Proving Theorems 2-3. EXAMPLE 1 Write a two-column proof GIVEN: m 1 = m 3 PROVE: m EBA = m DBC...
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Transcript of Proving Theorems 2-3. EXAMPLE 1 Write a two-column proof GIVEN: m 1 = m 3 PROVE: m EBA = m DBC...
![Page 1: Proving Theorems 2-3. EXAMPLE 1 Write a two-column proof GIVEN: m 1 = m 3 PROVE: m EBA = m DBC 1. m 1 = m.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649dd95503460f94acf446/html5/thumbnails/1.jpg)
Proving TheoremsProving TheoremsProving TheoremsProving Theorems
2-32-3
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![Page 3: Proving Theorems 2-3. EXAMPLE 1 Write a two-column proof GIVEN: m 1 = m 3 PROVE: m EBA = m DBC 1. m 1 = m.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649dd95503460f94acf446/html5/thumbnails/3.jpg)
![Page 4: Proving Theorems 2-3. EXAMPLE 1 Write a two-column proof GIVEN: m 1 = m 3 PROVE: m EBA = m DBC 1. m 1 = m.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649dd95503460f94acf446/html5/thumbnails/4.jpg)
![Page 5: Proving Theorems 2-3. EXAMPLE 1 Write a two-column proof GIVEN: m 1 = m 3 PROVE: m EBA = m DBC 1. m 1 = m.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649dd95503460f94acf446/html5/thumbnails/5.jpg)
![Page 6: Proving Theorems 2-3. EXAMPLE 1 Write a two-column proof GIVEN: m 1 = m 3 PROVE: m EBA = m DBC 1. m 1 = m.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649dd95503460f94acf446/html5/thumbnails/6.jpg)
![Page 7: Proving Theorems 2-3. EXAMPLE 1 Write a two-column proof GIVEN: m 1 = m 3 PROVE: m EBA = m DBC 1. m 1 = m.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649dd95503460f94acf446/html5/thumbnails/7.jpg)
![Page 8: Proving Theorems 2-3. EXAMPLE 1 Write a two-column proof GIVEN: m 1 = m 3 PROVE: m EBA = m DBC 1. m 1 = m.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649dd95503460f94acf446/html5/thumbnails/8.jpg)
![Page 9: Proving Theorems 2-3. EXAMPLE 1 Write a two-column proof GIVEN: m 1 = m 3 PROVE: m EBA = m DBC 1. m 1 = m.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649dd95503460f94acf446/html5/thumbnails/9.jpg)
EXAMPLE 1 Write a two-column proof
Write a two-column proof
GIVEN:m∠1 = m∠3
PROVE:m∠EBA = m∠DBC
1.m∠1 = m∠3
2.m∠EBA = m∠3 + m∠2
3.m∠EBA = m∠1 + m∠2
1. Given
2. Angle Addition Postulate
3. Substitution Property of Equality
STATEMENT REASONS
![Page 10: Proving Theorems 2-3. EXAMPLE 1 Write a two-column proof GIVEN: m 1 = m 3 PROVE: m EBA = m DBC 1. m 1 = m.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649dd95503460f94acf446/html5/thumbnails/10.jpg)
EXAMPLE 1 Write a two-column proof
5.m∠EBA = m∠DBC
4.m∠1 + m∠2 = m∠DBC4. Angle Addition Postulate
5. Transitive Property of Equality
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GUIDED PRACTICE for Example 1
GIVEN : AC = AB + AB
PROVE : AB = BC
1. Four steps of a proof are shown. Give the reasons for the last two steps.
1. AC = AB + AB
2. AB + BC = AC
3. AB + AB = AB + BC
4. AB = BC
1. Given
2. Segment Addition Postulate
STATEMENT REASONS
3. ?
4. ?
![Page 12: Proving Theorems 2-3. EXAMPLE 1 Write a two-column proof GIVEN: m 1 = m 3 PROVE: m EBA = m DBC 1. m 1 = m.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649dd95503460f94acf446/html5/thumbnails/12.jpg)
GUIDED PRACTICE for Example 1
GIVEN : AC = AB + AB
PROVE : AB = BC
ANSWER
1. AC = AB + AB
2. AB + BC = AC
3. AB + AB = AB + BC
4. AB = BC
1. Given
2. Segment Addition Postulate
3. Transitive Property of Equality
4. Subtraction Property of Equality
STATEMENT REASONS
![Page 13: Proving Theorems 2-3. EXAMPLE 1 Write a two-column proof GIVEN: m 1 = m 3 PROVE: m EBA = m DBC 1. m 1 = m.](https://reader035.fdocuments.in/reader035/viewer/2022062304/56649dd95503460f94acf446/html5/thumbnails/13.jpg)
EXAMPLE 2 Name the property shown
Name the property illustrated by the statement.
a. If R T and T P, then R P.
b. If NK BD , then BD NK .
SOLUTION
Transitive Property of Angle Congruencea.
b. Symmetric Property of Segment Congruence
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GUIDED PRACTICE for Example 2
2. CD CD
3. If Q V, then V Q.
Reflexive Property of Congruence
ANSWER
Symmetric Property of Congruence
ANSWER
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EXAMPLE 3 Use properties of equality
Prove this property of midpoints: If you know that M is the midpoint of AB ,prove that AB is two times AM and AM is one half of AB.
GIVEN: M is the midpoint of AB .
PROVE: a. AB = 2 AM
b.AM = AB21
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STATEMENT REASONS
EXAMPLE 3 Use properties of equality
1. M is the midpoint of AB.
2. AM MB
3. AM = MB
4. AM + MB = AB
1. Given
2. Definition of midpoint
3. Definition of congruent segments
4. Segment Addition Postulate
5. AM + AM = AB 5. Substitution Property of Equality
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EXAMPLE 3 Use properties of equality
6. 2AM = ABa.
AM = AB217.b.
6. Distributive Property
7. Division Property of Equality
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EXAMPLE 4 Solve a multi-step problem
Walking down a hallway at the mall, you notice the music store is halfway between the food court and the shoe store. The shoe store is halfway between the music store and the bookstore. Prove that the distance between the entrances of the food court and music store is the same as the distance between the entrances of the shoe store and bookstore.
Shopping Mall
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EXAMPLE 4 Solve a multi-step problem
SOLUTION
STEP 1 Draw and label a diagram.
STEP 2 Draw separate diagrams to show mathematical relationships.
STEP 3 State what is given and what is to be proved for the situation.Then write a proof.
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EXAMPLE 4 Solve a multi-step problem
GIVEN: B is the midpoint of AC .C is the midpoint of BD .
PROVE: AB = CD
STATEMENT REASONS
1. B is the midpoint of AC .C is the midpoint of BD .
1. Given
2. Definition of midpoint2. AB BC
3. BC CD 3. Definition of midpoint
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EXAMPLE 4 Solve a multi-step problem
5. AB = CD
4. AB CD 4. Transitive Property of Congruence
5. Definition of congruent segments