1. Angle Relationships

2. POD
If A + B + C = D, and X + C = D, then write an equation that defines X, or shows what just X equals.

3. Some angle basics review
What is an angle?
An angle is formed when
Two rays intersect at a
Common point.
What does it mean for
A diagram to be “Not drawn to scale”?
All of the diagrams we will look at this unit will “not be drawn to scale”
Ray
Ray
Vertex: the point that the two rays which form the angle, share.

4. Naming Angles: Angles are either named by three letters, where the middle letter has to be the vertex, or, sometimes using only one letter.
Why are angles are never named with two letters? B B A
<BDA or
<ADB
<BDC or
<CDB A D C

5. Basic Angle Relationships
Adjacent angles: Angles that share a vertex and one ray or line. When 2 (or more) angles form a straight angle
they have a sum of
180 degrees, and are
called Supplementary.
Vertical Angles: angles formed by the
Intersection of two lines, opposite each other, and share a vertex. B A
So then… <A + <B = 180 A B
So then… <A = <B

6. Using angle relation ships to find missing angles.
When we don’t know something, but we know how it is related to something we do know, we can use this relationship to find what is missing.
If I don’t know <q but I do know <r, I can use how they are related (vertical angles), to find the missing measure of <q.
Find <q if <r = 35 degrees
We can see they are vertical, so we know the equation
<q = <r, and we can substitute 35 for r giving us
<q = 35, which gives us our answer
<q = 35 degrees q r

7. Let’s try another, with algebraic expressions
We can observe that <c and <d are supplementary.
Therefor
<c + <d = 180
Substitute what we know, and solve for what we don’t
x x = 180
2x + 24 = 180
2x = , x = 78
Now because <c is given as an expression, substitute for x into what <c equals to find <c.
<c = x + 24
<c =
<c = 102
Find <c
if <c = x + 24, and <d = x c d

9. Practice solving for missing angles
Practice solving for missing angles. All problems use the same diagram, so you may want to redraw for each problem. The angle measure do not carry through from one problem to the next. c
Group C
#1) Find <d if <b = 42/6 + 9
#2) Find <d if <c = 123 – x, and <d = x
#3) Find <a if <a = 47 - x
and <c = 3x + 31
#4) Find <c if <a = x, and <c = 50x + 101
#5) Find <b if <b = 4.2x, and <d = x + 8
#6) Find <a if <a = 1.7x - 14
and < b = x + 5 b d a
Group B
#1) Find <d if <b =
#2) Find <d if <c = 123 – 9
#3) Find <a if <a = 47 - x
and <c = 3x + 31
#4) Find <c if <a = x, and <c = 50x + 101
#5) Find <b if <b = 5x, and <d = x + 25
#6) Find <a if <a = 3x + 5
and < b = 91 - x
Group A
#1) Find <d if <b = 53
#2) Find <d if <c = 108
#3) Find <a if <d = 47
#4) Find <c if <a = 143
#5) Find <b if <b = 2x, and <d = x + 25
#6) Find <a if <a = 3x + 5
and < b = x + 7

16. Two Parallel lines cut by a transversal.
A line that intersects two or more other lines, is referred to as a transversal line.
When the two lines intersected by a transversal are parallel, the angles produced by these intersections, have a variety of special relationships to one another. Understanding and applying these relationships allows us to find solutions to problem solving situations.

19. Name those relationships:
We will categorize all of these relationships as either a type of congruent relationship (discuss congruent), or a type of supplementary relationship (discuss supplementary)

20. Congruent Relationships: Angles are congruent if they have the exact same measure.
Here are the different congruent relationships formed from a transversal line intersecting 2 parallel lines.
Vertical angles: Angles, opposite from one another, formed by 2 intersecting lines.
<1=<4
<2=<3
<5=<8
<6=<7
Alternate Exterior: Angles, opposite sides of the transversal, on the outside of the parallel lines.
<1=<8
<2=<7
Alternate Interior: Angles, opposite sides of the transversal, on the inside of the parallel lines.
<3=<6
<4=<5
Corresponding: Angles, same side of the transversal, and both either above or below the parallel lines.
<1=<5
<2=<6
<3=<7
<4=<8

