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Splash Screen. Five-Minute Check (over Lesson 2–7) NGSSS Then/Now Postulate 2.10: Protractor Postulate Postulate 2.11: Angle Addition Postulate Example 1: Use the Angle Addition Postulate Theorems 2.3 and 2.4 Example 2: Real-World Example: Use Supplement or Complement

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 2–7) NGSSS Then/Now Postulate 2.10: Protractor Postulate Postulate 2.11: Angle Addition Postulate Example 1: Use the Angle Addition Postulate Theorems 2.3 and 2.4 Example 2: Real-World Example: Use Supplement or Complement Theorem 2.5: Properties of Angle Congruence Proof: Symmetric Property of Congruence Theorems 2.6 and 2.7 Proof: One Case of the Congruent Supplements Theorem Example 3: Proofs Using Congruent Comp. or Suppl. Theorems Theorem 2.8: Vertical Angles Theorem Example 4: Use Vertical Angles Theorems 2.9–2.13: Right Angle Theorems Lesson Menu

  3. A B C D Justify the statement with a property of equality or a property of congruence. A. Transitive Property B. Symmetric Property C. Reflexive Property D. Segment Addition Postulate 5-Minute Check 1

  4. A B C D Justify the statement with a property of equality or a property of congruence. A. Transitive Property B. Symmetric Property C. Reflexive Property D. Segment Addition Postulate 5-Minute Check 2

  5. A B C D Justify the statement with a property of equality or a property of congruence.If H is between G and I, then GH + HI = GI. A. Transitive Property B. Symmetric Property C. Reflexive Property D. Segment Addition Postulate 5-Minute Check 3

  6. A B C D State a conclusion that can be drawn from the statements given using the property indicated.W is between X and Z; Segment Addition Postulate. A.WX > WZ B.XW + WZ = XZ C.XW + XZ = WZ D.WZ – XZ = XW 5-Minute Check 4

  7. A B C D State a conclusion that can be drawn from the statements given using the property indicated. ___ ___ LMNO A. B. C. D. 5-Minute Check 5

  8. A B C D ___ Given B is the midpoint of AC, which of the following is true? A.AB + BC = AC B.AB + AC = BC C.AB = 2AC D.BC = 2AB 5-Minute Check 5

  9. MA.912.G.8.5 Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, two-column, and indirect proofs. NGSSS

  10. You identified and used special pairs of angles. (Lesson 2–7) • Write proofs involving supplementary and complementary angles. • Write proofs involving congruent and right angles. Then/Now

  11. Concept

  12. Concept

  13. Use the Angle Addition Postulate CONSTRUCTIONUsing a protractor, a construction worker measures that the angle a beam makes with a ceiling is 42°. What is the measure of the angle the beam makes with the wall? The ceiling and the wall make a 90 angle. Let 1 be the angle between the beam and the ceiling. Let 2 be the angle between the beam and the wall. m1 + m2 = 90 Angle Addition Postulate 42 + m2 = 90 m1 = 42 42 – 42 + m2 = 90 – 42 Subtraction Property of Equality m2 = 48 Substitution Example 1

  14. Use the Angle Addition Postulate Answer: The beam makes a 48° angle with the wall. Example 1

  15. A B C D Find m1 if m2 = 58 and mJKL = 162. A. 32 B. 94 C. 104 D. 116 Example 1

  16. Concept

  17. Use Supplement or Complement TIMEAt 4 o’clock, the angle between the hour and minute hands of a clock is 120º. When the second hand 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? UnderstandMake a sketch of the situation. The time is 4 o’clock and the second hand bisects the angle between the hour and minute hands. Example 2

  18. 60 + 60 = 120  Use Supplement or Complement PlanUse the Angle Addition Postulate and the definition of angle bisector. Solve Since the angles are congruent by the definition of angle bisector, each angle is 60°. Answer: Both angles are 60°. CheckUse the Angle Addition Postulate to check your answer. m1 + m2 = 120 Example 2

  19. A B C D QUILTINGThe 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. A. 20 B. 30 C. 40 D. 50 Example 2

  20. Concept

  21. Concept

  22. Concept

  23. Concept

  24. Given: Prove: Proofs Using Congruent Comp. or Suppl. Theorems Example 3

  25. Proof: Statements Reasons 1. m3 + m1 = 1801 and 4 form a linear pair. 1. Given 2. 1 and 4 aresupplementary. 2. Linear pairs are supplementary. 3. 3 and 1 are supplementary. 3. Definition of supplementary angles 4. 3  4 4. s suppl. to same  are . Proofs Using Congruent Comp. or Suppl. Theorems Example 3

  26. In the figure, NYR andRYA form a linear pair, AXY and AXZ form a linear pair, and RYA and AXZ are congruent. Prove that NYR and AXY are congruent. Example 3

  27. Statements Reasons 1. 1. Given linear pairs. 2. 2. If two s form a linear pair, then theyaresuppl.s. 3. Given 3. ? 4. ____________ 4.NYR  AXY Which choice correctly completes the proof? Proof: Example 3

  28. A B C D A. Substitution B. Definition of linear pair C. s supp. to the same  or to  s are . D. Definition of supplementary s Example 3

  29. Concept

  30. Use Vertical Angles If 1 and 2 are vertical angles and m1 = d – 32 and m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. Example 4

  31. Use Vertical Angles m1 = d – 32 m2 = 175 – 2d = 69 – 32 or 37 = 175 – 2(69) or 37 Answer:m1 = 37 and m2 = 37 Example 4

  32. A B C D A. B. C. D. Example 4

  33. Concept

  34. End of the Lesson

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