1. Segment Addition Segment addition problems can sometimes seem confusing, but as long as you remember that the two smaller parts make up the whole, you will be in good shape! This means: one small part + other small part = whole segment

2. Segment Addition one small part + other small part = whole segment A B C AB BC AC AB + BC = AC This is the Segment Addition Postulate

3. Segment Addition A B C AB BC AC AB + BC = AC Example 1: If AB = 23 and BC = 47, find AC

4. Segment Addition A B C 23 47 AC AB + BC = AC Example 1: If AB = 23 and BC = 47, find AC 23 + 47 = AC 70 = AC

5. Segment Addition A B C 4x + 5 15 60 AB + BC = AC Example 2: If AB = 4x + 5, BC = 15, and AC = 60 find x. 4x + 5 + 15 = 60 4x + 20 = 60 4x = 40 x = 10

6. Segment Addition A B C 28 ? 58 AB + BC = AC Example 3: If AB = 28 and AC = 58 find BC. 28 + BC = 58 + x = 58 x = 30 Which means BC = 30

7. + - x + 24 = 5x - 12 x + 16 = 5x – 12 16 = 4x – 12 Segment Addition A B C 2x - 8 -x + 24 5x - 12 AB + BC = AC Example 4: If AB = 2x – 8, BC = - x + 24, and AC = 5x – 12, find AC. 2x - 8 28 = 4x 7 = x Now that we know x = 7 we can easily find AC by replacing 7 with x in 5x – 12. 5x – 12 = AC 5(7) – 12 = AC 35 – 12 = AC 23 units = AC

8. Midpoint of a Segment Midpoint: We have learned that if A, B, and C are collinear then AB + BC = AC If AB = BC then B is called the midpoint of AC, and we can put in the tic marks to show that AB and BC are congruent ACB ACB

9. Midpoint of a Segment ACB AB + BC = AC Example 1: B is the midpoint of AC. If AC = 115 and AB = 5x – 10, find x. AB + BC = AC 5x – 10 + BC = 115 5x – 10 + 5x – 10 = 115 10x – 20 = 115 10x = 135 x = 13.5

10. Midpoint of a Segment ACB Example 2: B is the midpoint of AC. Find x, AB, BC, and AC. 4x + 12 5x - 3 AB = BC 4x + 12 = 5x – 3 4x + 15 = 5x 15 = x

11. Midpoint of a Segment ACB Example 2: B is the midpoint of AC. Find x, AB, BC, and AC. 4x + 12 5x - 3 AB = BC 4x + 12 = 5x – 3 4x + 15 = 5x 15 = x 4x + 12 = AB 4(15) + 12 = AB 60 + 12 = AB 72 = AB 72 = BC AB + BC = AC 72 + 72 = AC 144 = AC

12. Segment Bisector Biplane has TWO wings that are the same size. Bicycle has TWO wheels that are the same size. So what do a biplane and a bicycle have in common with a segment bisector?

13. Segment Bisector A segment Bisector ensures the segment has been divided into TWO parts that are the same size. The bisector goes through the segment midpoint and we know that: A B C m Line m is the segment bisector B is the midpoint of AC The marks indicate that each part is congruent

14. Perpendicular Segment Bisector A perpendicular segment Bisector is a segment bisector that runs perpendicular to the segment and passes through the midpoint of the segment. A B C m Line m is the perpendicular segment bisector B is the midpoint of AC The marks indicate that each part is congruent

15. Segment Addition with Bisectors A B C 2x - 4 50 Example 4: If m is a bisector of AC, AB = 2x – 4, AC = 50, find BC. BC m Recall: AB + BC = AC AB + BC = AC 2x – 4 + BC = 50

16. Segment Addition with Bisectors A B C 2x - 4 50 Example 4: If m is a bisector of AC, AB = 2x – 4, AC = 50, find BC. BC m Recall: AB + BC = AC AB + BC = AC 2x – 4 + BC = 50 2x – 4 + 2x – 4 = 50 4x – 8 = 50 4x = 58 x = 14.5

17. Segment Addition with Bisectors A B C 2x - 4 50 Example 4: If m is a bisector of AC, AB = 2x – 4, AC = 50, find BC. BC m Recall: AB + BC = AC AB + BC = AC 2x – 4 + BC = 50 2x – 4 + 2x – 4 = 50 4x – 8 = 50 4x = 58 x = 14.5 2x – 4 = BC 2(14.5) – 4 = BC 29 – 4 = BC 25 = BC

18. Example 5: Given, find JK. mJK = mLM -2x + 33 = 6x - 23 33 = 8x – 23 56 = 8x 7 = x JK = -2x + 33 JK = -2(7) + 33 JK = 19