Proving Triangles Congruent 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 SECTION 1 1. AC  AB 2. C  B 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? 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 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? 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….

Two – Column Proof Statements Justifications D A E B C 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? 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 Section 2 A D 1 3 B 4 2 C Section 3 Midpoint Theorem. All Vertical Angles are Congruent SSS Postulate SAS Postulate ASA Postulate Angle Bisector Theorem Segment Bisector Theorem Corresponding Angles Theorem Section 1 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.

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What are the first steps in a proof? 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

Read and Understand the Problem 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 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 The first two steps to solve a proof are……. Read and Understand the problem. 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 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.

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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 Statements Justifications S 1 3 T R 2 4 P

Enter the Given as #1 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 Review The first four steps to solve a proof are……. Read and Understand the problem. Analyze the given information. Draw and Label Columns. Enter Given Statement.

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

CHECK YOURSELF 1. SECTION 2 SECTION 1 1. Statements Justifications 2. 1. AB & 12 1. Given Statements Justifications Statements Justifications 2. 1. AB bisects DC & ABDC 1. Given

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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 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 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 Review The first six steps to solve a proof are……. Read and Understand the problem. Analyze the given information. Draw and Label Columns. Enter Given Statement. Determine Assumptions. Enter Assumptions into chart.

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

CHECK YOURSELF 1. SECTION 2 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()

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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 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 Chart 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 Review All of the steps to solve a proof are……. Read and 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. Stay here to complete your final assessment in your print materials. This way you may refer to the steps. Good Luck 

CHECK YOURSELF SECTION 1 2. 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.

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