 m2 180
Conjecture: If two angles form a linear pair,then they are supplementary.
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 AnglesVertical angles are two angles that are created by two intersecting lines and are opposite each
other.
Conjecture: If two angles are vertical angles,then they are congruent.
< 1 & <3 are a vertical pair of angles< 2 & <4 are a vertical pair of angles
Angles Around a PointAngles 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.
m< C = 360° − 298°
m < C = 62°
Example: What is angle “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°
C
ReviewState whether the following are acute, right, or obtuse.
1.
2.
3.
4.
5.
??
Complementary and Supplementary
6. Two angles are complementary. One measures 65 degrees.
7. Two angles are supplementary. One measures 140 degrees.
Find the missing angle.
Complementary and Supplementary
Find the missing angle. You do not need a protractor.Use the clues in the pictures.
8. 9.x55°
165°x
Vertical Angles & Linear Pair
10. Find the missing angles. You do not need a protractor. Use the clues in the pictures.
58°x
y
More drawings
C
J
D
EF
G
H
Box in the corner indicates a right angle.
11. Find the measure of all the angles with vertex C
Final Drawing
12. Find the measure of all the angles with vertex G.
B
A
F E
D
C
G