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Transcript
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. 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. Standardized Test Practice: Which statement is true, given that K is between J and L? A C JK + KL = JL LJ + JK = LK B D JL + LK = JK JK KL 5-Minute Check on Lesson 2-7 Transparency 2-8 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. Standardized Test Practice: Which statement is true, given that K is between J and L? A C JK + KL = JL LJ + JK = LK B D JL + LK = JK JK KL Objectives • Write proofs involving supplementary and complementary angles • Write proofs involving congruent and right angles Vocabulary • No new vocabulary 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 mPQR + mRQS = mPQS and if mPQR + mRQS = mPQS, then R is in the interior of PQS 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. 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. Angle Proof Given: 1 and 3 are vertical angles m1 = 3x + 5, m3 = 2x + 8 Prove: m1 = 14 2 1 Statement Reason 1 and 3 are vertical angles Given m1 m3 Vertical Angles Theorem m1 = m3 Congruence Definition m1 = 3x + 5 Given m3 = 2x + 8 Given 3x + 5 = 2x + 8 Substitution 3x – 2x + 5 = 2x – 2x + 8 Subtraction x+5=8 Substitution x+5–5=8–5 Subtraction x=3 Substitution m1 = 3 (3) + 5 = 14 Substitution (twice) 3 4 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. QUILTING The diagram below shows one square for a particular quilt pattern. If mBAC = mDAE = 20, and BAE is a right angle, find mCAD. Answer: 50 If 1 and 2 form a linear pair, and m2 = 166, find m1. Solution: m1 + m2 = 180 Supplement Theorem m2 = 166 Given m1 + 166 = 180 Substitution m1 + 166 – 166 = 180 – 166 Subtraction Property m1 = 14 Substitution Answer: 14 If 1 and 2 are complementary angles and m1 = 62, find m2. Answer: 28 Given: 1 and 4 form a linear pair m3 + m1 = 180. Prove: 3 4 Proof: Statements Reasons 1. 1. Given 2. 2. Linear pairs are supplementary. 3. Definition of supplementary angles 3. 4. 4. Subtraction Property 5. 5. Substitution 6. 6. Definition of congruent angles Given: NYR and RYA form a linear pair, AXY and AXZ form a linear pair, RYA AXZ. Prove: RYN AXY 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. If 1 and 2 are vertical angles and m1 = d - 32 and m2 = 175 – 2d, find m1 and m2. 1 2 Vertical Angles Theorem m1 = m2 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: m1 = 37 and m2 = 37 If A and Z are vertical angles and mA = 3b -23 and mZ = 152 – 4b, find mA and mZ. Answer: mA = 52; mZ = 52 Summary & Homework • Summary: – Properties of equality and congruence can be applied to angle relationships • Homework: – pg 112-3: 16-23, 27-32, 41