24. A color coded example of different congruent angle pairs: Each pair with the same color are congruent, and named by the term above each diagram

25. 8.) Find <1 if <1 = x + 100, and <3 = x + 10
A: #1-3, B: #2-5, C: #4-7, 10-12
For #’s 8-12, compose an equations based on the angle relationship you can use to find x, which can then be used to find the angles measurements. Justify the validity of your equation by naming the angle relationship you observe.
8.) Find <1 if <1 = x + 100, and <3 = x + 10
9.) Find <5, if <5 = x + 82, and <1 = x
10.) Find <2, if <1 = 8x – 10, and <8 = 10x – 40
11.) Find <5 if <1=3x + 17, and <2= x + 15
12.) Find <4 if <3= 3x + 50, and <6=15x - 10

29. 1.) Find <4 if <4 = 2x + 80, and <2 = x + 10
B: #2-4
C: #3-5
Compose an equation you can use to find x, which can then be used to find the angles measurements. Justify the validity of your equation by naming the angle relationship you observe.
1.) Find <4 if <4 = 2x + 80, and <2 = x + 10
2.) Find <1, if <5 = x + 86, and <1 = 202 – x
3.) Find <8, if <1 = 12x – 15, and <8 = 15x – 60
4.) Find <5 if <1=9x + 56, and <2= 3x + 43
5.) Find <4 if <5= 6x + 66, and <6=30x - 66

34. Group A: 1.) Find <3 if <1 = 2x, and <3 = x
All Groups: When you form an equation to help you solve for x, write the name of the angle relationship which allows you to form that equation, next to the equation.
Group C:
1.) Find <4 if
<3= 3.2x , and
<6=15.2x – 9.97
2.) Find <3 if,
<2 = (1/2)( x), and
<8 = (89/4) + 5x + (41/2)
3.) When a transversal line intersects two parallel lines, how can you use the concept of acute and obtuse angles to determine which angles are congruent to other angles, and which angles are supplementary to other angles? Thoroughly explain your answer, and provide examples to support your claim(s).
4.) We know that “alternate” interior/exterior angles are congruent. Make a claim about how “same-side” interior/exterior angles are related in this diagram. Justify your claim, using examples
Group B:
1.) Find <1, if <1 = 9x – 12, and <8 = 11x – 42
2.) Find <5 if <1=3x + 17, and <2= x + 15
3.) Many of the congruent relationships and supplementary relationships we have studied, which happen when a transversal line intersects, only exist when the two lines intersected are parallel. If the transversal line intersected two non-parallel lines, determine an angle pair which will still be congruent, and another angle pair which will no longer be congruent. Justify/explain your claim.
Group A:
1.) Find <3 if
<1 = 2x, and
<3 = x
2.) Find <4, if
<4 = x + 72, and
<8 = 208 – x
3.) Describe what “alternate” and “interior” and “exterior” refer to, when we speak of “alternate interior” and “alternate exterior” angle pairs.

35. Complementary Angles: When the sum of 2 or more angles is equal to 90 degrees.
How can we use memory devices to remember the difference between supplementary and complementary angles? A B C D

36. POD: How can we name all of the angles we see in this diagram
POD: How can we name all of the angles we see in this diagram? Write the names of each of the 3 angles we see. A D B C

37. Triangle Angle (interior/exterior) RelationshipsB C D
<ABC + <CAB + <BCA = 180 Sum of interior angles of a triangle always equals degrees. How can we demonstrate a proof of this?
<ACD + <BCA = The sum of two or more adjacent angles, which form a straight angle, always equals Therefore these angles are supplementary.
<ABC + <BAC = <ACD The sum of two interior angles of a triangle, is equal to the measure of the non-adjacent exterior angle.

38. Triangle Angle (interior/exterior) RelationshipsB C D
<A + <B + <C = Sum of interior angles of a triangle always equals degrees. How can we demonstrate a proof of this?
<C + <D = The sum of two or more adjacent angles, which form a straight angle, always equals Therefore these angles are supplementary.
<A + <B = <D The sum of two interior angles of a triangle, is equal to the measure of the non-adjacent exterior angle.

