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Geometry 2.2 And Now From a New Angle 2.2 Special Angles and Postulates: Day 1 Objectives Calculate the complement and supplement of an angle Classify adjacent angles, linear pairs, and vertical angles Differentiate between postulates and theorems Differentiate between Euclidean and nonEuclidean geometries Problem 1: Supplements and Complements Supplementary Angles Two angles that have a sum of 180𝑜 Use a protractor for #1 and #2 (2 Minutes) Calculate the measure in #3 (15 seconds) Problem 1: Supplements and Complements Complementary Angles Two angles that have a sum of 90𝑜 Use a protractor for #4 and #5 (2 Minutes) Calculate the measure in #6 (15 seconds) Problem 1: Supplements and Complements Collaborate #7 (5 Minutes) Problem 1: Supplements and Complements Problem 1: Supplements and Complements Problem 1: Supplements and Complements Problem 1: Supplements and Complements Collaborate #8 (6 Minutes) Summary Day 1 What are complementary angles? What are supplementary angles? 2.2 Special Angles and Postulates: Day 2 Objectives Calculate the complement and supplement of an angle Classify adjacent angles, linear pairs, and vertical angles Differentiate between postulates and theorems Differentiate between Euclidean and nonEuclidean geometries Summary Day 1 What are complementary angles? What are supplementary angles? Problem 2: Angle Relationships Collaborate Adjacent #1 (5 Minutes) Angles: Share a vertex and a side Problem 2: Angle Relationships Collaborate Linear #2 (5 Minutes) Pair: Two adjacent angles that form a line Problem 2: Angle Relationships Collaborate #3 (5 Minutes) Need Protractors for part d Vertical Angles: Nonadjacent angles formed by intersecting lines Vertical Angles are congruent Problem 2: Angle Relationships We are going to do #5a together Given Hypothesis: After the “If” Prove Statements Statements Conclusion: After the “then” G D E F Problem 2: Angle Relationships Collaborate 4-5 (6 Minutes) Formative Assessment Day 2 Performance Task Unit 2 Use protractor for #1 and #2 Take home to finish Must be turned in by tomorrow You may turn in today if you finish We will complete the student rubric on Monday Formative 16 points Problem 3 Postulates and Theorems Postulate A statement that is accepted without proof Theorem A statement that can be proven Euclidean Geometry 1. A straight line segment can be drawn joining any two points 2. Any straight line segment can be extended indefinitely in a straight line 3. Given any straight line segment, a circle can be drawn that has the segment as its radius and one endpoint as center 4. All right angles are congruent Euclidean Geometry 5. If two lines are drawn that intersect a third line in such a way that the sum of the inner angles of one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. (Parallel Postulate) Euclid’s Elements The 1. 2. 3. 4. 