Download Aim 4 - Notes - mod 1 - Manhasset Public Schools

CC Geometry R
Aim #4: How do we use deductive reasoning to write angle proofs?
Do Now: Use the "Basic Properties Reference Chart" to draw correct conclusions:
Addition Property of Equality
If equal quantities are added to equal quantities, the sums are equal.
1=3-2
6=2+4
conclusion:
conclusion:
Subtraction Property of Equality
If equal quantities are aubtracted from equal quantities, the differences are equal.
A quantity may be substituted for its
equal in an equation or inequality.
conclusion:
If two quantities are equal to the same
quantity, they are equal to each other.
conclusion:
A quantity is equal to itself.
≮A = ≮B + ≮C
≮D = ≮A
3x = 12
12 = y
conclusion:
conclusion:
1. Reflexive Property of Equality
≮A = ≮B + ≮C
≮C = ≮D
3x = 12
x = y
conclusion:
Transitive Property of Equality
≮A = ≮B
≮C = ≮D
12 = 10+2
8 = 9-1
conclusion:
Substitution Property of Equality
≮A = ≮B
≮C = ≮D
≮E = ≮F
conclusion:
m≮A =
Recall the statement of the Exterior Angle Theorem:
"An exterior angle of a triangle equals the sum of the two opposite interior angles."
a) Write the statement of the Exterior Angle Theorem in If-then form:
≮a. m
b) Using the Exterior Angle Theorem, derive
TWO-COLUMN PROOF of the Exterior Angle Theorem
[An exterior angle of a triangle equals the sum of the two opposite interior angles.]
Given
Prove:
Specific facts
or conclusions relating
to given diagram
General statement, such as
a definition, postulate, or
theorem
Reasons
Statements
0
1. m≮x + m≮y + m≮w = 180
1. Sum of the angle measures in a triangle is
180
0.
0
2. m≮w + m≮z = 180
2. Angles that form a linear pair are supplementary.
3. m≮x + m≮y + m≮w = m≮w + m≮z
3.
4. m≮w = m≮w
4. Reflexive Postulate
5. m≮x + m≮y = m≮z
5.Subtraction Property of Equality
Substitution Property of Equality
Recall the statement of the Exterior Angle Theorem:
"An exterior angle of a triangle equals the sum of the two opposite interior angles."
a) Write the statement of the Exterior Angle Theorem in If-then form:
≮a. m
b) Using the Exterior Angle Theorem, derive
TWO-COLUMN PROOF of the Exterior Angle Theorem
[An exterior angle of a triangle equals the sum of the two opposite interior angles.]
Given:
Prove:
Specific facts
or conclusions relating
to given diagram
General statement, such as
a definition, postulate, or
theorem
Reasons
Statements
0
1. m≮x + m≮y + m≮w = 180
1.
0
2. m≮w + m≮z = 180
2.
3. m≮x + m≮y + m≮w = m≮w + m≮z
3.
4. m≮w = m≮w
4.
5. m≮x + m≮y = m≮z
5.
Each step in a proof is justified by a previously known a
fct.
Reaching a conclusion based on knownfacts is deductive reasoning.
Exercises: Complete the following two-column proofs.
1. Prove: Vertical angles are congruent.
Plan: What do you know about ≮w and ≮x? ≮y and ≮x?
Complete the proof.
Reasons
Statements
0
m≮w + m ≮x = 1800
1.
2.
m≮y
+ m≮x = 180
1. Angles that form a linear pair sum to 180 .
0
0
2. Angles that form a linear pair sum to 180 .
3.
m≮w + m≮x = m≮y + m≮x
3. Substitution Property of Equality
4.
m≮w = m≮y
4. Subtraction Property of Equality
2. Given the diagram on the right, prove m w + m x + m z = 180
Plan: What do you know about
Statements
x,
y, and
z?
Reasons
1.
1.
2.
2.
3.
3.
3. Given the diagram on the right, prove that m w = m y + m z
Plan: What do you know about ≮w, ≮x, and ≮z?
Reasons
Statements
1.
1.
2.
2.
3.
3.
4. Given: AB ll CD and BC ll DE
Prove: m ABC = m CDE
Reasons
Statements
1. AB ll CD, BC ll DE
1.
2.
2.
3.
3.
5. Prove: m y + m z = m w + m x
a
Statements
Reasons
1. m≮w + m≮x + m≮a = 180
1.
2. m≮y + m≮z + m≮a = 180
2.
3.
3. Substitution Property of Equality
4.
4.