Download 2.2 Angles and Postulates

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