Download Lesson 9A: Proofs of Unknown Angles Basic Properties Reference

Lesson 9A
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Name:___________________________________
Period:________ Date:__________
Lesson 9A: Proofs of Unknown Angles
Learning Target: I can prove theorems about lines and angles.
“One of the main goals in studying geometry is to develop your ability to reason
critically, to draw valid conclusions based upon observations and proven facts.”
Warmup
Find the measure of
. Explain the reasoning behind your answer.
Discussion: Why do we need proofs?
Watch: http://www.youtube.com/watch?v=o30UY_flFgM&feature=youtu.be
Basic Properties Reference Chart
Property
Reflexive Property
Meaning
A quantity is equal to itself.
Geometry Example
If x = y, then y = x
Example: Suppose fish = tuna, then tuna = fish
If two quantities are equal to the same
quantity, then they are equal to each other.
If
and
, then
Algebra: If x = y and y = z, then x = z
Transitive Property
Example: Suppose John's height = Mary's height and
Mary's height = Peter's height, then John's height =
Peter's height
Symmetric Property
If a quantity is equal to a second quantity,
then the second quantity is equal to the
first.
Geometry: If
then
Algebra : If x = x
Example: 2 = 2 or I am equal to myself
.
Lesson 9A
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Name:___________________________________
Period:________ Date:__________
Addition
Property of
Equality
If equal quantities are added to equal
quantities, then the sums are equal.
Subtraction
Property of
Equality
If equal quantities are subtracted from equal If
quantities, the differences are equal.
Multiplication
Property of
Equality
If equal quantities are multiplied by equal
quantities, then the products are equal.
If
then
Division
Property of
Equality
If equal quantities are divided by equal
quantities, then the quotients are equal.
If
Substitution
Property of
Equality
A quantity may be substituted for its equal.
If
and
,
then
.
Algebra : is x+ y = z and y = a them
x+a=z
Partition Property
(includes “Angle Addition
Postulate,” “Segments
add,” “Between of
Points,” etc.)
A whole is equal to the sum of its parts.
If point
If
then
and
.
, then
then
is on ̅̅̅̅, then
To prove it, we use letters instead of numbers to talk about the general case.
Why is it true?
and
.
.
Now, we will prove the rule about the exterior angle of a triangle.
What is true?
,
.
.
Lesson 9A
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Name:___________________________________
Period:________ Date:__________
Lesson
Example 1. Similarly, we can also prove that vertical angles are congruent.
Make a plan:


What do you know about
and
?
and
?
What conclusion can you draw based on both pieces of knowledge?
What is true?
Why is it true?
Example 2. Given the diagram on the right, prove that
(Make a plan first. What do you know about
Statements
Reasons
1)
1)
2)
2)
3)
3)
4)
4)
.
, and
?)
Lesson 9A
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Name:___________________________________
Period:________ Date:__________
Example 3. Given the diagram on the right, prove that
.
Statements
Reasons
1)
1) The measure of the exterior angle of a triangle is equal
to the sum of measures of the remote interior angles.
2)
2) Vertical angles have equal measures.
3)
3) Substitution property of equality
Example 4. In the diagram on the right, prove that
.
(You will need to write in a label in the diagram that is not yet labeled for this proof.)
Statements
Reasons
1.
2.
3. Substitution property of equality
3.
4.
You Try
Example 5. In the figure on the right,
Prove that
.
Statements

and

Reasons
.
Lesson 9A
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
GEOMETRY
Name:___________________________________
Period:________ Date:__________
Lesson 9A: Proofs of Unknown Angles
Exit Ticket/Homework
Prove that the sum of the labeled angles is 180o.
x
y
z
Statements
Reasons