GEOMETRY 2.5 Proving Statements

about Segments and Angles

September 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

ESSENTIAL QUESTION

How can I prove a geometric statement?

September 19, 2016 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 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

REVIEW: ANGLE ADDITION POSTULATE

September 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

A B

CD

If B is in the interior of ADC, then

mADB + mBDC = mADC

REVIEW: SEGMENT ADDITION POSTULATE

September 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

If B is between A and C, then AB + BC = AC.

If AB + BC = AC, then B is between A and C.

A B C

AB BC

AC

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 19, 2016

GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

IN SYMBOLS:

September 19, 2016

GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

If 𝐴𝐵 ≅ 𝐶𝐷, then AB = CD.

If RS = TV, then 𝑅𝑆 ≅ 𝑇𝑉.

(Don’t forget this…)

WRITING A TWO-COLUMN PROOF

September 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

• We use deductive reasoning:

• One of the formats for a proof is a two-column proof.

Definitions, properties, postulates, and theorems

Statements Reasons1.2...

1.2...

EXAMPLE 1

What is the measure of the entire angle?

September 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

40°

30°

70°

EXAMPLE 2

September 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

M N P

If MN = 10, and MP = 24.5, find NP.

Solution

By SAP, MN + NP = MP

so 10 + NP = 24.5

and NP = 14.5

EXAMPLE 3

September 19, 2016

GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

𝑚∠1 = 𝑚∠3

𝑚∠1 +𝑚∠2

𝑚∠𝐶𝐵𝐷

𝑚∠𝐸𝐵𝐴 = 𝑚∠𝐶𝐵𝐷

YOUR TURN

September 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

Seg. Add. Prop.

Trans. Prop. of Equality

Subtr. Prop. of Equality

EXAMPLE 4

September 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

Write a two-column proof.

Given:

Prove: Statements1.

2.

3.

4.

5.

Reasons1. Given

2. Angle Addition Postulate

3. Substitution

4. Angle Addition Postulate

5. Transitive Property

D E

REMEMBER THESE FROM 2.4?

September 19, 2016 2.4 ALGEBRAIC REASONING

Algebraic

Properties of

Equality

Geometric Properties of

Congruence

Real Numbers Segments Angles

Reflexive a = a 𝐴𝐵 ≅ 𝐴𝐵 A ≅ A

Symmetric If a = b, then b = a

If 𝐴𝐵 ≅ 𝐶𝐷, then 𝐶𝐷 ≅ 𝐴𝐵

If A ≅ B, then B ≅ A

Transitive If a = b, and b = c, then a = c

If 𝐴𝐵 ≅ 𝐶𝐷, and 𝐶𝐷 ≅ 𝐸𝐹,then 𝐴𝐵 ≅ 𝐸𝐹

If A ≅ B, and B ≅ C, then A ≅ C

THEOREM 2.1

September 19, 2016

GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

Remember: a THEOREM is a statement that is proven to be true.

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

THEOREM 2.2

September 19, 2016 GEOMETRY 2.6 PROVING STATEMENTS ABOUT ANGLES 17

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.

Properties of Angle Congruence.

EXAMPLE 6

September 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

September 19, 2016

GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

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!

Food for Thought:

PROOF: SYMMETRIC PROPERTY

September 19, 2016

GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

1. Given

2. AB = CD 2. Def. seg.

3. CD = AB 3. Symm. Prop.

4. Def. seg.

1. AB CD

4. CD AB

Latin: quod erat demonstrandum

“That which was to be demonstrated.”

Statements Reasons

Given: 𝐴𝐵 ≅ 𝐶𝐷. Prove: 𝐶𝐷 ≅ 𝐴𝐵.

2. We just had this.

Step 3, although seemingly trivial and unnecessary, is important: we need it to show that segment congruence is symmetric just as in algebra.

IS ALL THIS NECESSARY?

September 19, 2016

GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

EXAMPLE 7

September 19, 2016

GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

Given AB = 20, M is the midpoint of AB.

Prove: AM = 10.A M B

Statements Reasons1. AB = 20 1. Given2. M is midpt of AB 2. Given3. AM MB 3. Def. of midpoint

5. AM + MB = AB 5. Seg. Add. Post. (SAP)

4. AM = MB 4. Def. of congruent seg.

6. AM + AM = 20 6. Substitution (4,5 & 1,5)

7. 2AM = 20 7. Simplify

8. AM = 10 8. Division Property

QED

EXAMPLE 8

September 19, 2016 GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

Given: 𝐴𝐵 ≅ 𝐶𝐷,

B is the midpoint of 𝐴𝐶.

Prove: 𝐵𝐶 ≅ 𝐶𝐷

Statements Reasons

3. 𝐴𝐵 ≅ 𝐵𝐶

2. B is the midpoint of 𝐴𝐶

1. 𝐴𝐵 ≅ 𝐶𝐷

4. 𝐵𝐶 ≅ 𝐴𝐵

5. 𝐵𝐶 ≅ 𝐶𝐷 5. Trans. Prop. Of Seg. ≅

2. Given

3. Def. of Midpoint

4. Sym. Prop. of Seg. ≅

1. Given

EXAMPLE 9: USING ALGEBRA

September 19, 2016

GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES

Solve for x. AC = 110.

Statements Reasons

1. AC = 110 1. Given

2. AB = 3x + 8, BC = 6x + 12 2. Given3. AB + BC = AC 3. Seg. Add. Post. (SAP)

5. 9x + 20 = 110 5. Simplify

4. (3x + 8) + (6x + 12) = 110 4. Substitution (2,3 & 1,3)

6. 9x = 90 6. Subtraction Property

7. x = 90 7. Division Property

QED

A B C

3x + 8 6x + 12

ASSIGNMENT

September 19, 2016

GEOMETRY 2.5 PROVING STATEMENTS ABOUT SEGMENTS AND ANGLES