Properties from Algebra

1. Properties from Algebra Section 2-2 p. 37 Section 2-3 p. 43 Section 2-4 p. 50

2. Addition Property • If a = b and c = d, then a + c = b + d • Subtraction Property • If a = b and c = d, then a – c = b – d • Multiplication Property • If a = b then ca = cb • Division Property • If a = b and c ≠ 0, then a/c = b/c Properties of Equality

3. Substitution Property • If a = b, then either a or b may be substituted for the other in any equation or inequality • Reflexive Property • a = a (reflection in the mirror) • Symmetric Property • If a = b, then b = a • Transitive Property • If a = b and b = c, then a = c Properties of Equality (continued)

4. Reflexive Property • Symmetric Property • If • If • Transitive • If • If F __ __ __ __ __ __ __ __ __ __ __ __ Properties of Congruence

5. Midpoint Theorem • If M is the midpoint of AB, then AM = ½ AB and MB = ½ AB • Given: M is the midpoint of AB • Prove: AM = ½ AB; MB = ½ AB • Statements Reasons • M is the midpoint of AB Given • AM = MB Definition of Midpoint • AM + MB = AB Segment Addition Postulate • AM + AM = AB Substitution • MB + MB = AB Substitution • AM = ½ AB Division Property of Equality • MB = ½ AB Division Property of Equality __ __ __ __ __ __ __ __ __ __ Proving Theorems 2-3

6. Angle Bisector Theorem • If BX is the bisector of , then m = ½ m and m = ½ m • Given: BX is the bisector of • Prove: m = ½ m and m = ½ m • Statements Reasons • BX is the bisector of Given • m = m Def of Angle Bisector • m + m = m Angle Add. Postulate • m + m = m Substitution • Or 2*m = m Substitution • m = ½ m Mult.or Div. Prop. of Equality • m = ½ m Substitution → → → Proving Theorems 2-3

7. Complementary Angles • Two angles whose measures add up to 90 • Supplementary Angles • Two angles whose measures add up to 180 • Note that angles do NOT have to be adjacent to be complementary or supplementary! X W 30° 60° Y Z W 180-X° X° X Y Z Special Pairs of Angles 2-4

8. Vertical Angle Theorem • Vertical angles are congruent.

9. Two lines that intersect to form right angles Lines that intersect to form one right angle, form three additional right angles For two intersecting lines, if one angle is a right angle, then the two lines are perpendicular Lines, rays, and segments can be perpendicular Perpendicular Lines

10. 2 1 3 4 If these two lines are perpendicular, what are the measurements of angles 1, 2, 3, and 4?

11. 2 1 3 4 If angle 1 is a right angle, what are the measurements of angles 2, 3, and 4? What can be said about these two lines?

12. Theorem 2-4 If two lines are perpendicular, then they form congruent adjacent angles • Theorem 2-5 If two lines form congruent adjacent angles, then the lines are perpendicular • Do you see something similar about these two theorems? • They are converses of each other • Theorem 2-6 If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary More Theorems

13. Theorem 2-7 If two angles are supplements of congruent angles (or of the same angle) then the two angles are congruent. • Theorem 2-8 If two angles are complements of congruent angles (or of the same angle) then the two angles are congruent More Theorems p.61

14. Parts of a Proof • Statement of the Theorem (conditional statement; typically If-then statement) • Diagram showing given information • List of what is Given • List of what you are trying to Prove • Series of Statements and Reasons (lead from given information to the statement you are proving) • Remember that postulates are accepted without proof, but you have to prove theorems using definitions, postulates, and given information Planning a Proof

15. Gather as much info as you can. • Reread what is given. What does it tell you? • Look at the diagram. What other info can you conclude? • Develop a plan to get from a to b (what you are given to what you are trying to prove). Planning a Proof- Method 1

16. Work backward. Start with the conclusion (what you are trying to prove) • Answer the question: This statement would be true if ________? • Continue back to the Given statement, continuing to ask the same question: This statement would be true if ________? • This becomes the plan for your proof. Planning a Proof- Method 2