39. Collaborative POD:
a + b = ?
b + c = ?
c + a = ? e a b c d f

40. Lets Model #1: All: #103 Group A: #1-3 Group B: 3-5 Group C: 4-6 A B C
Find <BAC, if <BAC = 2x, <ABC = 2x + 8, <ACB = x + 2

41. Lets Model: All: #103 Group A: #1-3 Group B: 3-5 Group C: 4-6 A
#2: Find <ACD, if <ABC = 41, <BAC = 61 B C D
#3: Find <ABC, if <BAC = 6x - 2, <ABC = 3x + 16, <ACD = 2x + 100

42. Practice using the angle relationships to find missing angles.
All Groups: #1-3
A: 4-6
B: 6 – 8
C: 7-10 B C D
1.) Find <ABC, if <BAC = 42, and <ACB = 67
2.) Find <ACB, if <ACD = 138
3.) Find <ACD, if <ABC = 43, and <BAC = 83
4.) Find <ACB and <CBA, if <BAC=40, <ACB=X+20, and <CBA=3x
5.) Find <ABC, if <BAC = 49, and < ACD = 136
6.) Find <ABC, if <BAC = X + 20, <ABC = 2X – 30, <ACD = 113
7.) Find <ACB, if <ABC = x, <BAC = 2x, <ACB = 2x - 15
8.) Find <ACD, if <ACD = 3x, <BAC = x + 10, <ABC = x + 39
9.) Find <ABC, if <BAC = 4x – 5, <ACB = 5x, <ACD = 11x + 20
10.) Find <BAC, if <ABC = 2X – 15, <ACD = 126, <ACB = 2X + 10

43. POD: Determine the measurement of the missing angle x.B C 27 32

44. Don’t forget, when you are stuck or unsure, and Mr
Don’t forget, when you are stuck or unsure, and Mr. Gibbons is currently unavailable, you can…
Reference your notes
Reference the power point on our class website.
Do a google key word search on the topic at hand.
Reference a text book
Start/join a group discussion; ask a fellow student for help
Check your solutions against a classmate’s.

45. All Groups: You must justify the equations you form using terms or phrases that refer to how the angles are
related! D
Group C:
1.) Find <ABC, if <ABC = 0.2x – 2.7,
<BCA = 0.6x , <CAB = 0.5x – 1.9
2.) Find <ABC,
if <ABC=(1/2)X + (34/4),
<CAB = (5/4)x + (71/2)
<ACD = (3/2)x + (237/4)
3.) Students were asked to write their own missing angle problems and have another student solve them. Susan gave Fatima a problem to solve for the missing <ABC, but Fatima is completely stuck. Susan gave her the following given information:
<ABC = 2x + 42, <CAB = 6x + 26, and <ACD = 4(2x + 16).
Fatima now thinks she is terrible at math!! Investigate why you think Fatima is having trouble solving for the missing angle, and explain why she is in fact not terrible at math. C B A
Group B:
1.) Find <CAB, if <ABC = x,
<BCA = x + 17, <CAB = 2x – 5
2.) Find <ACD, if <BCA = 4(x + 8),
<ACD = 3x + 29
3.) Find <ABC, if <ABC=2X + 27,
<CAB = 10x + 41, <ACD = 40x – 16
4.) Two students, John and Jay, disagree about the following problem: In an obtuse-Isosceles triangle ABC,
<A = 154, so what are the measurements of <B, and <C? John says its impossible to solve, but Jay says it can be solved. Who is correct? Justify your answer, and include a diagram of this triangle.
Group A:
1.) Find <BAC, if <BCA = 81, <ABC = 33.
2.) Find <ACD, if <BCA = 2x + 11,
<ACD = 3x + 24
3.) Find <ABC, if <ABC=X + 30,
<CAB = 71, <ACD = 104
4.) A classmate gives you a missing angle problem to solve:
“Find <BCA, if <CAB=54,
<ABC=61, and <ACD = 114.”
However, there is something wrong with the problem they wrote. Explain what is wrong with this problem, and why.

