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Geometric Proofs

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Geometric Proofs

Proving

Triangles

Congruent

- Before we begin, let’s see how much you already know.
- In your print materials there is a Entry-Test.
- Complete it now.

SECTION 3

1. Congruence

2. Perpendicular

3. Equal

4. Parallel

5. Angle

6. Triangle

7. Line Segment AB

8. Measure of Angle

SECTION 1

1. AC AB

2. C B

3. Isosceles Triangle

SECTION 2

1. AB CD

2. AB BE

3. Right Triangle

- Excellent - 12 - 14 correct
- Great - 10 -12 correct
- Good - 8 - 10 correct

If you fall into any of these categories…

continue to next page.

- Apply Geometric Marking Symbols
- Identify Geometric Postulates, Definitions, and Theorems.
- Identify Two-Column Proof Method.

Angles- using arcs on each angle.

example:1 2

A

Segments- using slash marks on each segment.

example: AB AC

1

2

B

C

Parallel Lines – using an arrow on each line.

example: AD || BC

A

D

Perpendicular lines – using a right angle box.

example: AB BC

B

C

- SSS Postulate - If the sidesof one triangle are congruent to the sides of another triangle, then the triangles are congruent.
- SAS Postulate - If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

SSS

SAS

- ASA Postulate - If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
- AAS Theorem - If two angles and a non includedside are congruent to the corresponding two angles and side of another triangle, then the triangles are congruent.

ASA

AAS

- Let’s take the following paragraph proof and transform it into a two-column proof….

Two – Column Proof

Given: E is the midpoint of segment AC and segment BD

Prove: ABE CED.

Statements

Justifications

D

A

1. E is the midpoint of AC and BD

1. Given

2. AE EC and BE ED

2. Midpoint Theorem

E

B

C

. Paragraph Proof

Since E is the midpoint of segment AC, segment AE is congruent to EC by midpoint theorem. Since E is the midpoint of segment BD, segment BE is congruent to segment ED by midpoint theorem. Angle AEB and angle CED are vertical angles by definition. Therefore angle AEB is congruent to angle CED because all vertical angles are congruent. Triangle ABE is congruent to triangle CED by the side-angle-side postulate.

3. AEB and CED are vertical angles.

3. Definition of Vertical Angles

4. AEBCED

4. All Vertical angles are congruent.

5. ABE CED

5. SAS Postulate.

- The symbols used to mark figures.
Arcs, Slashes, Arrows, and Boxes

- The Postulates and Theorems used to prove triangles are congruent.
SSS, SAS, ASA, and AAS

- What a Two-Column proof looks like.
Column 1 is mathematical statements. Column 2 is justifications of those statements.

- In your print materials there is a Unit 1 Assessment.
- Stop and Complete it now.

Section 1

- Section 2
- Midpoint Theorem.
- All Vertical Angles are Congruent
- SSS Postulate
- SAS Postulate
- ASA Postulate
- Angle Bisector Theorem
- Segment Bisector Theorem
- Corresponding Angles Theorem

A

D

1

3

B

4

2

C

Statements

Justifications

Section 3

1. M is midpoint of AB

1. Given

2. AM = MB

2. Defn. of midpoint

3. AM MB

3. Midpoint Theorem.

- Excellent - 12 - 15 correct
- Great - 10 -12 correct
- Good - 8 - 10 correct

If you fall into any of these categories…

continue to next page.

If not, click here…

Read and understand the problem.

Analyze the given information by…

Locate and label the diagram with the given information.

Determine the relationship between the given, prove, and diagram

Example:

Given:1 &2 are rt. And ST TP.

Prove: STR PTR

1. Re-state the given statement.

Angle one and angle two are right angles. Segment ST is congruent to segment TP.

S

2. What is supposed to be proved?

1

3

T

R

Triangle STR is congruent to triangle PTR.

2

4

P

Example:

Given:1 &2 are rt. And ST TP.

Prove: STR PTR

1. Mark the diagram with the given information.

2. Determine the relationship between the given, prove, and diagram.

Angle 1 and angle 2 are congruent because all right angles are congruent. Segment TR is congruent to itself.

S

1

3

T

R

2

4

P

The first two steps to solve a proof are…….

- Readand Understand the problem.
- Analyze the given information by marking the diagram and determining the relationship between the statements and the diagram.

- In your print materials there is a Unit 2 Assessment.
- Stop and Complete it now.

SECTION 2

1.

SECTION 1

1. Segment EF is congruent to segment GH and segment EH is congruent to GF.

2. Triangle EFH is congruent to triangle GHF.

Angles YPH and HPX are right angles and they are congruent. Segment HP is congruent to itself.

2.

Segments AE and ED are congruent. Angles AEB and CED are vertical and congruent.

