1. CHAPTER 2: DEDUCTIVE REASONING Section 2-3: Proving Theorems

2. MIDPOINT Recall Definition of Midpoint: The point that divides a segment into 2 congruent segments. *If M is the midpoint of AB, then AM = MB AMB

3. MIDPOINT THEOREM Theorem 2-1 Midpoint Theorem: If M is the midpoint of AB, then AM = ½ AB and MB = ½ AB.

4. EXAMPLES Method 1 AM = ½ AB 3x + 5 = ½ (46) 3x + 5 = 23 3x = 18 x = 6 Method 2 AM = MB So, AM + MB = AB (3x + 5) + (3x + 5) = 46 6x + 10 = 46 6x = 36 x = 6 M is the midpoint of AB. If AM = 3x + 5 and AB = 46, find x. AMB

5. EXAMPLES Method 1 GR = ½ GH 3x = ½ (4x + 10) 3x = 2x + 5 x = 5 Method 2 GR = RH GR + RH = GH 3x + 3x = 4x + 10 6x = 4x + 10 2x = 10 X = 5 R is the midpoint of GH. GH = 4x + 10 and GR = 3x. Find x. GRH

6. EXAMPLES Method 1 XT = ½ XY 6x – 20 = ½ (2x) 6x – 20 = x 6x = x + 20 5x = 20 x = 4 Method 2 XT = TY XT + TY = XY (6x – 20) + (6x – 20) = 2x 12x – 40 = 2x 12x = 2x + 40 10x = 40 x = 4 T is the midpoint of XY. XT = 6x - 20 and XY = 2x. Find x. XTY

7. ANGLE BISECTOR Recall Definition of Angle Bisector: a ray that divides an angle into 2 ≡ adjacent angles. If BX is the bisector of ABC, then m ABX = m XBC B A X C

8. ANGLE BISECTOR THEOREM Theorem 2-2: Angle Bisector Theorem: If BX is the bisector of ABC, then m ABX = ½ m ABC and m XBC = ½ m ABC

9. EXAMPLES Method 1 m ABX = ½ m ABC 2x + 3 = ½ (38) 2x + 3 = 19 2x = 16 x = 8 Method 2 m ABX = m XBC m ABX + m XBC = m ABC (2x + 3) + (2x + 3) = 38 4x + 6 = 38 4x = 32 x = 8 BX is the bisector of ABC. If m ABC = 38 and m ABX = 2x + 3, find x.

10. EXAMPLE Method 1 m DMN = ½ m DME 5x = ½ (6x + 12) 5x = 3x + 6 2x = 6 x = 3 Method 2 m DMN = m NME m DMN + m NME = m DME 5x + 5x = 6x + 12 10x = 6x + 12 4x = 12 x = 3 MN is the bisector of DME. If m DMN = 5x, m DME = 6x + 12, find x.

11. CLASSWORK/HOMEWORK Classwork: Pg. 45, Classroom Exercises 2-10 Even Homework: Pg. 46-47, Written Exercises 2-16 Even