Download 2x=20 x=10

G.CO.9 STUDENT NOTES
&
PRACTICE WS
#7/#2
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7
Proving things to be true is a common task for geometry students. To prove something is to logically establish
the connections from what you know to what you want to prove all the while providing accurate reasoning for
each conclusion. This process is often difficult for new geometry students - it is hard to clearly explain what
you know and why you know it. One format for a proof is to provide it in a paragraph form. To simply write it
you would say it. This can be a comfortable style for many students. The key is to after each conclusion or
deduction to state the reason for knowing it. lf you do this the proof will flow naturally and correctly.
as
Prove Vertical Angles are Congruent.
Our knowledge of rotations will help us here so first I want to look back at how we defined an 180o rotation.
When we defined a rotation we looked at the properties of the special rotation of 180o.
A rotation of 1800 maps A to A' such
that:
aA
rotation)
b) OA = OA' (from definition of rotationl
al m/-AOA' = 180o (from definition of
c) Ray
OA
Tdisthe
and Aay
O,{
same line
,,.'
J
are opposite rays. (They form a
line.)
At
asZi'
a
,"'
O
This will help us prove the relationship between two vertical angles. First
of all, vertical angles are the two non-adjacent angles formed by
intersecting lines. So in the diagram
and
14
Zt and Z3 are vertical angles and 22
are vertical angles as well.
To Prove that VerticalAngles are Congruent we use the properties of an
180o rotation.
Prove: IDEA= ZBEC
A rotation of L80o about point E, maps D onto opposite ray EE
.
O'lies on
fA . nrotation of 180" about point E, maps A onto opposite ray E. A' liut
onEd . ID'EA' = lBECbecause the angles use the same rays and vertex.
Thus using the transitive property, /DEA= IBEC.
Using a similar argument we could also prove,
Find
Z!
mZL=
& 12
34o (vertical
IDEC= ZBEA.
Find x
/.=l
mlT = 180 - 34 (linear pair)
mZZ= 146"
2x+ !6 = 724 (vertical Z
2x = 108
x=54
Find mZFEG
=)
5x-4
= 3x + 16 (vertical
2x=20
x=10
5(10)-4=46"=nZFEG
Z
=)
G.CO.9 STUDENT NOTES
NYTS
&
PRACilCE WS #7/#2
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2
(Now You Trv Somel
1..Find
ZL& Z2
2. Find x
'vcrh
t$o-{ | '= l3?a
*l r- g)
(r.rr.,rAC Pfre)
?x
3. Find x and
vl? e 1( 1vohe.-t
3r-1/
€9
rr)
nICAB
Zx=x+qo
G.p
rar-c*c;;llio
Prove when a transversal crosses parallel lines, alternate interior angles are congruent and
corresponding angles are congruent.
l^r
To prove this relationship we are also going to go back to the properties of a translation of an angle along one
of its rays.
B
vector E2 maps all points such that
L. IABC = IA'B' C' (lsometry)
2. B, A, B'and A' are collinear (translation on angle ray)
A translatio n
of
ZABC by
Because the two angles are equal and formed on the same ray, then:
nc il pc'
Parallel lines are formed when we translate an angle along one of its
rays. lf we extend those rays into lines we form a few more angles.
When lines are parallelwe use arrowheads to denote which Iines are
parallelto each other. So in the diagram, line g I I line h.
/-7, 23, Z5 & 27 along the transversal line
give us congruent corresponding angles, 12, 24, 16 & 18.
The translation of angles
m
This angle relationship is called CORRESPONDING ANGLES and because
of the properties of the isometric translation, CORRESPONDING
ANGLES MUST BE CONGRUENT.
/.1,= 12,
/3= 14, 15 = 16 & /7 = l8
l^
C
G.CO.9 STUDENT NOTES
&
PRACTICE WS
#7/#2
-
geometrycommoncore.com
15 & Z2
and Z7 & 14 are called ALTERNATE EXTERIOR ANGLES.
Alternate because they are on alternating sides of the transversal and
exterior because they are on the outside of the parallel lines.
PROVE: ALTERNATE EXTERIOR ANGLES ARE CONGRUENT
PROVE: Z4=./7 & 12=./5
14 = Z3
because corresponding angles are congruent and Z3 =
because vertical angles are congruent. Thus using the transitive
property, /-4 = l7
.
Z7
We could use a similar argument to prove ZZ =
2.5.
An alternate way of writing it.....
PROVE: Z4= Z7 & Z2= Z5
Earlier we established that opposite angles are equal due to the
rotation of 180"... thus 17 = 13 because they are opposite angles. 23
Z4 because we established that corresponding angles are congruent
=
due to the translationZF. Using the transitive property,then ./.4=
17. We could use a similar argument to prove 22 = 15.
Z3 & l8 and 16 & lL are called ALTERNATE
INTERIOR ANGLES.
Alternate because they are on alternating sides of the transversal and
interior because they are on the interior of the parallel lines.
PROVE: ALTERNATE INTERIOR ANGLES ARE CONGRUENT
PROVE: /3= ZB & 16=
ll
Z3 = Z4 because corresponding angles are congruent and 14 = l8
because vertical angles are congruent. Thus using the transitive property,
13 = 18. We could use a similar argument to prove ZG = lt.
An alternate way of writing it.....
PROVE: ./3 = ZB & 16 = ZL
Earlier we established that opposite angles are equal due to the rotation of
180"... thus 13 17 because they are opposite angles. 17 = l8 because
=
we established that corresponding angles are congruent due to the
translation 7E . Uring the transitive property, then Z3 =
use a similar argument to prove 26 = 11,.