- Excellent - 4 correct
- Great – 3 correct
- Good – 2 correct

If you fall into any of these categories…

continue to next page.

If not click here…

Draw and Label Columns

Enter the Given statement as number 1 in both columns

Example:

Given:1 &2 are rt. And ST TP.

Prove: STR PTR

Statements

Justifications

S

1

3

T

R

2

4

P

Example:

Given:1 &2 are rt. And ST TP.

Prove: STR PTR

Statements

Justifications

1. 1 &2 are rt. & ST TP.

1. Given

S

1

3

T

R

2

4

P

The first four steps to solve a proof are…….

- Readand Understand the problem.
- Analyze the given information.
- Draw and Label Columns.
- Enter Given Statement.

- In your print materials there is a Unit 3 Assessment.
- Stop and Complete it now.

SECTION 2

1.

SECTION 1

1.

Statements

Justifications

1. AB & 12

1. Given

Statements

Justifications

Statements

Justifications

2.

1. AB bisects DC & ABDC

1. Given

- Excellent - 3 correct
- Great – 2 correct
- Good – 1 correct

If you fall into any of these categories…

continue to next page.

If not click here…

Determine what can be assumed from the diagram and the theorem or postulate that allows the assumption.

Enter next step into chart.

- Remember the previous relationship step.
- Angle 1 and angle 2 are congruent because all right angles are congruent.Segment TR is congruent to itself.
- These are the assumptions!
- Re-write them with symbols and justifications.
- 12: all right’s are .
- TRTR: Reflexive Property()

Example:

Given:1 &2 are rt. And ST TP.

Prove: STR PTR

S

1

3

T

R

2

4

P

Example:

Given:1 &2 are rt. And ST TP.

Prove: STR PTR

Statements

Justifications

1. 1 &2 are rt. & ST TP.

2.12

3. TRTR

- Given
- All Rt. ’s are .
- Reflexive Prop.()

S

1

3

T

R

2

4

P

The first six steps to solve a proof are…….

- Readand Understand the problem.
- Analyze the given information.
- Draw and Label Columns.
- Enter Given Statement.
- Determine Assumptions.
- Enter Assumptions into chart.

- In your print materials there is a Unit 4 Assessment.
- Stop and Complete it now.

SECTION 2

1.

SECTION 1

- Angles two and four are vertical angles by definition. They are also congruent because all vertical angles are congruent.
- Segments MN and NP are congruent by definition of bisector. Segment NO is congruent to itself by reflexive property of equality.

Statements

Justifications

- 12
- 2&4 are vertical.
- 24

- Given
- Defn. of vert. ’s
- All vert. ’s are .

2.

Statements

Justifications

- MO PO and MO bisects MP
- MN NP
- NO No

- Given
- Defn. of Bisector
- Reflexive prop()

- Excellent – 4 correct
- Great – 3 correct
- Good – 2 correct

If you fall into any of these categories…

continue to next page.

If not click here…

Ask yourself “Is the last step listed the prove statement?”

If the answer is yes, then you are finished.

If the answer is no, then Determine the next assumption from the present information and enter it into the chart.

Example:

Given:1 &2 are rt. And ST TP.

Prove: STR PTR

Statements

Justifications

1. 1 &2 are rt. & ST TP.

2. 12

3.TRTR

- Given
- All Rt. ’s are .
- Reflexive Prop.()

S

1

3

No, What assumption could be made next?

T

R

2

4

By looking at the diagram, I see that the triangles are congruent by the side-angle-side postulate.

P

Example:

Given:1 &2 are rt. And ST TP.

Prove: STR PTR

Statements

Justifications

1. Given

2. All Rt. ’s are .

3. Reflexive Prop.()

4. SAS Postulate

1. 1 &2 are rt. & ST TP.

2. 12

3. TRTR

4.STRPTR

S

1

3

T

R

2

4

Now, the proof is complete since the last statement is the prove YEAH

P

Stay here to complete your final assessment in your print materials.

This way you may refer to the steps.

Good Luck

All of the steps to solve a proof are…….

- Readand Understand the problem.
- Analyze the given information.
- Draw and Label Columns.
- Enter Given Statement.
- Determine Assumptions.
- Enter Assumptions into chart.
- “Is the last statement the prove?” If not return to step 5.

2.

SECTION 1

1.

Statements

Justifications

- RLDC & LCRD
- DL DL
- MGK RGK

- Given
- Reflexive prop.()
- SSS Postulate.

Statements

Justifications

- GK MR & GK bisects MR.
- GK GK
- MK KR
- GKM & GKR are rt.
- GKM GKR
- MGKRGK

- Given
- Reflexive Prop().
- Defn. of bisect.
- Defn. of perpendicular.
- All rt. Angles are .
- SAS postulate.

CONGRATULATIONS

You have officially completed this module on proofs!!!!