Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To...
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Transcript of Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To...
![Page 1: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/1.jpg)
Geometry 9/5/14 - Bellwork
![Page 2: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/2.jpg)
2.6 Prove Statements about Segments and Angles
Objectives:
1. To understand the role of proof in a deductive system
2. To write proofs using geometric theorems
![Page 3: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/3.jpg)
Premises in Geometric Arguments
The following is a list of premises that can be used in geometric proofs:
1. Definitions and undefined terms
2. Properties of algebra, equality, and congruence
3. Postulates of geometry
4. Previously accepted or proven geometric conjectures (theorems)
![Page 4: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/4.jpg)
Amazing
Usually we have to prove a conditional statement. Think of this proof as a maze, where the hypothesis is the starting point and the conclusion is the ending.
p
q
![Page 5: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/5.jpg)
Amazing
Your job in constructing the proof is to link p to q using definitions, properties, postulates, and previously proven theorems.
p
q
![Page 6: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/6.jpg)
Example 1
Construct a two-column proof of:If m1 = m3, then mDBC = mEBA.
![Page 7: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/7.jpg)
Example 1
Given: m1 = m3Prove: mDBC = mEBA
Statements Reasons
![Page 8: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/8.jpg)
Example 1
Given: m1 = m3Prove: mDBC = mEBA
Statements Reasons
1. m1 = m3
![Page 9: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/9.jpg)
Example 1
Given: m1 = m3Prove: mDBC = mEBA
Statements Reasons
1. m1 = m3 1.Given
![Page 10: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/10.jpg)
Example 1
Given: m1 = m3Prove: mDBC = mEBA
Statements Reasons
1. m1 = m3 1.Given
2. m1 + m2 = m3 + m2
![Page 11: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/11.jpg)
Example 1
Given: m1 = m3Prove: mDBC = mEBA
Statements Reasons
1. m1 = m3 1.Given
2. m1 + m2 = m3 + m2 2.Addition Property
![Page 12: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/12.jpg)
Example 1
Given: m1 = m3Prove: mDBC = mEBA
Statements Reasons
1. m1 = m3 1.Given
2. m1 + m2 = m3 + m2 2.Addition Property
3. m1 + m2 = mDBC
![Page 13: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/13.jpg)
Example 1
Given: m1 = m3Prove: mDBC = mEBA
Statements Reasons
1. m1 = m3 1.Given
2. m1 + m2 = m3 + m2 2.Addition Property
3. m1 + m2 = mDBC 3.Angle Addition Postulate
![Page 14: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/14.jpg)
Example 1
Given: m1 = m3Prove: mDBC = mEBA
Statements Reasons
1. m1 = m3 1.Given
2. m1 + m2 = m3 + m2 2.Addition Property
3. m1 + m2 = mDBC 3.Angle Addition Postulate
4. m3 + m2 = mEBA
![Page 15: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/15.jpg)
Example 1
Given: m1 = m3Prove: mDBC = mEBA
Statements Reasons
1. m1 = m3 1.Given
2. m1 + m2 = m3 + m2 2.Addition Property
3. m1 + m2 = mDBC 3.Angle Addition Postulate
4. m3 + m2 = mEBA 4.Angle Addition Postulate
![Page 16: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/16.jpg)
Example 1
Given: m1 = m3Prove: mDBC = mEBA
Statements Reasons
1. m1 = m3 1.Given
2. m1 + m2 = m3 + m2 2.Addition Property
3. m1 + m2 = mDBC 3.Angle Addition Postulate
4. m3 + m2 = mEBA 4.Angle Addition Postulate
5. mDBC = mEBA
![Page 17: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/17.jpg)
Example 1
Given: m1 = m3Prove: mDBC = mEBA
Statements Reasons
1. m1 = m3 1.Given
2. m1 + m2 = m3 + m2 2.Addition Property
3. m1 + m2 = mDBC 3.Angle Addition Postulate
4. m3 + m2 = mEBA 4.Angle Addition Postulate
5. mDBC = mEBA 5.Substitution Property
![Page 18: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/18.jpg)
Two-Column Proof
Notice in a two-column proof, you first list what you are given (hypothesis) and what you are to prove (conclusion).
The proof itself resembles a T-chart with numbered statements on the left and numbered reasons for those statements on the right.
Before you begin your proof, it is wise to try to map out the maze from p to q.
![Page 19: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/19.jpg)
Generic Two-Column Proof
Given: ____________
Prove: ____________
Statements Reasons
1. 1.
2. 2.
3. 3.
Insert illustration here
![Page 20: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/20.jpg)
Theorems of Congruence
Congruence of SegmentsSegment congruence is reflexive, symmetric,
and transitive.
![Page 21: Geometry 9/5/14 - Bellwork. 2.6 Prove Statements about Segments and Angles Objectives: 1.To understand the role of proof in a deductive system 2.To write.](https://reader035.fdocuments.us/reader035/viewer/2022062805/5697bfd51a28abf838cad54b/html5/thumbnails/21.jpg)
Congruence of AnglesAngle congruence is reflexive, symmetric, and
transitive.
Theorems of Congruence
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Assignment
• Textbook PP. 116-119: 3,4, 10-13, 16, 21, 22