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Segment and Angle Addition Postulates and Proofs Unit 3 Lesson 14 Notes Line Segment A line segment consists of two points called endpoints of the segment and all the points between them. A D A piece of spaghetti is a physical model of a line segment. H In order for you to say that a point B is between two points A and C, all three points must lie on the same line, and AB + BC = AC. Given: Plane Figure R Conclusion: D C Using Segment Addition Postulate, answer the following questions: Example 1: If DT = 60, find the value of x. Then find DS and ST. Segment Addition Continued: Example 2: If EG = 100, find the value of x. Then find EF and FG. Example 3: Using the Segment Addition Postulate M is between N and O. Find NO. Example 4: Draw a picture and solve for the missing segment. B is between A and C, AC = 14 and BC = 11.4. Find. AB Example 5: Draw a picture and solve for the missing segment. Find RT if S is between R and T. RS = 2x + 7, ST = 28 and RT = 4x. Midpoint of a segment: The midpoint of a segment is the point that bisects, or divides, the segment into two congruent segments. Example: C is the midpoint of BD and AF Segment bisector: Example 6: BC = 3x + 2 and CD = 5x – 10. Solve for x. What is the length of BD? Segment bisector: Example 7: AC = 5x - 16 and CF = 2x – 4. Solve for x. What is the length of AF? Angle Addition Postulate First, let’s recall some previous information from last week…. We discussed the Segment Addition Postulate, which stated that we could add the lengths of adjacent segments together to get the length of an entire segment. For example: J K L JK + KL = JL If you know that JK = 7 and KL = 4, then you can conclude that JL = 11. The Angle Addition Postulate is very similar, yet applies to angles. It allows us to add the measures of adjacent angles together to find the measure of a bigger angle… Angle Addition Postulate If B lies on the interior of AOC, then mAOB + mBOC = mAOC. B A mAOC = 115 50 O 65 C Example 1: D Example 2: G 114 K 95 19 H Given: mGHK = 95 mGHJ = 114. Find: mKHJ. J 134° A 46° B C This is a special example, because the two adjacent angles together create a straight angle. Predict what mABD + mDBC equals. ABC is a straight angle, therefore mABC = 180. The Angle Addition Postulate tells us: mABD + mDBC = mABC mGHK + mKHJ = mGHJ 95 + mKHJ = 114 mKHJ = 19. Plug in what you know. Solve. mABD + mDBC = 180 So, if mABD = 134, 46 then mDBC = ______ R Given: mRSV = x + 5 mVST = 3x - 9 mRST = 68 V Find x. S T Set up an equation using the Angle Addition Postulate. mRSV + mVST = mRST x + 5 + 3x – 9 = 68 Solve. 4x- 4 = 68 4x = 72 x = 18 Plug in what you know. Extension: Now that you know x = 18, find mRSV and mVST. mRSV = x + 5 mRSV = 18 + 5 = 23 mVST = 3x - 9 mVST = 3(18) – 9 = 45 Check: mRSV + mVST = mRST 23 + 45 = 68 B C mBQC = x – 7 mCQD = 2x – 1 mBQD = 2x + 34 Find x, mBQC, mCQD, mBQD. mBQC = x – 7 mBQC = 42 – 7 = 35 Q D mBQC + mCQD = mBQD x – 7 + 2x – 1 = 2x + 34 3x – 8 = 2x + 34 x – 8 = 34 x = 42 mCQD = 2x – 1 mCQD = 2(42) – 1 = 83 mBQD = 2x + 34 mBQD = 2(42) + 34 = 118 Check: mBQC + mCQD = mBQD 35 + 83 = 118 x = 42 mCQD = 83 mBQC = 35 mBQD = 118 Theorem 2.2 Properties of Angle Congruence Angle congruence is reflexive, symmetric, and transitive. Examples: Reflexive: For any angle A, A ≅ A. Symmetric: If A ≅ B, then B ≅ A Transitive: If A ≅ B and B ≅ C, then A ≅ C. Ex. 1: Transitive Property of Angle Congruence Prove the Transitive Property of Congruence for angles C Given: A ≅ B, B ≅ C Prove: A ≅ C B A Ex. 1: Transitive Property of Angle Congruence Statement: 1. A ≅ B, B ≅ C 2. mA = mB 3. mB = mC 4. mA = mC 5. A ≅ C Reason: 1. Given 2. Def. Cong. Angles 3. Def. Cong. Angles 4. Transitive property 5. Def. Cong. Angles Ex. 2: Using the Transitive Property Given: m3 ≅ 40, 1 ≅ 2, 2 ≅ 3 Prove: m1 = 40 1 4 2 3 Ex. 2: Statement: 1. m3 ≅ 40, 1 ≅ 2, 2 ≅ 3 2. 3. 4. 1 ≅ 3 m1 = m 3 m1 = 40 Reason: 1. Given Trans. Prop of Cong. 3. Def. Cong. Angles 4. Substitution 2. Theorem 2.3 All right angles are congruent. Example 3: Proving Theorem 2.3 Given: 1 and 2 are right angles Prove: 1 ≅ 2 Ex. 3: Statement: 1. 1 and 2 are right angles 2. m1 = 90, m2 = 90 3. m1 = m2 4. 1 ≅ 2 Reason: 1. Given Def. Right angle 3. Transitive property 4. Def. Cong. Angles 2. Properties of Special Pairs of Angles Theorem 2.4: Congruent Supplements. If two angles are supplementary to the same angle (or to congruent angles), then they are congruent. If m1 + m2 = 180 AND m2 + m3 = 180, then 1 ≅ 3. Congruent Complements Theorem Theorem 2.5: If two angles are complementary to the same angle (or congruent angles), then the two angles are congruent. If m4 + m5 = 90 AND m5 + m6 = 90, then 4 ≅ 6. Proving Theorem 2.4 Given: 1 and 2 are supplements, 3 and 4 are supplements, 1 ≅ 4 Prove: 2 ≅ 3 1 2 3 4 Ex. 4: Statement: 1. 2. 3. 4. 5. 6. 7. 1 and 2 are supplements, 3 and 4 are supplements, 1 ≅ 4 m 1 + m 2 = 180; m 3 + m 4 = 180 m 1 + m 2 =m 3 + m 4 m 1 = m 4 m 1 + m 2 =m 3 + m 1 m 2 =m 3 2 ≅ 3 Reason: 1. Given 2. Def. Supplementary angles 3. Transitive property of equality 4. Def. Congruent Angles 5. Substitution property 6. Subtraction property 7. Def. Congruent Angles Postulate 12: Linear Pair Postulate If two angles form a linear pair, then they are supplementary. 1 2 m 1 + m 2 = 180 Example 5: Using Linear Pairs In the diagram, m8 = m5 and m5 = 125. Explain how to show m7 = 55 5 6 7 8 Solution: Using the transitive property of equality m8 = 125. The diagram shows that m 7 + m 8 = 180. Substitute 125 for m 8 to show m 7 = 55. Vertical Angles Theorem Vertical angles are congruent. 2 1 3 4 1 ≅ 3; 2 ≅ 4 Proving Theorem 2.6 Given: 5 and 6 are a linear pair, 6 and 7 are a linear pair Prove: 5 7 5 6 7 Ex. 6: Proving Theorem 2.6 Statement: 1. 5 and 6 are a linear pair, 6 and 7 Reason: 1. Given 2. Linear Pair postulate 3. Congruent Supplements Theorem are a linear pair 2. 3. 5 and 6 are supplementary, 6 and 7 are supplementary 5 ≅ 7