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Lesson 2-4 Special Pairs of Angles (page 50) Essential Question Can you justify the conclusion of a conditional statement? Complementary Angles: … two angles whose measures have the sum 90 degrees. A 60º Example: mÐA + mÐB = 90º 30º C B \ ÐA & ÐB are complementary angles. Supplementary Angles: … two angles whose measures have the sum 180 degrees. Example: 1 2 mÐ1 + mÐ2 = 180º \ Ð1 & Ð2 are supplementary angles. In the diagram, ∠CXD is a right angle. Example #1. B• A • E X • C • •D Name another right angle. ÐAXD _________________________________ In the diagram, ∠CXD is a right angle. Example #1. B• A • E X • C • •D Name complementary angles. ÐAXE & ÐEXD _________________________________ In the diagram, ∠CXD is a right angle. Example #1. B• A • E X • C • •D Name 2 congruent supplementary angles. ÐAXD & ÐCXD _________________________________ In the diagram, ∠CXD is a right angle. Example #1. B• A • E X • C • •D Name 2 noncongruent supplementary angles. ÐAXE & ÐEXC ; ÐAXE & ÐAXB ; ÐAXB & ÐBXC ; ÐCXE & ÐCXB … ________________________________________________ Example # 2. ∠C and ∠ D are complementary, m∠C = 3x - 5 and m∠D = x + 15. Find the value of x, m∠C, and m∠D. mÐC + mÐD = 90º 20 3x -5 + x +15 = 90 55º 4x +10 = 90 = 90º 4x = 80 35º x = 20 mÐC = 3x -5 mÐD = x +15 mÐC = 3(20)- 5 mÐD = 20 +15 mÐC = 55º mÐD = 35º x = ______ m∠C = _____ m∠D = _____ Example # 3. ∠P and ∠ K are supplementary, m∠P = 4x + 10 and m∠K = x + 20. Find the value of x, m∠P, and m∠K. mÐP + mÐK = 180º 30 4x +10+ x + 20 =180 130º 5x + 30 =180 = 180º 5x =150 50º x = 30 mÐP = 4x +10 mÐK = x + 20 mÐP = 4(30)+10 mÐK = 30+ 20 mÐP =130º mÐK = 50º x = ______ m∠P = _____ m∠K = _____ Example #4. A supplement of an angle is seven times a complement of the angle. Find the measures of the angle, its complement, and its supplement. measure of angle x 75º 90 - x 15º 180 - x 105º = __________ = ________ measure of complement = __________ = ________ measure of supplement = __________ = ________ 180 - x = 7(90 - x) 180 - x = 630 - 7x 6x = 450 x = 75 Example #5. A complement of an angle is 15 less than twice the measure of the angle. Find the measures of the angle, its complement, and its supplement. measure of angle x 35º 90 - x 55º 180 - x 145º = __________ = ________ measure of complement = __________ = ________ measure of supplement = __________ = ________ 90 - x = 2x - 15 105 = 3x 35 = x x = 35 Vertical Angles: … two angles whose sides form two pairs of opposite rays. Example: Ð1 & Ð3 _______________ 4 and 1 2 3 Ð2 & Ð4 _______________ Theorem 2-3 Vert. ∠’s R ≅ Vertical angles are congruent. Given: ∠1 and ∠2 are vertical angles Prove: ∠1 ≅ ∠2 1 3 2 Given: ∠1 and ∠2 are vertical angles Prove: ∠1 ≅ ∠2 Statements 1 3 2 Reasons 1. ∠1 ___________________ and ∠2 are vertical angles ___________________ Given 2. ___________________ ___________________ Angle Add. Post. m∠1 + m∠3 = 180º ___________________ m∠2 + m∠3 = 180º 3. m∠1 ___________________ ___________________ Substitution Prop. + m∠3 = m∠2 + m∠3 4. ______________________ ___________________ Reflexive Prop. m∠3 = m∠3 5. ___________________ ___________________ Subtraction Prop. m∠1 = m∠2 6. ___________________ ___________________ ∠1 ≅ ∠2 Def. of ≅ ∠’s Example #4. Find the value of “x”. 35º (2x-5)º 20 x = _______ 2x - 5 = 35 2x = 40 x = 20 Example #5. Find the value of “x”. xº 35º xº 55 x = _______ x + 35 = 90 x = 55 Example #6. Find the value of “x”. 4xº (x+30)º 10 x = _______ 4x = x + 30 3x = 30 x =10 Example #7. Find the values of “x” and “y”. (2x+30)º 4yº 2x + 30 =120 2x = 90 x = 45 (x+y)º 120º 45 15 y = _______ x = _______ 4y +120 =180 4y = 60 y = 15 Example #8. Find the measure of each angle. C• • A 90º-60º • D = 30º 60º 90º B 60º • E 30º •G •F Assignment Written Exercises on pages 52 to 54 RECOMMENDED: 1 to 17 odd numbers REQUIRED: 19 to 31 odd numbers, 32 to 35 Assignment Worksheet on Lesson 2-4 Prepare for a Quiz on Lesson 2-4: Special Pairs of Angles Can you justify the conclusion of a conditional statement?