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Lesson 2-8

Lesson 2-8. Proving Angle Relationships. Transparency 2-8. 5-Minute Check on Lesson 2-7. Justify each statement with a property of equality or a property of congruence. 1. If AB  CD and CD  EF , then AB  EF . 2. RS  RS 3. If H is between G and I , then GH + HI = GI.

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Lesson 2-8

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  1. Lesson 2-8 Proving Angle Relationships

  2. Transparency 2-8 5-Minute Check on Lesson 2-7 Justify each statement with a property of equality or a property of congruence. 1. If ABCD and CDEF, then ABEF. 2. RSRS 3. If H is between G and I, then GH + HI = GI. State a conclusion that can be drawn from the statements given using the property indicated. 4. W is between X and Z; Segment Addition Postulate 5. LMNO and NOPQ; Transitive Property of Congruence 6. Which statement is true, given that K is between J and L? Standardized Test Practice: A JK + KL = JL JL + LK = JK B C LJ + JK = LK JK  KL D

  3. Transparency 2-8 5-Minute Check on Lesson 2-7 Justify each statement with a property of equality or a property of congruence. 1. If ABCD and CDEF, then ABEF. Transitive Property 2. RSRS Reflexive Property 3. If H is between G and I, then GH + HI = GI. Segment Addition Postulate State a conclusion that can be drawn from the statements given using the property indicated. 4. W is between X and Z; Segment Addition Postulate XW + WZ = XZ 5. LMNO and NOPQ; Transitive Property of Congruence LMPQ 6. Which statement is true, given that K is between J and L? Standardized Test Practice: A JK + KL = JL JL + LK = JK B C LJ + JK = LK JK  KL D

  4. Objectives • Write proofs involving supplementary and complementary angles • Write proofs involving congruent and right angles

  5. Vocabulary • No new vocabulary

  6. Postulates Postulate 2.10, Protractor Postulate: Given ray AB and a number r between 0 and 180, there is exactly one ray with endpoint A, extending on either side of ray AB, such that the angle formed measures r°. Postulate 2.11, Angle Addition Postulate: If R is in the interior of PQS, then mPQR + mRQS = mPQS and if mPQR + mRQS = mPQS, then R is in the interior of PQS

  7. Theorems Theorem 2.3, Supplement Theorem: if two angles form a linear pair, then they are supplementary angles. Theorem 2.4, Complement Theorem: if the non-common sides of two adjacent angles form a right angle, then the angles are complementary angles. Theorem 2.5, Angles supplementary to the same angle or to congruent angles are congruent. Theorem 2.6, Angles complementary to the same angle or to congruent angles are congruent. Theorem 2.7, Vertical Angles Theorem: If two angles are vertical angles, then they are congruent.

  8. Theorems Theorem 2.9, Perpendicular lines intersect to form four right angles. Theorem 2.10, All right angles are congruent. Theorem 2.11, Perpendicular lines form congruent adjacent angles. Theorem 2.12, If two angles are congruent and supplementary, then each angle is a right angle. Theorem 2.13, If two congruent angles form a linear pair, then they are right angles.

  9. Angle Proof 23 14 Given:1 and 3 are vertical angles m1 = 3x + 5, m3 = 2x + 8Prove: m1 = 14

  10. TIME At 4 o’clock, the angle between the hour and minute hands of a clock is 120º. If the second hand stops where it bisects the angle between the hour and minute hands, what are the measures of the angles between the minute and second hands and between the second and hour hands? Solution: If the second hand stops where the angle is bisected, then the angle between the minute and second hands is one-half the measure of the angle formed by the hour and minute hands, or ½(120º) = 60º. By the Angle Addition Postulate, the sum of the two angles is 120º, so the angle between the second and hour hands is also 60º. Answer: They are both 60º by the definition of angle bisector and the Angle Addition Postulate.

  11. QUILTING The diagram below shows one square for a particular quilt pattern. If mBAC = mDAE = 20, and BAE is a right angle, find mCAD. Answer: 50

  12. If 1 and 2 form a linear pair, and m2 = 166, find m1. Solution: m1 + m2 = 180 Supplement Theorem m2 = 166 Given m1 + 166 = 180 Substitution m1 + 166 – 166 = 180 – 166 Subtraction Property m1 = 14 Substitution Answer: 14

  13. If 1 and 2 are complementary angles and m1 = 62, find m2. Answer: 28

  14. Proof: Statements Reasons 1. 1. Given 2. 2. Linear pairs are supplementary. 3. 3. Definition of supplementary angles 4. 4. Subtraction Property 5. 5. Substitution 6. 6. Definition of congruent angles Given: 1 and 4form a linear pair m3 + m1 = 180.Prove: 3 4

  15. Proof: Statements Reasons 1. 1. Given linear pairs. 2. 2. If two s form a linear pair, then they are suppl. s. 3. 3. Given 4. 4. Given: NYR and RYA form a linear pair, AXY and AXZ form a linear pair, RYA  AXZ.Prove: RYN  AXY

  16. If 1 and 2are vertical angles and m1 = d - 32 andm2 = 175 – 2d, find m1 and m2. 1 2 Vertical Angles Theorem m1 = m2 Definition of congruent angles d – 32 = 175 – 2d Substitution 3d – 32 = 175 Add 2d to each side. 3d = 207 Add 32 to each side. d = 69 Divide each side by 3. Answer: m1 = 37 and m2 = 37

  17. If A and Z are vertical angles and mA = 3b -23 and mZ = 152 – 4b, find mA and mZ. Answer: mA= 52; mZ= 52

  18. Summary & Homework • Summary: • Properties of equality and congruence can be applied to angle relationships • Homework: • pg 112-3: 16-23, 27-32, 41

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