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Essential Geometry Concepts: Proofs, Postulates, and Angle Relationships

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hoshi
15 diapositives · 276 Views · Sep 17, 2014

This guide outlines fundamental geometry concepts, including the definitions and applications of proofs, postulates, and theorems. Key terms like the two-column proof format are explained, emphasizing logical argumentation in geometry. The Segment Addition and Angle Addition Postulates are detailed, presenting how to prove relationships between angles and segments. Examples include using properties of equality to solve for unknowns, demonstrating congruence, and applying the transitive property. This resource is perfect for students seeking clarity in geometry proofs and terminology.

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Essential Geometry Concepts: Proofs, Postulates, and Angle Relationships
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  1. 4.5 Segment and Angle Proofs

  2. Basic geometry symbols you need to know

  3. Vocabulary • Proof – a logical argument that shows a statement is true • Two – column proof – numbered statements in one column, corresponding reason in other • Statement Reasons

  4. Postulate – a rule that is accepted without proof. Write forwards and backwards – reverse it and it‘s still true • Segment Addition Postulate -if B is between A and C, then AB + BC = AC converse….. Reverse it…. - if AB + BC = AC, then B is between A and C Draw it….

  5. Angle Addition Postulate – • - If P is the interior (inside) of <RST then the measure of < RST is equal to the sum of the measures of <RSP and <PST. • Draw it -

  6. Theorem – a statement that can be proven • Theorem 4.1 – Congruence of Segments • Reflexive – • Symmetric – • Transitive – • Theorem 4.2 – Congruence of Angles • Reflexive • Symmetric • Transitive

  7. Write a two-column proof for the situation in Example 4 from Lesson 2.5. 4. m∠ 1+m∠ 2=m∠ DBC m∠ 1=m∠ 3 GIVEN: m∠ EBA=m∠ DBC PROVE: 4. Angle Addition Postulate REASONS m∠ EBA= m∠ DBC STATEMENT 5. 5. Transitive Property of Equality 1. 1. m∠ 1=m∠ 3 Given 2. Angle Addition Postulate 2. m∠ EBA=m∠ 3+m∠ 2 3. Substitution Property of Equality 3. m∠ EBA=m∠ 1+m∠ 2 EXAMPLE 1 Write a two-column proof

  8. ANSWER GIVEN :AC = AB + AB PROVE :AB = BC REASONS STATEMENT 1. 1. AC = AB + AB Given 2. 2. AB + BC = AC Segment Addition Postulate 3. 3. AB + AB = AB + BC Transitive Property of Equality 4. 4. AB = BC Subtraction Property of Equality for Example 1 GUIDED PRACTICE

  9. a. IfRTandTP, then RP. b. IfNKBD, thenBDNK. a. Transitive Property of Angle Congruence b. Symmetric Property of Segment Congruence EXAMPLE 2 Name the property shown Name the property illustrated by the statement. SOLUTION

  10. 2. CD CD ANSWER Reflexive Property of Congruence 3. If Q V, then V Q. ANSWER Symmetric Property of Congruence for Example 2 GUIDED PRACTICE Name the property illustrated by the statement.

  11. Solving for x. • Based on the properties learned, if you know 2 “parts” are congruent, you set them equal to each other and solve. • m<A=2x+15, m<B=4x-3 • 2x+15=4x-3 • 15=2x-3 • 18=2x • 9=x • 2(9)+15=18+15=33 B A

  12. Prove this property of midpoints: If you know that Mis the midpoint of AB,prove that ABis two times AMand AMis one half of AB. a. AB= 2AM PROVE: AM=AB b. 1 2 EXAMPLE 3 Use properties of equality GIVEN: Mis the midpoint of AB.

  13. STATEMENT REASONS 1. 1. Mis the midpoint of AB. Given AMMB 2. 2. Definition of midpoint AM= MB 3. 3. Definition of congruent segments AM + MB = AB 4. 4. Segment Addition Postulate 5. AM + AM = AB 5. Substitution Property of Equality 1 a. 6. 2AM = AB 6. Distributive Property 2 b. 7. AM=AB 7. Division Property of Equality EXAMPLE 3 Use properties of equality

  14. GIVEN: Bis the midpoint of AC. Cis the midpoint of BD. 4. Transitive Property of Congruence PROVE: AB = CD REASONS STATEMENT 1. 1. Bis the midpoint of AC. Given Cis the midpoint of BD. ABBC 2. 2. Definition of midpoint 3. 3. BCCD Definition of midpoint ABCD 4. 5. AB = CD 5. Definition of congruent segments EXAMPLE 4 Solve a multi-step problem