Download GEOMETRY 2.5 Proving Statements about Segments and Angles

GEOMETRY
2.5 Proving Statements
about Segments and Angles
September 29, 2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES
ESSENTIAL QUESTION
How can you prove a mathematical statement?
September 29, 2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES
REVIEW!
Today we are starting proofs.
This means we will be using ALL of the theorems
and postulates you have learned this year.
Let’s review.
September 29, 2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES
REVIEW: ANGLE ADDITION POSTULATE
B
A
D
C
If B is in the interior of ADC,
then mADB + mBDC = mADC
September 29, 2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES
REVIEW: SEGMENT ADDITION POSTULATE
If B is between A and C, then AB + BC = AC.
If AB + BC = AC, then B is between A and C.
A
B
AB
C
BC
AC
September 29, 2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES
EXAMPLE 1
What is the measure of the entire angle?
70°
40°
30°
September 29, 2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES
EXAMPLE 2
M
P
N
If MN = 10, and MP = 24.5, find NP.
Solution
By SAP, MN + NP = MP
so 10 + NP = 24.5
and NP = 14.5
September 29, 2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES
EXAMPLE 3
𝑚∠1 = 𝑚∠3
𝑚∠1 + 𝑚∠2
𝑚∠𝐶𝐵𝐷
𝑚∠𝐸𝐵𝐴 = 𝑚∠𝐶𝐵𝐷
September 29, 2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES
YOUR TURN
Seg. Add. Prop.
Trans. Prop. of Equality
Sub. Prop. of Equality
September 29, 2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES
REVIEW: DEF. OF CONGRUENT SEGMENTS
Two segments are congruent if and only if they
have the same length.
This is a biconditional:
1) If two segments are congruent, then they have
the same length.
2) If two segments have the same length, then they
are congruent.
September 29,
2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT
SEGMENTS AND ANGLES
IN SYMBOLS:
If 𝐴𝐵 ≅ 𝐶𝐷, then AB = CD.
If RS = TV, then 𝑅𝑆 ≅ 𝑇𝑉.
(Don’t forget this…)
September 29,
2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT
SEGMENTS AND ANGLES
WRITING A TWO-COLUMN PROOF
• We use deductive reasoning:
**Definitions, properties, postulates, and theorems**
• One of the formats for a proof is a twocolumn proof.
Statements
1.
2.
.
.
September 29, 2015
Reasons
1.
2.
.
.
GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES
EXAMPLE 4
Write a two-column proof.
D
Given:
Prove:
Statements
1.
Reasons
1. Given
2.
2. Angle Addition Postulate
3.
3. Substitution
4.
4. Angle Addition Postulate
5.
5. Transitive Property
September 29, 2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES
E
EXAMPLE 5
Write a two-column proof.
Statements
Given: T is the midpoint of
Prove:
Reasons
1. T is the midpoint of 𝑆𝑈.
1. Given
2.
2. Def. of Midpoint
3.
3. Def. of Congruent Segments
4.
4. Substitution
5.
5. Subtraction Property
6.
6. Division Property
September 29, 2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES
.
REMEMBER THESE FROM 2.4?
Algebraic
Properties of
Equality
Reflexive
Real Numbers
Segments
Angles
a=a
𝐴𝐵 ≅ 𝐴𝐵
A ≅ A
If 𝐴𝐵 ≅ 𝐶𝐷,
then 𝐶𝐷 ≅ 𝐴𝐵
If A ≅ B,
then B ≅ A
If 𝐴𝐵 ≅ 𝐶𝐷,
and 𝐶𝐷 ≅ 𝐸𝐹,
then 𝐴𝐵 ≅ 𝐸𝐹
If A ≅ B,
and B ≅ C,
then A ≅ C
Symmetric If a = b,
then b = a
Transitive
If a = b,
and b = c,
then a = c
September 29, 2015
Geometric Properties of
Congruence
2.4 ALGEBRAIC REASONING
THEOREM 2.1
Properties of Segment Congruence.
Segment congruence is reflexive, symmetric, and
transitive.
Reflexive:
𝐴𝐵 ≅ 𝐴𝐵
Symmetric: If 𝐴𝐵 ≅ 𝐶𝐷, then 𝐶𝐷 ≅ 𝐴𝐵
Transitive:
If 𝐴𝐵 ≅ 𝐶𝐷, and 𝐶𝐷 ≅ 𝑅𝑆,
then 𝐴𝐵 ≅ 𝑅𝑆
Remember: a THEOREM is a statement that is proven
to be true.