5. five “common notions” Things that equal the same thing also equal one another If equals are added to equals, then the wholes are equal If equals are subtracted from equals, then the remainders are equal Things that coincide with one another equal one another The whole is greater than the part Problem 3 Postulates and Theorems Linear Pair Postulate If two angles form a linear pair, then the angles are supplementary Collaborate #1 (2 Minutes) Problem 3 Postulates and Theorems Segment Addition Postulate If point B is on 𝐴𝐶 and between points A and C, then AB + BC = AC Collaborate #2 (2 Minutes) Problem 3 Postulates and Theorems Angle Addition Postulate If point D lies in the interior of ∠𝐴𝐵𝐶, then 𝑚∠𝐴𝐵𝐷 + 𝑚∠𝐷𝐵𝐶 = 𝑚∠𝐴𝐵𝐶 Collaborate #3 (90 Seconds) Summary Addition Property of Equality 𝐼𝑓 𝑎, 𝑏, 𝑎𝑛𝑑 𝑐 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑎𝑛𝑑 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎 + 𝑐 = 𝑏 + 𝑐. Examples Angle Measures 𝐼𝑓 Segment Measures 𝐼𝑓 𝑚∠1 = 𝑚∠2, 𝑡ℎ𝑒𝑛 𝑚∠1 + 𝑚∠3 = 𝑚∠2 + 𝑚∠3 𝑚𝐴𝐵 = 𝑚𝐶𝐷, 𝑡ℎ𝑒𝑛 𝑚𝐴𝐵 + 𝑚𝐸𝐹 = 𝑚𝐶𝐷 + 𝑚𝐸𝐹 Distances 𝐼𝑓 𝐴𝐵 = 𝐶𝐷, 𝑡ℎ𝑒𝑛 𝐴𝐵 + 𝐸𝐹 = 𝐶𝐷 + 𝐸𝐹 Summary Subtraction Property of Equality 𝐼𝑓 𝑎, 𝑏, 𝑎𝑛𝑑 𝑐 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑎𝑛𝑑 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎 − 𝑐 = 𝑏 − 𝑐 Examples Angle Measures 𝐼𝑓 Segment Measures 𝐼𝑓 𝑚∠1 = 𝑚∠2, 𝑡ℎ𝑒𝑛 𝑚∠1 − 𝑚∠3 = 𝑚∠2 − 𝑚∠3 𝑚𝐴𝐵 = 𝑚𝐶𝐷, 𝑡ℎ𝑒𝑛 𝑚𝐴𝐵 − 𝑚𝐸𝐹 = 𝑚𝐶𝐷 − 𝑚𝐸𝐹 Distances 𝐼𝑓 𝐴𝐵 = 𝐶𝐷, 𝑡ℎ𝑒𝑛 𝐴𝐵 − 𝐸𝐹 = 𝐶𝐷 − 𝐸𝐹 Summary Reflexive Property 𝐼𝑓 𝑎 𝑖𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟, 𝑡ℎ𝑒𝑛 𝑎 = 𝑎 Examples Angle 𝑚∠1 Measures = 𝑚∠1 Segment 𝑚𝐴𝐵 Measures = 𝑚𝐴𝐵 Congruent ∠1 ≅ ∠1 Congruent 𝐴𝐵 Angles ≅ 𝐴𝐵 Segments Summary Substitution Property 𝐼𝑓 𝑎 𝑎𝑛𝑑 𝑏 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑎𝑛𝑑 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎 𝑐𝑎𝑛 𝑏𝑒 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒𝑑 𝑓𝑜𝑟 𝑏 Examples Angle Measures 𝐼𝑓 Segment Measures 𝐼𝑓 𝑚∠1 = 56𝑜 𝑎𝑛𝑑 𝑚∠2 = 56𝑜 , 𝑡ℎ𝑒𝑛 𝑚∠1 = 𝑚∠2 𝑚𝐴𝐵 = 4 𝑚𝑚 𝑎𝑛𝑑 𝑚𝐶𝐷 = 4 𝑚𝑚, 𝑡ℎ𝑒𝑛 𝑚𝐴𝐵 = 𝑚𝐶𝐷 Distances 𝐼𝑓 𝐴𝐵 = 12 𝑓𝑡 𝑎𝑛𝑑 𝐶𝐷 = 12 𝑓𝑡, 𝑡ℎ𝑒𝑛 𝐴𝐵 = 𝐶𝐷 Summary Transitive Property 𝐼𝑓 𝑎, 𝑏, 𝑎𝑛𝑑 𝑐 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠, 𝑎 = 𝑏, 𝑎𝑛𝑑 𝑏 = 𝑐, 𝑡ℎ𝑒𝑛 𝑎 = 𝑐 Examples Angle Measures 𝐼𝑓 Segment Measures 𝐼𝑓 𝐴𝐵 = 𝑚𝐶𝐷 𝑎𝑛𝑑 𝑚𝐶𝐷 = 𝑚𝐸𝐹, 𝑡ℎ𝑒𝑛 𝑚𝐴𝐵 = 𝑚𝐸𝐹 Congruent Angles 𝐼𝑓 𝑚∠1 = 𝑚∠2 𝑎𝑛𝑑 𝑚∠2 = 𝑚∠3, 𝑡ℎ𝑒𝑛 𝑚∠1 = 𝑚∠3 ∠1 ≅ ∠2 𝑎𝑛𝑑 ∠2 ≅ ∠3, 𝑡ℎ𝑒𝑛 ∠1 ≅ ∠3 Congruent Segments 𝐼𝑓 𝐴𝐵 ≅ 𝐶𝐷 𝑎𝑛𝑑 𝐶𝐷 ≅ 𝐸𝐹, 𝑡ℎ𝑒𝑛 𝐴𝐵 ≅ 𝐸𝐹 Summary Parallel Lines and Angles If 2 lines are parallel, ………………… Corresponding Angle Postulate Alternate Alternate Then Corresponding Angles Congruent Interior Angle Theorem Then Alternate Interior Angles Congruent Exterior Angle Theorem Then Alternate Exterior Angles Congruent Same-Side Then Same-Side Interior Angles are Supplementary Same-Side Interior Angle Theorem Exterior Angle Theorem Then Same-Side Exterior Angles are Supplementary Formative Assessment Day 3 Skills Practice 2.2 Vocabulary – All Problem Set Need a protractor (1-16) - SKIP Do all of the ODD problems (17-25) Do all (27-50) End of Chapter Test for Review Quiz We Tomorrow will discuss the review before you leave today 2.2 Special Angles and Postulates: Day 4 Get out Skills Practice 2.2 Vocabulary (17-50) Odd Formative Assessment Quiz 2.2 (10 Points) You may write on the test Please scan when you turn in Assignments 2.2 Please pick-up when you are finished with the quiz You will need a protractor also