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