geometry 2.5 proving statements about segments and · pdf filereview: segment addition...

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?


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.


REVIEW: ANGLE ADDITION POSTULATE


A B

CD

If B is in the interior of ADC,

then mADB + mBDC = mADC

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 C

AB BC

AC

EXAMPLE 1

What is the measure of the entire angle?


40°

30°

70°

EXAMPLE 2


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


𝑚∠1 = 𝑚∠3

𝑚∠1 +𝑚∠2

𝑚∠𝐶𝐵𝐷

𝑚∠𝐸𝐵𝐴 = 𝑚∠𝐶𝐵𝐷

YOUR TURN


Seg. Add. Prop.

Trans. Prop. of Equality

Sub. Prop. of Equality

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:

September 29, 2015


If 𝐴𝐵 ≅ 𝐶𝐷, then AB = CD.

If RS = TV, then 𝑅𝑆 ≅ 𝑇𝑉.

(Don’t forget this…)

WRITING A TWO-COLUMN PROOF


• 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 4


Write a two-column proof.

Given:

Prove: Statements

1.

2.

3.

4.

5.

Reasons1. Given

2. Angle Addition Postulate

3. Substitution

4. Angle Addition Postulate

5. Transitive Property

D E

EXAMPLE 5


Write a two-column proof. Given: T is the midpoint of .

Prove: Statements

1. T is the midpoint of 𝑆𝑈.

2.

3.

4.

5.

6.

Reasons

1. Given

2. Def. of Midpoint

3. Def. of Congruent Segments

4. Substitution

5. Subtraction Property

6. Division Property

REMEMBER THESE FROM 2.4?

September 29, 2015 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 29, 2015


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 29, 2015 GEOMETRY 2.6 PROVING STATEMENTS ABOUT ANGLES 18

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


YOUR TURN


PROOF: SYMMETRIC PROPERTY

September 29, 2015


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.

IS ALL THIS NECESSARY?

September 29, 2015


EXPLANATION

September 29, 2015


1. Given

2. AB = CD 2. Def. seg.

3. CD = AB 3. Symm. Prop.

4. Def. seg.

1. AB CD

4. CD AB

Statements Reasons

Given: AB CD. Prove: CD AB.

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

EXAMPLE 7

September 29, 2015


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

Prove: AM = 10.

Statements Reasons

1. AB = 20 1. Given

2. M is midpt of AB 2. Given

3. 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. Simplify8. AM = 10 8. Division Property

QED

A M B

EXAMPLE 8


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

September 29, 2015


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:

EXAMPLE 9: USING ALGEBRA

September 29, 2015


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 = 10 7. Division Property

QED

A B C

3x + 8 6x + 12

ASSIGNMENT

September 29, 2015


geometry 2.5 proving statements about segments and · pdf filereview: segment addition...

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