Download Angle Relationships

Angle Relationships
POD
• If A + B + C = D, and X + C = D, then write an
equation that defines X, or shows what just X
equals.
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
Vertex: the point that the
two rays which form the
angle, share.
A diagram to be “Not drawn to scale”?
All of the diagrams we will look at this unit will “not be
drawn to scale”
• Naming Angles: Angles are either named by
three letters, where the middle letter has to
be the vertex, or, sometimes using only one
letter.
B
<BDC or
<CDB
<BDA or
<ADB
A
A
D
B
C
– Why are angles are never named with two letters?
Basic Angle Relationships
• Adjacent angles: Angles that share a vertex and
one ray or line. When 2 (or more) angles form a
straight angle
B
A
they have a sum of
180 degrees, and are
So then…
<A + <B = 180
called Supplementary.
• Vertical Angles: angles formed by the
Intersection of two lines, opposite each other, and
share a vertex.
A
So then…
<A = <B
B
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
q
r
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
Let’s try another, with algebraic expressions
• Find <c
if <c = x + 24, and
<d = x
c
d
• 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 + 24 + x = 180
2x + 24 = 180
-24 -24
2x = 156 , x = 78
2
2
• Now because <c is given as an
expression, substitute for x into
what <c equals to find <c.
<c = x + 24
<c = 78 + 24
<c = 102
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.
• Group C
c
d
a
• 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
b
• Group B
#1) Find <d if <b = 42 + 9
#2) Find <d if <c = 123 – 9
#3) Find <a if <a = 47 - x
and <c = 3x + 31
#4) Find <c if <a = 143 + 8x,
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
#1) Find <d if <b = 42/6 + 9
#2) Find <d if <c = 123 – x,
and <d = 25 + 5x
#3) Find <a if <a = 47 - x
and <c = 3x + 31
#4) Find <c if <a = 159 - 8x,
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
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.
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)
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
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
1.) Find <3 if <2 = 82
2.) Find <8 if <6 = 48
3.) Find <1 if <5 = 103
4.) Find <1 if <6 = 34
5.) Find <7 if <2 = 67
6.) Find <4 if <5 = 98
7.) Find <4 if <6 = 57
A: #1-3, 8-9
B: #2-5, 9-10
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 = 198 - 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
A: #1, 2
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
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
Group B:
<3= 3.2x + 50.03, and
1.) Find <1, if <1 = 9x – 12,
<6=15.2x – 9.97
Group A:
and <8 = 11x – 42
2.) Find <3 if,
1.) Find <3 if
2.) Find <5 if <1=3x + 17,
<2 = (1/2)(10 + 13x), and
<8 = (89/4) + 5x + (41/2)
and <2= x + 15
<1 = 2x, and
3.) When a transversal line intersects
3.) Many of the congruent
<3 = x
two parallel lines, how can you use the
relationships and
2.) Find <4, if
concept of acute and obtuse angles to
supplementary relationships
<4 = x + 72, and
determine which angles are congruent
we have studied, which
<8 = 208 – x
to other angles, and which angles are
happen when a transversal
3.) Describe what
supplementary to other angles?
line intersects, only exist
“alternate” and
Thoroughly explain your answer, and
when the two lines
“interior” and
intersected are parallel. If the provide examples to support your
“exterior” refer to, transversal line intersected
claim(s).
when we speak of two non-parallel lines,
4.) We know that “alternate”
“alternate interior” determine an angle pair
interior/exterior angles are congruent.
and “alternate
which will still be congruent, Make a claim about how “same-side”
exterior” angle
and another angle pair which interior/exterior angles are related in
pairs.
this diagram. Justify your claim, using
will no longer be congruent.
examples
Justify/explain your claim.
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
D
C
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
C
B
Triangle Angle (interior/exterior)
Relationships
A
B
C
D
<ABC + <CAB + <BCA = 180 Sum of interior angles of a triangle always equals 180
degrees. How can we demonstrate a proof of this?
<ACD + <BCA = 180 The sum of two or more adjacent angles, which form a
straight angle, always equals 180. 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.
Triangle Angle (interior/exterior)
Relationships
A
B
C
D
<A + <B + <C = 180 Sum of interior angles of a triangle always equals 180
degrees. How can we demonstrate a proof of this?
<C + <D = 180
The sum of two or more adjacent angles, which form a
straight angle, always equals 180. 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.
Collaborative POD:
a+b=?
b+c=?
e
a
c+a=?
b
f
c
d
Lets Model #1:
A
B
C
D
Find <BAC, if <BAC = 2x, <ABC = 2x + 8, <ACB = x + 2
Lets Model:
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
A
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
POD: Determine the measurement of
the missing angle x.
A
D
C
32
X
27
B
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.
All Groups: You must
justify the equations
you form using terms or
phrases that refer to
how the angles are
related!
B
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.
D
C
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 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.
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.
POD
17
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.
1
5
9
10
13 14
2
6
3
4
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.
Note: All 4
interior <‘s of a
quadrilateral
add to = 360
17
1
5
9
10
13 14
2
6
All:
1.) Find <17, if <10 = 49, <11 = 57
2.) Find <4, if <13=68, <17 = 85
3
7
4
• Then Group A do BLUE
• Group B do 2 from BLUE, 2
from GREEN
• Group C do GREEN
8
11
12
15 16
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
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
Homework
• #22: p214-215, read definitions, diagrams,
examples, do quickchecks #1-3
• #23: p216 #5-7, #12-15
• #24: p218-219 read, do quickchecks #1-3
• #25: p220, #1-4, 14-19
• #26: p221, #29-32
• #27: p220-221, #20-24
• #28: p221, #25-28
homework
• #27: p220-221, #20-24
• #28: p221, #25-28
• #29: read all of p232-233,
do quickchecks #1, 2
• #30: p234, #6-9, 12
• #31: p235, #13, 14
• #32: p235, #15-19
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, alternateinterior, & 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.