51. City Planning
Fatma and Brian are civil engineers, and are charged with making improvements in road repair, after many snow storms had damaged the roads and sidewalks.
Depending on the angles formed by the intersection of the streets, they need different equipment. At intersections creating an angle greater than 125 degrees, they need a special sidewalk curb molder. There are too many intersections to repair to measure each angle physically, so here’s what they know: The intersection of Broadway and 36th street, creates an angle which is 20 more than 3 times a number, and another angle that is 40 more than 9 times that same number. Will the curb at the corner of the NY Real Estate Institute, formed by the intersection of Broadway and 36th Street, require the use of a special curb molder? Show all relevant, necessary work, and justify your conclusion.

52. POD
Which answer choice explains the best way to find <17 if you are given <4, and <9.
a.) Subtract the measure of <4 from <9.
b.) <9 is congruent to <2 because they are alternate exterior, and <4 and <3 are supplementary so subtract <4 from 180 to find <3. Add <2 and <3 and subtract from 180.
c.) <9 is corresponding to <1, which is supplementary to <2, so subtract <1 from 180 to find <2. <4 and <3 are supplementary so subtract <4 from 180 to find <3. Add <2 and <3 and subtract from 180.
d.) <9 and <4 are alternate exterior, so they are both supplementary to <2, and <3. Add <2 and <3 and subtract from 180. 17 1 2 3 4 6 5 9 10 13 14

53. Note: All 4 interior <‘s of a quadrilateral add to = 36017
All:
1.) Find <17, if <10 = 49, <11 = 57
2.) Find <4, if <13=68, <17 = 85
Then Group A do BLUE
Group B do 2 from BLUE, 2 from GREEN
Group C do GREEN 1 2 3 4 6 7 8 5 12 9 10 11 13 14 15 16
GREEN:
1.) Find < 12 if <12= 5x – 15 ,
<5= x + 4, <17= 4x + 1
2.) Find <16 if <17 = 35,
<10= x + 25, <3 = 3x + 10
3.) find <10 if <11= 2x + 15,
<3 = 4x – 9, <17 = 7x
4.) Find <4 if <17=2x + 10,
<13 = 2x + 30, <15= 12x - 20
5.) Find <10, if <13=x + 50,
<1= 4x + 5, <16=x + 15, <4= 2x + 90
BLUE:
1.) Find < 12 if <12= 5x – 5 , <11= x + 65
2.) Find <8 if <17 = 40, <2= x + 30, <3 = 3x + 10
3.) find <2 if <11= 2x, <3 = 4x – 50, <17 = 48
4.) Find <4 if <17=2x + 10, <2 = 2x + 30, <4= 12x - 20

57. Homework
#22: p , read definitions, diagrams, examples, do quickchecks #1-3
#23: p216 #5-7, #12-15
#24: p read, do quickchecks #1-3
#25: p220, #1-4, 14-19
#26: p221, #29-32
#27: p , #20-24
#28: p221, #25-28

58. homework
#27: p , #20-24
#28: p221, #25-28
#29: read all of p ,
do quickchecks #1, 2
#30: p234, #6-9, 12
#31: p235, #13, 14
#32: p235, #15-19

59. Test This Tuesday (3/21/17)
For the test on Angle Relationships students must…
Identify, with appropriate vocabulary, different kinds of angle relationships which appear when a transversal line intersects 2 parallel lines (including supplementary, vertical, corresponding, alternate-interior, & alternate-exterior)
Create and solve appropriate algebraic equations, based on the angle relationships listed above, which can be used to solve for a missing angle measurement.
Identify any of the relationships found between different combinations of interior and exterior angles of a triangle. (The sum of all interior angles is equal to 180 degrees. The sum of 2 interior angles is equal to the measure of the non-adjacent exterior angle. Adjacent interior and exterior angles of a triangle are supplementary.)
Use the relationships between angles of a triangle to set up algebraic equations which can be used to solve for a missing angle.