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## Segments

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**Segments**Module 1 Section 2 Image courtesy of Microsoft**In this lesson you will explore:**• Segments • Congruent Segments • Segment Addition • Midpoint of a Segment • Segment Bisector • Perpendicular Bisector Common Core Standards: G-CO.1, G-CO.12, G-GPE.6**The Segment**m A B Properties of a Segment: • A segment is part of a line. • A segment has two definite endpoints and consists of those two endpoints and all points between those endpoints. Here the segment has endpoints A and B and is a part of line m. • A segment is named using its two endpoints and has a drawn above the letters that name the segment. We would name this segment: AB Note: There are an infinite number of segments on any given line.**Identifying and Naming Segments**B A D C E F G H How many segments can you name? Pause this presentation and make a list, then continue the presentation.**Identifying and Naming Segments**B A D C E F G H How many segments can you name?**Identifying and Naming Segments**B A D C E F G H AB, AD, AE, BC, BF, CD, CG, DH, EF, EH, FG, GH**Identifying and Naming Segments**B A D C E F G H AB, AD, AE, BC, BF, CD, CG, DH, EF, EH, FG, GH**Identifying and Naming Segments**B A D C E F G H AB, AD, AE, BC, BF, CD, CG, DH, EF, EH, FG, GH**Identifying and Naming Segments**B A D C E F G H AB, AD, AE, BC, BF, CD, CG, DH, EF, EH, FG, GH AH, DE, AF, BE, BG, CF, CH, DG, AG, BH, AC, BD, EG, and FH**Congruency**We can call two values equal because we are only talking about a measure (or a number value). We may use “equal” when talking about length or a degree. However, when we want to discuss figures that are the same size and the same shape, we call them congruent. Congruent figures are noted using a (congruent sign is an = sign with a ~ on top).**Congruent Segments**Congruent Segments: • Segments that have the same measure (same length). • They usually have different endpoints, but they may share an endpoint. • Notation: • Segment AB is 3 units long can be written as AB = 3 • Segment CD is 3 units long can be written as CD = 3**Congruent Segments**Congruent Segments: • Segments that have the same measure (same length). • They usually have different endpoints, but they may share an endpoint. C A Given: AB = 3 CD = 3 D B**C**Figure 1: 3 cm A D B 3 cm C A Figure 2: 3 cm B 3 cm Figure 3: A C K 3 cm 3 cm B D**Figure 1:**Figure 2: A B C D Segment AB, segment AC, segment AD, segment BC, segment BD, segment CD.**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**Segment Addition**one small part + other small part = whole segment AC C B A AB BC AB + BC = AC This is the Segment Addition Postulate**Segment Addition**AC C B A AB BC AB + BC = AC Example 1: If AB = 23 and BC = 47, find AC**Segment Addition**AC C B A 23 47 AB + BC = AC Example 1: If AB = 23 and BC = 47, find AC + 47 = AC 70 = AC 23**Segment Addition**60 C B A 4x + 5 15 AB + BC = AC Example 2: If AB = 4x + 5, BC = 15, and AC = 60 find x. + 15 = 60 4x + 5 4x + 20 = 60 4x = 40 x = 10**Segment Addition**58 C B A 28 ? AB + BC = AC Example 3: If AB = 28 and AC = 58 find BC. + BC = 58 + x = 58 28 28 x = 30 Which means BC = 30**Segment Addition**AB + BC = AC 5x - 12 C B A 2x - 8 -x + 24 Example 4: If AB = 2x – 8, BC = - x + 24, and AC = 5x – 12, find AC. 2x - 8 + - x + 24 = 5x - 12 x + 16 = 5x – 12 16 = 4x – 12 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 28 = 4x 7 = x**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 A B C A B C**Midpoint of a Segment**A B C 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**Midpoint of a Segment**Example 2: B is the midpoint of AC. Find x, AB, BC, and AC. A B C AB = BC 4x + 12 = 5x – 3 4x + 15 = 5x 15 = x 4x + 12 5x - 3**Midpoint of a Segment**Example 2: B is the midpoint of AC. Find x, AB, BC, and AC. A B C AB = BC 4x + 12 = 5x – 3 4x + 15 = 5x 15 = x 4x + 12 5x - 3 4x + 12 = AB 4(15) + 12 = AB 60 + 12 = AB 72 = AB 72 = BC AB + BC = AC 72 + 72 = AC 144 = AC**Segment Bisector**Bicycle has TWO wheels that are the same size. Biplane has TWO wings that are the same size. So what do a biplane and a bicycle have in common with a segment bisector?**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: Line m is the segment bisector B is the midpoint of AC m C B A The marks indicate that each part is congruent**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. • Line m is the perpendicular segment bisector • B is the midpoint of AC m C B A The marks indicate that each part is congruent**Segment Addition with Bisectors**m 50 C B A 2x - 4 BC Example 4: If m is a bisector of AC, AB = 2x – 4, AC = 50, find BC. Recall: AB + BC = AC AB + BC = AC 2x – 4 + BC = 50**Segment Addition with Bisectors**m 50 C B A 2x - 4 BC Example 4: If m is a bisector of AC, AB = 2x – 4, AC = 50, find BC. Recall: AB + BC = AC AB + BC = AC 2x – 4 + BC = 50 2x – 4 + 2x – 4 = 50 4x – 8 = 50 4x = 58 x = 14.5**Segment Addition with Bisectors**m 50 C B A 2x - 4 BC Example 4: If m is a bisector of AC, AB = 2x – 4, AC = 50, find BC. 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**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