Geometry Chapter 2: Reasoning and Proof Section 2.6-­‐ Prove Statements about Segments and Angles SWBAT: write proofs using geometric theorems. Common Core: G.CO.9; G.CO.10; G.CO.10; G.CO.11 Vocabulary: Proof-­‐ a proof is a _________________________ argument that shows a statement is true. Two-­‐column proof-­‐ a two-­‐column proof has _____________________________ statements and __________________________________ reasons that show an argument in logical order. Theorem-­‐ a theorem is a statement that can be ________________________________. Example 1: Write a two-­‐column proof Use the diagram to prove m∠1 = m∠4. Given m∠2 = m∠3, m∠AXD = m∠AXC Prove m∠1= m∠4 Statements Reasons 1. m∠AXC = m∠AXD 1. 2. m∠AXD = m∠_______ + m∠_______ 2. Angle Addition Postulate 3. m∠AXC = m∠_______ + m∠_______ 3. Angle Addition Postulate 4. m∠l + m∠2 = m∠3 + m∠4 4. 5. m∠2 = m∠3 5. 6. m∠l + m∠_______ = m∠3 + m∠4 6. Substitution Property of Equality 7. m∠l = m∠4 7. Geometry Chapter 2: Reasoning and Proof THEOREM 2.1 CONGRUENCE of SEGMENTS Segment congruence is reflexive, symmetric, and transitive. Reflexive For any segment AB, _____________________________________. Symmetric If 𝐴𝐵 ≅ 𝐶𝐷, then ________________________________________. Transitive If 𝐴𝐵 ≅ 𝐶𝐷 and 𝐶𝐷 ≅ 𝐸𝐹, then ____________________________________. THEOREM 2.2 CONGRUENCE of ANGLES Angle congruence is reflexive, symmetric, and transitive. Reflexive For any angle A, ________________________________________. Symmetric If ∠A ≅ ∠B, then _______________________________________. Transitive If ∠A ≅ ∠B and ∠B ≅ ∠C, then _______________________________________________. Example 2: Name the property illustrated by the statement If ∠5 ≅ ∠3, then ∠3 ≅ ∠5. ___________________________________________________________________ Checkpoint Complete the following exercises. 1. Three steps of a proof are shown. Give the reasons for the last two steps. Given BC = AB Prove AC = AB + AB Statements Reasons 1. BC = AB 2. AC = AB + BC 1.
2.
Given _____________________________________ 3. AC =AB + AB 3.
_____________________________________ 2. Name the property illustrated by the statement. If ∠H ≅ ∠T and ∠T ≅ ∠B, then ∠H ≅ ∠B. Geometry Chapter 2: Reasoning and Proof Example 3 Use properties of equality If you know that 𝑩𝑫 bisects ∠ ABC, prove that m∠ABC is two times m∠l. Given: 𝐵𝐷 bisects ∠𝐴𝐵𝐶. Prove: 𝑚∠𝐴𝐵𝐶 = 2 ∙ 𝑚∠1 Statements Reasons 1. BD bisects ∠ABC. 1.
_____________________________________________ 2. ______________________________________ 2. Definition of angle bisector 3. ______________________________________ 3. Definition of congruent angles 4. m∠l + m∠2 = m∠ABC 4.
____________________________________________ 5. m∠1 + m∠___ = m∠ABC 5.
Substitution Property of Equality 6. ______________________________________ 6. Simplify Geometry Chapter 2: Reasoning and Proof CONCEPT SUMMARY: WRITING A TWO-­‐COLUMN PROOF Proof of the Symmetric Property of Segment Congruence Copy or draw diagrams and label information to help develop proofs. Given AB ≅ CD Prove CD ≅ AB Statements based on facts that you know or conclusions from deductive reasoning Statement 1. AB ≅ CD
1. ___________________________________ 2. _______________ 2. Definition of congruent segments 3. _______________ 4. CD
≅ AB
The number of statements will very. Reasons 3. Symmetric Property of Equality 4. Definition of congruent segments Remember to give a reason for the last statements Definitions, postulates, or proven theorems that allow you to state the corresponding statement. Geometry Chapter 2: Reasoning and Proof Example 4: Solve a multi-­‐step problem Interstate There are two exits between rest areas on a stretch of interstate. The Rice exit is halfway between rest area A and the Mason exit. The distance between rest area B and the Mason exit is the same as the distance between rest area A and the Rice exit. Prove that the Mason exit is halfway between the Rice exit and rest area B. Solution Step 1 Draw a diagram. Step 2 Draw diagrams showing relationships. Step 3 Write a proof. Given R is the midpoint of 𝐴𝑀, MB = AR. Prove M is the midpoint of 𝑅𝐵. Statements Reasons 1. R is the midpoint of 𝐴𝑀, MB = AR. 1.
2. _______________________________________ 2. Definition of midpoint ____________________________________________________ 3. _______________________________________ 3. Definition of congruent segment 4. MB = RM 4. ______________________________________________________ 5. ______________________________________ 5. Definition of congruent segments 6. M is the midpoint of 𝑅𝐵 6. ______________________________________________________ Homework: Pgs. 116 – 117 #’s 3 – 12, 15 – 18, 21 – 22