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Chapter 9
Congruence, Symmetry, and Similarity
Section 9.2
Applications of Transformations
Congruent Angles
When 2 or 3 lines intersect in certain patterns congruent angles are formed. These
particular types of angles come up so often we give them their own names.
Vertical Angles
When two lines in a plane intersect at a point they form two pairs of vertical angles.
Vertical angles have a common vertex (the point of intersection of the lines) but no
side in common. The pairs of vertical angles are congruent.
A
B
AOB and COD are vertical angles
O
C
AOC and BOD are vertical angles
AOB  COD and AOC  BOD
D
The congruence of vertical angles can be established by formal deduction as in van
Hiele level 4 as follows:
mAOC + mAOB = 180
and
mAOB + mBOD = 180
mAOC + mAOB = mAOB + mBOD
mAOC = mBOD
Corresponding and Alternate Interior Angles
When a pair of parallel lines is intersected by another line, the line that intersects
the two parallel lines is called a transversal (or transversal line). When this happens
several pairs of congruent angles are formed. The pairs of congruent angles are
referred to so often we give them special names. In the picture below to the right
line l is parallel to line m (i.e. l || m). The line t is the transversal line.
t
The orange angles are all congruent (i.e.
1
2
1,3,5,7). The green angles are all
l
4
3
congruent (i.e. 2,4,6,8).
m
5
8
6
7
Pairs of Alternating Interior Angles:
Pairs of Corresponding Angles:
3 and 5
4 and 6
1 and 5
2 and 6
Pairs of Alternating Exterior Angles:
3 and 7
4 and 8
1 and 7
2 and 8
The intuitive reasons for why this is true come from translating (i.e. applying a
translation transformation) the angle shaped formed by lines l and t along t until it
exactly coincides (matches) with the angle formed by the lines m and t.
The importance of these relationships is that it allows a student studying geometry
to be able to reason a great deal of information about a shape even though they
were given only a small amount of information about the shape to start with. This
is illustrated by the example below.
AB
Given
parallel to
CD
x
A
B
w 42
v
C
78
58
y
z
D
v = 42
This is because the 42 angle and the v
angle are vertical angles.
w = 78
This is because the 78 angle and the w
angle are corresponding angles.
x = 60
The sum of interior angles of a triangle is
180. We get: 78 + 42 + x = 180
y = 42
This is because the v angle and the y angle are alternate interior angles.
Z = 80
The sum of interior angles of a triangle is 180. We get: 58 + 42 + x = 180
Congruent Shapes and Parts
Shapes that are congruent have the parts that correspond and are congruent.
Below consider the example of the two congruent triangles.
A
B
D
C
E
F
We say ABC  DEF and we have the following parts of the triangles that
correspond () and are congruent ():
Vertices
Sides
Angles
AD
AB  DE and AB  DE
CAB  FDE and CAB  FDE
BE
AC  DF and AC  DF
ABC  DEF and ABC  DEF
CF
BC  EF and BC  EF
BCA  EFD and BCA  EFD
In shapes that are congruent corresponding parts are congruent (usually we
think of this as sides and angles in polygons) and if all corresponding parts are
congruent then so are the shapes.