18.
We could
3
G.CO.9 STUDENT NOTES & PRACTICE WS #7/#2
13 & ZG and Z7 & ZB are called
- geometrycommoncore.com
4
CONSECUTIVE INTERIOR ANGLES (OR
SAME SIDE INTERIOR ANGLES). I prefer same side.... Same Side because
they are on the same side of the transversal and interior because they are
on the interior of the parallel lines.
PROVE: SAME SIDE INTERIOR ANGLES ARE SUPPLEMENTARY
PROVE: mlL + m./8 = 180o & mZ3 + mZG = L8O"
ml! + ml7 = L80o because they are a linear pair and ml7 :
m/.8
because corresponding are congruent. lf we substitute, we get mZL
ml8 = L80o.
We could use a similar argument to prove
ml3 + ml6
= 18O".
An alternate proof using transformations.
PROVE: m,/l + ml$ = 180o & ml3 + mZG = L8O"
l2
Z8 are a linear pair. Thus mZZ + ml8 = 180' by definition. lt is
also true that 12 = 1L (ml2 = m/-Ll because a translatio n of E) maps
12 onto lt. So if we substitute these values we get mlt + ml9 = t80".
We could use a similar argument to prove ml3 + ml6 = L8Oo.
and
23 & Z6
and
17 & l8 are called
CONSECUTIVE EXTERIOR ANGLES (OR
SAME SIDE EXTERIOR ANGLES). I prefer same side.... Same Side because
they are on the same side of the transversal and exterior because they are
on the exterior of the parallel lines.
PROVE: SAME SIDE EXTERIOR ANGLES ARE SUPPLEMENTARV
PROVE: mlz + m,/7 = 180" & mZA + mZS = 1,SOo
mlt + ml7 = L80o because they are a linear pair and mlL = ml2
because corresponding are congruent. lf we substitute, we get
mlz
+
mZ7 = t80".
We could use a similar argument to prove mZ4 + ml5 = 180".
CONGRUENT
SUPPLEMENTARY
Corresponding angles are congruent.
Alternate interior angles are congruent.
Alternate exterior angles are congruent.
Consecutive (Same Side) interior angles are supplementary.
Consecutive (Same Side) exterior angles are supplementary.
&
G.CO.9 STUDENT NOTES
PRAC:nCE WS
#7/#2
-
geometrycommoncore.com
4. Provide the name of the following relationships.
al
Zl
&
Z5
c)
t5
&
t4 Alr.
Cocrra^ESIUnOlr./6
tnrrtrapq
t7
b)
t2
d\
t4 a zo )l*1C.
(o
&
r.r S
AL?, wtEg@a
Ecrl?t vE.
S t oE,
trsruDv&
tn{Tl9tc€.
5. Find the measure of the angle and give a reason for knowing it.
(reason)
(measurel
almZL= 46o
b)mZ2=
cl
ml3
6. Solve
a) x=
=
for
8Zo
46"
e
Co oaesel,,rDt rvc ro 92
Ltr,
txrrt'oltqL To
unknown values.
il"'*"YoruE)'
b)
"
16"
x= C
5x-4
5x -1
=
lb
(Ar-r,'rr'-)
5x =ro
x =lO
20
r<1-{
+x*t,
A.,rgl,.l.f
Zlxfst=t8os
At<= lU
1x = ll"l '
tr5
C) Points on a perpendicular bisector of a line segment are exactly those equidistant
from
the segment's endpoints.
i.
lc
As defined a perpendicular bisector is the perpendicular
line that passes through the midpoint of a segment.
We have also learned that the perpendicular bisector is the
line of reflection for AS
.
r-&.-fu
A
fuI BA
tr
B
G.CO.9 STUDENT NOTES
PROVE: ,qC
=
&
PRACilCE WS #1/#2
- geometrycommoncore.com
6
nC
A reflection over
MC
maps A onto B because of the
definition of a reflection and C onto C and M onto M
because they are on the line of reflection.
Because
AC
by the isometric
^^psonto.BC
transformation reftection AC = BC
v
.
Another way to prove this might be to prove the two triangles are congruent. The common side, the bisected
segment and the right angle give us a SAS relationship
CONCEPT 2
- PAIRS OF ANGLES
It is very common for two lines to intersect in the plane. When two lines intersect a point is formed and also
number of angles. In the diagram to the right, the intersection of line m and line n is point A. The angles
formed have many different names and relationships.
The diagram to the right has some Adjacent Angles.
that share a vertex and a ray and no interior
points. So in the diagram to the right lt & 22 are adjacent angles. There are
other examples of adjacent angles in the diagram such as Z4 & 11.
ADJACENT ANGLES are angles
The diagram to the right has some Linear Pairs.
A LINEAR PAIR are two angles that are adjacent and sum to 180o. ln this particular diagram
more specifically called a linear pair. 12 & 23, 23 &
24,
and
l4
lt
&
12
are
& 21, are also a linear pairs.
The diagram to the right has some Vertical Angles.
VERTICAL ANGLES are a pair
labeled
Zt
l4
& Z3 and 12 &
SUPPLEMENTARY ANGLES
of non-adjacent angles formed by the intersection of two lines. The angles
-
COMPLEMENTARY ANGLES
are vertical angles.
Two angles are supplementary if the sum of their measures is 180o.
-
Two angles are complementary if the sum of their measures is 90".
a