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ColLinear & CoPlanar

Points and lines are coplanar if they are together on the same plane.

1. Are B,C and E coplanar?

Yes

2. Are B, C, A, and D coplanar?

No

Quick Review

- Name the plane that is on the bottom of the box.

ex. EFG

- Name the blue plane.

ex. CGH

- Which plane is ADE?

The front

Segment and Angle addition postulates

<AOB = 43 and <BOC = 15

What is the measure of <AOC?

<AOB + <BOC = <AOC

43 + 15 = <AOC

58 = <AOC

Segment and Angle addition postulates

XZ = 17 and YZ = 11

What is the length of segment XY?

XY + YZ = XZ

XY + 11 = 17

XY = 6

Segment and Angle addition postulates

<ABC = 124. Find the measure of x.

Top < + Right < = Total <

<ABD + <DBC = <ABC

(3x+1) + (4x-3) = 124

7x – 2 = 124

7x = 126

x = 18

Midpoints

M is the midpoint of AB. AM = 3x-6 and

MB = 5x-12. Find the measure of x.

3x – 6 = 5x – 12

6 = 2x

3 = x

5x-12

3x-6

Naming an Angle

When two angles share the same vertex you must use three letters to name the angle.

What is the name of the angle on the left?

<BAC or <CAB

What is the name of the angle on the right?

<CAD or <DAC

C

B

D

A

Review – Find the length and midpoint of AB

Use distance and midpoint formulas

Length – 10 units

Midpoint – (0,-1)

Find the area and perimeter of the shape.

Since the diameter wraps around the circle 3.14 or π times the formula for perimeter is πd.

The perimeter = 10 π or 31.4

The Area = πr2 = π(5)2 = 25 π or 78.5

Write the converse for the statement.

If it is August, then it is summertime.

If it is summertime, then it is August.

What is the truth value of this statement?

False

Are the following definitions good definitions?

A dog is an animal with a tail.

No

Wednesday is the midpoint of the week.

Yes

February is a month during the year.

No

Example of a Biconditional

Ex. Wednesday is the midpoint of the week.

Conditional:Ifit is Wednesday, thenit is the midpoint of the week. (True)

Converse:Ifit is the midpoint of the week, thenit is Wednesday. (True)

Biconditional:It is Wednesday if and only if it is the midpoint of the week.

Working backwards – Write the biconditional as two sentences.

It is a right angle if and only if it has 90 degrees.

- Ifit is a right angle, thenit has 90 degrees.
- If it has 90 degrees, thenit is a right angle.

Law of Syllogism

Let’s start with an example:

If it is a dog, then it is a mammal.

If it is a mammal, then it is an animal.

By using the Law of Syllogism, we can combine the two sentences. We get:

If it is a dog, then it is an animal.

Sometimes the sentences are reversedBE CAREFUL!

Example:

If you are in math class, then you are in school.

If you are in Geometry, then you are in math class.

The Law of Syllogism allows us to write the following:

If you are in Geometry, then you are in school.

Law of Detachment

Don’t think too long when doing this law. It is easier than you think.

Example:

If it is Friday, the lunch room doesn’t serve meat.

“Hey, It’s Friday. What are they serving for lunch?”

… The lunch room isn’t serving meat.

Law of Detachment

Another example:

If water is frozen, then its temperature is less than 32 degrees.

“There is ice (frozen water) on the pond”

… It’s temperature must be less then 32 degrees.

Find the measure of the angles

- <CAD =
- <DAE =
- <CAB =
- <BAF =
- <CAE =

Naming Angles

- An angle supplementary to <BAD
- A pair of vertical angles.
- Two angles complementary to <DAE
- An angle adjacent to <CAD

Properties of Equality

In Algebra, we use conditionals all the time.

Properties:

Addition Prop of = If a=b, then a+c=b+c

Subtraction Prop of = If a=b, then a–c=b-c

Mult Prop of = If a=b, then

Division Prop of = If a=b, then a/c =b/c

More Properties

Reflexive Property a=a

Symmetric Property If a=b, then b=a

Transitive Property If a=b & b=c, then a=c

Substitution Property If a=b, then a can replace b in any expression.

Distributive Property

Identifying the Properties

- <ABC = <ABC

Reflexive Property

- If x + 5 = 11, then x = 6

Subtraction Prop of =

- If x + y = 6 and y = x+2, then x + (x+2) = 6

Substitution Property

- If AB = CD and CD = EF, then AB = EF

Transitive Property

Identifying More Properties

- 5(x – 3) = 5x – 15

Distributive Prop

6. If ½(TR) = 9, then TR = 18

Mult Prop of =

7. If AB = 9, then 9 = AB

Symmetric Prop

Reflexive, Symmetric, Transitive

Which combination of Reflexive, Symmetric, and Transitive are the following relationships.

… as happy as…

Reflexive, Symmetric, Transitive

… is faster than…

Transitive only

Write an equation going through (4,1) with a slope of 3. Then graph it.

Write an equation going through (-2,3) with a slope of 1/2. Then graph it.

Label each type of angle as congruent or supplementary, when lines are parallel

Corresponding:

Same Side Interior:

Same Side Exterior:

Alternate Interior:

Alternate Exterior:

How do you know if two lines parallel, perpendicular or neither based on their slopes?

Parallel lines have …

Perpendicular lines have …

Are the lines parallel, perpendicular, or neither?

- y = -2x y = -2x + 4
- y = -3/5x + 1 y = -3 + 5/3x
- 2x – 3y = 1 3x – 2y = 8

Which Triangle Congruence postulate would we use to show the triangles are congruent?

Which Triangle Congruence postulate would we use to show the triangles are congruent?

Do the following sides make a triangle?

- 4, 5, 7
- 3, 7, 11
- 5, 10, 15

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