September 29,
2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT
SEGMENTS AND ANGLES
THEOREM 2.2
Properties of Angle Congruence.
Angle congruence is reflexive, symmetric
and transitive.
Reflexive: ABC  ABC
Symmetric:
If A  B, then B  A
Transitive:
If A  B, and B  C,
then A  C
The proofs are similar to those for segment
congruence and will not be given here.
September 29, 2015
GEOMETRY 2.6 PROVING STATEMENTS ABOUT ANGLES
18
EXAMPLE 6
September 29, 2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES
YOUR TURN
September 29, 2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES
PROOF: SYMMETRIC PROPERTY
Given: 𝐴𝐵 ≅ 𝐶𝐷. Prove: 𝐶𝐷 ≅ 𝐴𝐵.
Statements
1. AB  CD
Reasons
2. AB = CD
3. CD = AB
1. Given
 seg.
2. Def.
We just
had this.
3. Symm. Prop.
4. CD  AB
4. Def.  seg.
Latin: quod erat demonstrandum
“That which was to be demonstrated.”
September 29,
2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT
SEGMENTS AND ANGLES
IS ALL THIS NECESSARY?
September 29,
2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT
SEGMENTS AND ANGLES
EXPLANATION
Given: AB  CD. Prove: CD  AB.
Statements
1.
2.
3.
4.
AB  CD
AB = CD
CD = AB
CD  AB
Reasons
1.
2.
3.
4.
Given
Def.  seg.
Symm. Prop.
Def.  seg.
Step 3, although seemingly trivial and unnecessary, is
important: we need it to show that segment
congruence is symmetric just as in algebra.
September 29,
2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT
SEGMENTS AND ANGLES
EXAMPLE 7
Given AB = 20, M is the midpoint of AB.
A
Prove: AM = 10.
M
Statements
Reasons
1. AB = 20
2. M is midpt of AB
3. AM  MB
4. AM = MB
5. AM + MB = AB
6. AM + AM = 20
7. 2AM = 20
8. AM = 10
1. Given
2. Given
3. Def. of midpoint
4. Def. of congruent seg.
5. Seg. Add. Post. (SAP)
6. Substitution (4,5 & 1,5)
7. Simplify
8. Division Property
September 29,
2015
QED
GEOMETRY 2.5 PROVING STATEMENTS ABOUT
SEGMENTS AND ANGLES
B
EXAMPLE 8
Given: 𝐴𝐵 ≅ 𝐶𝐷,
B is the midpoint of 𝐴𝐶.
Prove: 𝐵𝐶 ≅ 𝐶𝐷
Statements
Reasons
1. 𝐴𝐵 ≅ 𝐶𝐷
1. Given
2. B is the midpoint of 𝐴𝐶
2. Given
3. 𝐴𝐵 ≅ 𝐵𝐶
3. Def. of Midpoint
4. 𝐵𝐶 ≅ 𝐴𝐵
4. Sym. Prop. of Seg. ≅
5. 𝐵𝐶 ≅ 𝐶𝐷
5. Trans. Prop. Of Seg. ≅
September 29, 2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES
Food for Thought:
There is no magical way to learn to do proofs.
Doing proofs requires hard thinking, serious
effort, memorization, a lot of writing, and
dedication. There are no shortcuts, there are no
quick easy answers.
To be successful at proof, you must know every
definition, postulate and theorem. Looking them
up in a book is no substitute.
Every year, millions of students across the
country learn proofs. You can do it, too!
September 29,
2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT
SEGMENTS AND ANGLES
EXAMPLE 9: USING ALGEBRA
Solve for x. AC = 110.
3x + 8
A
Statements
6x + 12
C
B
Reasons
1. Given
1. AC = 110
2. AB = 3x + 8, BC = 6x + 12 2. Given
3. AB + BC = AC
3. Seg. Add. Post. (SAP)
4. (3x + 8) + (6x + 12) = 110 4. Substitution (2,3 & 1,3)
5. Simplify
5. 9x + 20 = 110
6. 9x = 90
6. Subtraction Property
7. Division Property
7. x = 10
QED
September 29,
GEOMETRY 2.5 PROVING STATEMENTS ABOUT
2015
SEGMENTS AND ANGLES
ASSIGNMENT
September 29,
2015
GEOMETRY 2.5 PROVING STATEMENTS ABOUT
SEGMENTS AND ANGLES