# A, m∠A = m∠A.

```Bell Work 10/3/11
• 1) Solve for the following and •
provide a reason for each step in
the process
•
4(x – 5) = x + 4
• 2) Use your notes to give the
property that justifies each
statement
• A) If m∠1 = m∠2,
then m∠2 = m∠1
• B) m∠2 = m∠2
• C) If EF = GH and GH = IJ,
then EF = IJ
• D) If EF = 8 and EF = GH,
then 8 = GH
3) Determine if the argument
uses the Law of Detachment,
the Law of Syllogism, or neither
If I go to the movie, then I’ll eat
popcorn.
• If I eat popcorn, then I’ll enjoy
the movie
• 4) Determine if the argument
uses the Law of Detachment,
the Law of Syllogism, or neither
• If I miss my bus, then I’ll be late
for school.
• I miss the bus.
Outcomes
• I will be able to:
• 1) Use properties of measurement to justify
segment length
• 2) Use properties of measurement to justify angle
relationships
• 3) Justify statements about congruent segments
• 4) Write the steps of a proof
Properties
of Length
In the diagram below AC = BD. Use Segment Addition
to prove AB = CD
• AC = BD
• Given
• BC = BC
• Reflexive Property
• AC - BC = BD - BC
• Subtraction Property
• AB + BC = AC; BC + CD = BD
• AB + BC – BC = BC +CD – BC
• Substitution Property
• AB = CD
• Simplify
Properties of Angle Measure
• Reflexive Property
• For any angle A,
m∠A = m∠A.
• Symmetric Property
• If m∠A = m∠B, then
m∠B = m∠A.
• Transitive Property
• If m∠A = m∠B and
m∠B = m∠C,
then m∠A = m∠C.
Example
m∠ABC = m∠DBE
m∠ABC + m∠CBD = m∠DBE + m∠CBD
m∠CBE = m∠DBE + m∠CBD
m∠ABD = m∠CBE
Substitution
Properties of Angle Measures
Real-Life Example
• AUTO RACING: The Talladega Superspeedway racetrack
in Alabama has four banked turns, which are described
in the diagram below. Use the given information about
the maximum banking angle of the four turns to find
m&lt;4.
Given information:
•
How can we find angle 4?
Proofs
• Theorem – A true statement that follows as a result of
other true statements.
• Two-Column Proof – Two columns of statements and
reasons that show the logical order of an argument.
• Paragraph Proof – Writing statements and reasons in
complete sentences, showing the logical order of an
argument.
Properties of Segment Congruence
• Reflexive Property
For any segment AB, AB  AB
• Symmetric Property
If AB  CD, then CD  AB
• Transitive Property
If AB  CD and CD  EF ,
then AB  EF
Prove the Symmetric Property of
Segment Congruence
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Given: PQ  XY
Prove: XY  PQ
Statements
1. PQ  XY
2. PQ = XY
3. XY = PQ
4. XY  PQ
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Reasons
1. Given
2.Definition of Congruence
3. Symmetric Prop of Equality
4. Definition of Congruence
Proofs:
In the figure below, prove EG  FH if we are given
EF = GH
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Statements
1. EF = GH
2. EF + FG = GH + FG
3. EG = EF + FG,
FH = GH + FG
• 4. EG = FH
• 5. EG  FH
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Reasons
1. Given
Postulate
• 4. Substitution Prop.
• 5. Definition of
Congruent Segments
Proofs
Given : RT  WY and ST  WX
Statements:
1.
RT  WY
2. RT = WY
3. RT = RS + ST;
WY = WX + XY
4. RS + ST = WX + XY
5. ST = WX
6. RS = XY
7.
RS  XY
Prove: RS  XY
Reasons:
1. Given
2. Def of Congruence
Postulate
4. Substitution
5. Given
6. Subtraction Prop from 4
7. Def of congruent Segments
Extra Practice
K
J
L
LK = 5
JK = 5
Definition of Congruence
Extra Practice
Given
Definition of midpoint