Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Geometric Proofs
Document related concepts
Transcript
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 SECTION 1 1. AC AB 2. C B 3. Isosceles Triangle SECTION 2 1. AB CD 2. AB BE 3. Right Triangle 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? A Angles- using arcs on each angle. example:1 2 Segments- using slash marks on each segment. example: AB AC 1 B 2 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…. Given: E is the midpoint of segment AC and segment BD Prove: ABE CED. B Statements D A E Two – Column Proof 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. Justifications 1. E is the midpoint of AC and BD 1. Given 2. AE EC and BE ED 2. Midpoint Theorem 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. 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 2 Section 1 A B 1 3 4 2 Section 3 Statements D C Justifications 1. Midpoint Theorem. 2. All Vertical Angles are Congruent 3. SSS Postulate 4. SAS Postulate 5. ASA Postulate 6. Angle Bisector Theorem 1. M is midpoint of AB 1. Given 7. Segment Bisector Theorem 2. AM = MB 2. Defn. of midpoint 8. Corresponding Angles Theorem 3. AM MB 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 1 2 3 4 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? R 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 1 2 3 4 R 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 2 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. 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 1 2 3 4 R Statements Justifications Enter the Given as #1 Example: Given: 1 &2 are rt. And ST TP. Prove: STR PTR S T P 1 2 3 4 R Statements 1. 1 &2 are rt. & ST TP. Justifications 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 2 SECTION 1 1. Statements Justifications 1. Statements Justifications 1. AB & 12 Statements 1. Given Justifications 2. 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 •Remember the previous relationship step. Example: Angle 1 and angle 2 are congruent Given: 1 &2 are rt. because all right angles are congruent. Segment TR is congruent to itself. And ST TP. Prove: STR PTR • These are the assumptions! •Re-write them with symbols and justifications. S T P 1 2 3 4 R •12: all right’s are . •TRTR: Reflexive Property() Enter Assumptions into Chart Example: Given: 1 &2 are rt. And ST TP. Prove: STR PTR P 1. 1 &2 are rt. & ST TP. 2. 12 Justifications 1. Given 2. All Rt. ’s are . 3. Reflexive Prop.() 3. TRTR S T Statements 1 2 3 4 R Let’s Review The 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 YOURSELF SECTION 2 SECTION 1 1. 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. 1. 1. 12 1. Given 2. 2&4 are vertical. 2. Defn. of vert. ’s 3. 24 3. 2. 1. Statements Justifications Statements MO PO and MO bisects MP 2. MN NP 3. NO No All vert. ’s are . Justifications 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 P Justifications 1. 1 &2 are rt. & ST TP. 2. 12 1. Given 2. All Rt. ’s are . 3. Reflexive Prop.() 3. TRTR S T Statements 1 2 3 4 R 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 1 2 3 4 R Statements Justifications 1. 1 &2 are rt. & ST TP. 1. Given 2. 12 2. All Rt. ’s are . 3. TRTR 3. Reflexive Prop.() 4.STRPTR 4. SAS Postulate Now, 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 YOURSELF SECTION 1 1. Statements 1. GK MR & GK bisects MR. 2. Statements Justifications 1. Given 2. Reflexive Prop(). 2. GK GK 3. MK KR 3. Defn. of bisect. 4. GKM & GKR are rt. 4. Defn. of perpendicular. 5. GKM GKR 5. All rt. Angles are . 6. MGKRGK 6. SAS postulate. 1. 2. 3. Justifications RLDC & 1. Given LCRD 2. Reflexive DL DL prop.() MGK RGK 3. SSS Postulate. CONGRATULATIONS You have officially completed this module on proofs!!!!
