1. Geometry
2.2 And Now From a New Angle

2. 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

4. 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)

5. 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)

6. Problem 1: Supplements and Complements
Collaborate #7 (5 Minutes)

7. Problem 1: Supplements and Complements

8. Problem 1: Supplements and Complements

9. Problem 1: Supplements and Complements

10. Problem 1: Supplements and Complements
Collaborate #8 (6 Minutes)

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

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

15. 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?

16. 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

17. 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

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

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

24. 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

25. 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)

26. 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

27. Problem 3 Postulates and Theorems
Linear Pair Postulate
If two angles form a linear pair, then the angles are supplementary
Collaborate #1 (2 Minutes)

28. 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)

29. 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)

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

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