Notes on Correspondence in Congruent Triangles
Name_________________________
For this unit, it will be important to recall a certain amount of previously learned information in
geometry. The concepts of vertical angles and alternate interior angles are concepts from
middle school.
Fill in the blanks:
***If two segments are congruent, then they have the same ______________.***
***If two angles are congruent, then they have the same ________________.***
--------------------------------------------------------------------------------------------------------------------Vertical angles are the angles opposite each other when
two lines or two segments intersect.
In the figure to the left, 1 and __________ are vertical
angles. What is the other pair of vertical angles?
Vertical Angle Theorem - Vertical angles are congruent.
In the figure, 1 is congruent to 3 because they are vertical angles.
--------------------------------------------------------------------------------------------------------------------n
Suppose that lines l and m are parallel lines in the figure to the
right. Notice how they are both intersected by the transversal,
line n. As a result, the angles numbered 5 thru 8 are formed.
5 and 8 are known as alternate interior angles.
The term "alternate" is used because they are on opposite sides
of line n, and the term "interior" is used because they are in
between the two parallel lines.
5
7
l
6
8
m
What is the other pair of alternate interior angles in the diagram?
Alternate Interior Angle Theorem - Alternate interior angles are congruent.
If m8  56 , what is m5 ?
--------------------------------------------------------------------------------------------------------------------Question: What does it mean for two triangles to be congruent?
Answer: If two triangles are congruent, then the corresponding parts of the two congruent
triangles are congruent.
Let us define a couple of words so that this statement makes more sense.
First, the word parts refers to the sides and angles of a triangle.
Second, the word corresponding means "similar in position, purpose, or form." For example,
"Corresponding locations at the elementary school and the high school are the library and the
media center." While they go by different names and are at different locations, the library and
the media center serve similar purposes. In regards to congruent triangles, two corresponding
sides may lie on two different triangles, but they are corresponding because they have exactly
the same length.
--------------------------------------------------------------------------------------------------------------------For Questions 1-3, fill in the blank.
1.
The prime minister of the United Kingdom corresponds to the _______________ of the
United States.
2.
The video cassette tape of the 1980's corresponds to the _______________ of the 2000's.
3.
Suppose the two triangles below are congruent. Using your knowledge of
correspondence, side AB would likely correspond to side ___________.
D
A
B
C
F
E
--------------------------------------------------------------------------------------------------------------------Let's rewrite the statement from the front using these new definitions and ideas.
Before: If two triangles are congruent, then the corresponding parts of the two
congruent triangles are congruent.
After: If two triangles are congruent, then specific pairs of sides and angles in the
two congruent triangles are congruent.
--------------------------------------------------------------------------------------------------------------------Using the two congruent triangles above, we can write six congruence statements based on this
idea of correspondence. If
, then...
1) AB  DE
2) BC  EF
3) AC  ______
4) A  D
5) B  E
6) C  ________
The order of the letters in the triangle congruence statement (
) is important.
Corresponding angles must be written in the same spots. For example, since
, they
must be in identical positions in the statement. Thus, 'A' and 'D' are both listed first. The same
would go for the other letters.
--------------------------------------------------------------------------------------------------------------------4.
Suppose
. Write the six congruence statements as in the box above.
--------------------------------------------------------------------------------------------------------------------The abbreviation for the concept of congruent triangles having congruent corresponding parts is
abbreviated CPCTC (Corresponding Parts of Congruent Triangles are Congruent).
Homework on CPCTC
Name_________________________
For Questions 1-6, let RAT  YOU . Fill in the blanks below based on this triangle
congruence statement.
AT  _______
1.
R
2.
_______
3.
U  ______
RT  _______
YO  _______
A  ______
4.
5.
6.
--------------------------------------------------------------------------------------------------------------------7.
Let
. What are six other congruence statements that can be generated
based solely on this statement?
--------------------------------------------------------------------------------------------------------------------For the rest of the homework, some questions may NOT have an answer.
--------------------------------------------------------------------------------------------------------------------Consider parallelogram FGIH to the left.
G
F
1
8.
Fill in the blank: FG must be parallel to __________.
9.
Extend all of the solid lines in the figure to the left so
that the solid lines form a figure similar to one on the
first page of your notes.
10.
What type of angles are 1 and 2 ?
2
H
I
11.
Based on your answer in #10, what else do we know
about 1 and 2 (use a theorem on the first page of
your notes)?
--------------------------------------------------------------------------------------------------------------------B
For Questions 12-15, consider the figure to the left in which the
6
line that contains AB is parallel to the line that contains CD .
7
5
C
3
8
A
4
D
12.
What term describes angles 7 and 8?
13.
What term describes angles 3 and 5?
14.
What term describes angles 4 and 6?
15.
What term describes angles 4 and 5?
--------------------------------------------------------------------------------------------------------------------K
For Questions 16-19, consider quadrilateral KNML to the left.
13
L
9 10
11 12
16.
Which term describes angles 9 and 10?
17.
Which term describes angles 9 and 12?
18.
Which term describes angles 10 and 11?
19.
Which term describes angles 13 and 14?
N
14
M