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 non-Euclidean geometries

Problem 1: Supplements and Complements Supplementary Angles Two angles that have a sum of 180 𝑜 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 90 𝑜 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)

Problem 2: Angle Relationships Collaborate #1 (5 Minutes) Adjacent Angles: Share a vertex and a side

Problem 2: Angle Relationships Collaborate #2 (5 Minutes) Linear Pair: Two adjacent angles that form a line

Summary Day 1 What are complementary angles? What are supplementary angles? What does it mean for angles to be adjacent? What does it mean for angles to be a linear pair?

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 non-Euclidean geometries

Problem 2: Angle Relationships Collaborate #3 (5 Minutes) Vertical Angles: Nonadjacent angles formed by intersecting lines Vertical Angles are congruent Need Protractors for part d

Problem 2: Angle Relationships Collaborate 4-5 (8 Minutes)

Problem 3 Postulates and Theorems A statement that is accepted without proof Theorem A statement that can be proven

Euclidean Geometry A straight line segment can be drawn joining any two points Any straight line segment can be extended indefinitely in a straight line Given any straight line segment, a circle can be drawn that has the segment as its radius and one endpoint as center All right angles are congruent

Euclidean Geometry 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 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 (2 Minutes)

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 𝑚∠1=𝑚∠2, 𝑡ℎ𝑒𝑛 𝑚∠1+𝑚∠3=𝑚∠2+𝑚∠3 Segment Measures 𝐼𝑓 𝑚 𝐴𝐵 =𝑚 𝐶𝐷 , 𝑡ℎ𝑒𝑛 𝑚 𝐴𝐵 +𝑚 𝐸𝐹 =𝑚 𝐶𝐷 +𝑚 𝐸𝐹 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 𝑚∠1=𝑚∠2, 𝑡ℎ𝑒𝑛 𝑚∠1−𝑚∠3=𝑚∠2−𝑚∠3 Segment Measures 𝐼𝑓 𝑚 𝐴𝐵 =𝑚 𝐶𝐷 , 𝑡ℎ𝑒𝑛 𝑚 𝐴𝐵 −𝑚 𝐸𝐹 =𝑚 𝐶𝐷 −𝑚 𝐸𝐹 Distances 𝐼𝑓 𝐴𝐵=𝐶𝐷, 𝑡ℎ𝑒𝑛 𝐴𝐵−𝐸𝐹=𝐶𝐷−𝐸𝐹

Summary Reflexive Property 𝐼𝑓 𝑎 𝑖𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟, 𝑡ℎ𝑒𝑛 𝑎=𝑎 Examples Angle Measures 𝑚∠1=𝑚∠1 Segment Measures 𝑚 𝐴𝐵 =𝑚 𝐴𝐵 Congruent Angles ∠1≅∠1 Congruent Segments 𝐴𝐵 ≅ 𝐴𝐵

Summary Substitution Property 𝐼𝑓 𝑎 𝑎𝑛𝑑 𝑏 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑎𝑛𝑑 𝑎=𝑏, 𝑡ℎ𝑒𝑛 𝑎 𝑐𝑎𝑛 𝑏𝑒 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒𝑑 𝑓𝑜𝑟 𝑏 Examples Angle Measures 𝐼𝑓 𝑚∠1= 56 𝑜 𝑎𝑛𝑑 𝑚∠2= 56 𝑜 , 𝑡ℎ𝑒𝑛 𝑚∠1=𝑚∠2 Segment Measures 𝐼𝑓 𝑚 𝐴𝐵 =4 𝑚𝑚 𝑎𝑛𝑑 𝑚 𝐶𝐷 =4 𝑚𝑚, 𝑡ℎ𝑒𝑛 𝑚 𝐴𝐵 =𝑚 𝐶𝐷 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 𝑚∠1=𝑚∠2 𝑎𝑛𝑑 𝑚∠2=𝑚∠3, 𝑡ℎ𝑒𝑛 𝑚∠1=𝑚∠3 Segment Measures 𝐼𝑓 𝐴𝐵 =𝑚 𝐶𝐷 𝑎𝑛𝑑 𝑚 𝐶𝐷 =𝑚 𝐸𝐹 , 𝑡ℎ𝑒𝑛 𝑚 𝐴𝐵 =𝑚 𝐸𝐹 Congruent Angles 𝐼𝑓 ∠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 Then Corresponding Angles Congruent Alternate Interior Angle Theorem Then Alternate Interior Angles Congruent Alternate Exterior Angle Theorem Then Alternate Exterior Angles Congruent Same-Side Interior Angle Theorem Then Same-Side Interior Angles are Supplementary Same-Side Exterior Angle Theorem Then Same-Side Exterior Angles are Supplementary

Formative Assessment Skills Practice 2.2 Vocabulary – All Problem Set Do all of the ODD problems (17-50) Need a protractor (1-16) - SKIP