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 1

1. AC AB

2. C B

3. Isosceles Triangle

SECTION 2

1. AB CD

2. AB BE

3. Right Triangle

SECTION 3

1. Congruence

2. Perpendicular

3. Equal

4. Parallel

5. Angle

6. Triangle

7. Line Segment AB

8. Measure of Angle

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?

• Apply Geometric Marking Symbols

• Identify Geometric Postulates, Definitions, and Theorems.

• Identify Two-Column Proof Method.

How do you mark a figure?Angles- using arcs on each

angle.

example:1 2

Segments- using slash marks on each segment.

example: AB AC

A

B C

1 2

Parallel Lines – using an arrow on each line.

example: AD || BC

Perpendicular lines – using a right angle box.

example: AB BC

A

B C

D

What Postulates and Theorems are used

to prove Triangles Congruent? • SSS Postulate - If the sides of 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 included side 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?

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

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

Prove: ABE CED.

A

B E

D

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.

Statements Justifications

1. E is the midpoint of AC and BD

2. AE EC and BE ED

3. AEB and CED are vertical angles.

4. AEBCED

5. ABE CED

1. Given

2. Midpoint Theorem

3. Definition of Vertical Angles

4. All Vertical angles are congruent.

5. SAS Postulate.

Two – Column Proof

What Have We Learned So Far?

• 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

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

• Stop and Complete it now.

Check Yourself

Section 1

A

B

D

C

1 3

24

Section 2

1. Midpoint Theorem.

2. All Vertical Angles are Congruent

3. SSS Postulate

4. SAS Postulate

5. ASA Postulate

6. Angle Bisector Theorem

7. Segment Bisector Theorem

8. Corresponding Angles Theorem

Section 3Statements Justifications

1. M is midpoint of AB

2. AM = MB

3. AM MB

1. Given

2. Defn. of midpoint

3. Midpoint Theorem.

HOW DID YOU DO?

• 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.

• Analyze the given information by…

1. Locate and label the diagram with the given information.

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

Read and Understand the Problem

Example:

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

Prove: STR PTR

S

T

P

R12

34

1. Re-state the given statement.

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

2. What is supposed to be proved?

Triangle STR is congruent to triangle PTR.

Analyze the Given Information

Example:

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

Prove: STR PTR

S

T

P

R12

34

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.

Let’s Review

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

1. Read and Understand the problem.

2. Analyze the given information by marking the diagram and determining the relationship between the statements and the diagram.

Assessment Time

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

• Stop and Complete it now.

CHECK YOURSELF

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.

SECTION 2

1.

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 DO?

• 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

• Enter the Given statement as number 1 in both columns

Draw and Label Columns

Example:

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

Prove: STR PTR

S

T

P

R12

34

Statements Justifications

Enter the Given as #1

Example:

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

Prove: STR PTR

S

T

P

R12

34

Statements Justifications

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

1. Given

Let’s Review

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

1. Read and Understand the problem.

2. Analyze the given information.

3. Draw and Label Columns.

4. Enter Given Statement.

Assessment Time

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

• Stop and Complete it now.

CHECK YOURSELF

SECTION 11.

SECTION 2

1.

2.

Statements Justifications

Statements Justifications

1. AB & 12

1. Given

Statements Justifications

1. AB bisects DC & ABDC

1. Given

HOW DID YOU DO?

• 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 proof?

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

• Enter next step into chart.

Determine Assumptions

Example:

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

Prove: STR PTR

S

T

P

R12

34

•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()

Enter Assumptions into Chart

Example:

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

Prove: STR PTR

S

T

P

R12

34

Statements Justifications

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

2. 12

3. TRTR

1. Given

2. All Rt. ’s are .

3. Reflexive Prop.()

Let’s ReviewThe first six steps to solve a

proof are…….

1. Read and Understand the problem.

2. Analyze the given information.

3. Draw and Label Columns.

4. Enter Given Statement.

5. Determine Assumptions.

6. Enter Assumptions into chart.

Assessment Time

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

• Stop and Complete it now.

CHECK YOURSELFSECTION 11. Angles two and four are vertical angles by definition. They are also congruent because all vertical angles are

congruent.2. Segments MN and NP are congruent by definition of bisector. Segment NO is congruent to itself by reflexive

property of equality.

SECTION 2

1.

2.

Statements Justifications

1. 12

2. 2&4 are vertical.

3. 24

1. Given

2. Defn. of vert. ’s

3. All vert. ’s are .

Statements Justifications

1. MO PO and MO bisects MP

2. MN NP

3. NO No

1. Given

2. Defn. of Bisector

3. Reflexive prop()

HOW DID YOU DO?

• 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?

• 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.

Is The Last Statement the Prove?

Example:

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

Prove: STR PTR

S

T

P

R12

34

Statements Justifications

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

2. 12

3. TRTR

1. Given

2. All Rt. ’s are .

3. Reflexive Prop.()No, What assumption

could be made next?

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

Enter Assumptions into Chart

Example:

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

Prove: STR PTR

S

T

P

R12

34

Statements Justifications

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

2. 12

3. TRTR

4.STRPTR

1. Given

2. All Rt. ’s are .

3. Reflexive Prop.()

4. SAS PostulateNow, the proof is complete since the last statement is the prove YEAH

Let’s Review All of the steps to solve a

proof are…….1. Read and Understand

the problem.2. Analyze the given

information.3. Draw and Label

Columns.4. Enter Given Statement.5. Determine Assumptions.6. Enter Assumptions into

chart.7. “Is the last statement

the prove?” If not return to step 5.

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

This way you may refer to the steps.

Good Luck

CHECK YOURSELFSECTION 11.

Statements

Justifications

1. GK MR & GK bisects MR.

2. GK GK

3. MK KR

4. GKM & GKR are rt.

5. GKM GKR

6. MGKRGK

1. Given

2. Reflexive Prop().

3. Defn. of bisect.

4. Defn. of perpendicular.

5. All rt. Angles are .

6. SAS postulate.

2. Statements Justifications

1. RLDC & LCRD

2. DL DL

3. MGK RGK

1. Given

2. Reflexive prop.()

3. SSS Postulate.

CONGRATULATIONS

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