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Geometric Proofs. Proving Triangles Congruent. LET’S GET STARTED. Before we begin, let’s see how much you already know. In your print materials there is a Entry-Test . Complete it now. CHECK YOURSELF. SECTION 3 1. Congruence 2. Perpendicular 3. Equal 4. Parallel

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Geometric proofs

Geometric Proofs

Proving

Triangles

Congruent


Let s get started
LET’S GET STARTED

  • Before we begin, let’s see how much you already know.

  • In your print materials there is a Entry-Test.

  • Complete it now.


Check yourself
CHECK YOURSELF

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


How did you do
HOW DID YOU DO?

  • Excellent - 12 - 14 correct

  • Great - 10 -12 correct

  • Good - 8 - 10 correct

If you fall into any of these categories…

continue to next page.


What do you need to know in order to complete a proof
What do you need to know in order to complete a proof?

  • Apply Geometric Marking Symbols

  • Identify Geometric Postulates, Definitions, and Theorems.

  • Identify Two-Column Proof Method.


How do you mark a figure
How do you mark a figure?

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


What postulates and theorems are used to prove triangles congruent
What Postulates and Theorems are used to prove Triangles Congruent?

  • 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 Congruent? - 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


What is the two column proof method
What is the Two-Column Proof Method? Congruent?

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


Two – Column Proof Congruent?

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. AEBCED

4. All Vertical angles are congruent.

5. ABE  CED

5. SAS Postulate.


What have we learned so far
What Have We Learned So Far? Congruent?

  • 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.


Assessment time
Assessment Time Congruent?

  • In your print materials there is a Unit 1 Assessment.

  • Stop and Complete it now.


Check yourself1

Section 1 Congruent?

Check Yourself

  • 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.


How did you do1
HOW DID YOU DO? Congruent?

  • 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…


What are the first steps in a proof

Read and understand the problem. Congruent?

Analyze the given information by…

Locate and label the diagram with the given information.

Determine the relationship between the given, prove, and diagram

What are the first steps in a proof?


Read and understand the problem
Read and Understand the Problem Congruent?

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


Analyze the given information
Analyze the Given Information Congruent?

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


Let s review
Let’s Review Congruent?

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.


Assessment time1
Assessment Time Congruent?

  • In your print materials there is a Unit 2 Assessment.

  • Stop and Complete it now.


Check yourself2
CHECK YOURSELF Congruent?

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.


How did you do2
HOW DID YOU DO? 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…


What are next steps in a proof

Draw and Label Columns Congruent?

Enter the Given statement as number 1 in both columns

What are next steps in a proof?


Draw and label columns
Draw and Label Columns Congruent?

Example:

Given:1 &2 are rt. And ST  TP.

Prove: STR  PTR

Statements

Justifications

S

1

3

T

R

2

4

P


Enter the given as 1
Enter the Given as #1 Congruent?

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


Let s review1
Let’s Review Congruent?

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

  • Readand Understand the problem.

  • Analyze the given information.

  • Draw and Label Columns.

  • Enter Given Statement.


Assessment time2
Assessment Time Congruent?

  • In your print materials there is a Unit 3 Assessment.

  • Stop and Complete it now.


Check yourself3
CHECK YOURSELF Congruent?

SECTION 2

1.

SECTION 1

1.

Statements

Justifications

1. AB & 12

1. Given

Statements

Justifications

Statements

Justifications

2.

1. AB bisects DC & ABDC

1. Given


How did you do3
HOW DID YOU DO? Congruent?

  • 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…


What are next steps in a proof1

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

Enter next step into chart.

What are next steps in a proof?


Determine assumptions
Determine Assumptions theorem or postulate that allows the assumption.

  • 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.

  • 12: all right’s are .

  • TRTR: Reflexive Property()

Example:

Given:1 &2 are rt. And ST  TP.

Prove: STR  PTR

S

1

3

T

R

2

4

P


Enter assumptions into chart
Enter Assumptions into Chart theorem or postulate that allows the assumption.

Example:

Given:1 &2 are rt. And ST  TP.

Prove: STR  PTR

Statements

Justifications

1. 1 &2 are rt. & ST  TP.

2.12

3. TRTR

  • Given

  • All Rt. ’s are .

  • Reflexive Prop.()

S

1

3

T

R

2

4

P


Let s review2
Let’s Review theorem or postulate that allows the assumption.

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.


Assessment time3
Assessment Time theorem or postulate that allows the assumption.

  • In your print materials there is a Unit 4 Assessment.

  • Stop and Complete it now.


Check yourself4
CHECK YOURSELF theorem or postulate that allows the assumption.

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

  • 12

  • 2&4 are vertical.

  • 24

  • 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()


How did you do4
HOW DID YOU DO? theorem or postulate that allows the assumption.

  • 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…


What are next steps in a proof2

Ask yourself “ theorem or postulate that allows the assumption.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.

What are next steps in a proof?


Is the last statement the prove
Is The Last Statement the Prove? theorem or postulate that allows the assumption.

Example:

Given:1 &2 are rt. And ST  TP.

Prove: STR  PTR

Statements

Justifications

1. 1 &2 are rt. & ST  TP.

2. 12

3.TRTR

  • 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


Enter assumptions into chart1
Enter Assumptions into Chart theorem or postulate that allows the assumption.

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. 12

3. TRTR

4.STRPTR

S

1

3

T

R

2

4

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

P


Let s review3
Let’s Review theorem or postulate that allows the assumption.

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.


Check yourself5
CHECK YOURSELF theorem or postulate that allows the assumption.

2.

SECTION 1

1.

Statements

Justifications

  • RLDC & LCRD

  • 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

  • MGKRGK

  • Given

  • Reflexive Prop().

  • Defn. of bisect.

  • Defn. of perpendicular.

  • All rt. Angles are .

  • SAS postulate.


Congratulations

CONGRATULATIONS theorem or postulate that allows the assumption.

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


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