Download Corresponding Parts (CPCTC) and Identifying

Corresponding Parts (CPCTC)
and Identifying Minimal
Conditions
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Concept 1. Corresponding Parts (CPCTC) and Identifying Minimal Conditions
C ONCEPT
1
Corresponding Parts
(CPCTC) and Identifying Minimal
Conditions
Learning Objectives
• Define congruence in triangles.
• Create accurate congruence statements.
• Understand that if two angles of a triangle are congruent to two angles of another triangle, the remaining
angles will also be congruent.
Defining Congruence in Triangles
Two figures are congruent if they have exactly the same size and shape. Another way of saying this is that the two
figures can be perfectly aligned when one is placed on top of the other—but you may need to rotate or reflect (flip)
the figures to make them line up.
When figures have exactly the same size and shape, they are ___________.
When that alignment is done, the angles that are matched are called corresponding angles, and the sides that are
matched are called corresponding sides.
In congruent figures, the angles that match up are called ___________ angles and the matching sides are called
___________ sides.
Though the two triangles above may not look the same at first, when you rotate and flip triangle DEF
In the diagram above,
• Sides AC and DE each have one tic mark, indicating that they have the same length. Since they have the same
length and are in matching positions in the triangle, they are corresponding sides.
• Sides BA and DF each have two tic marks, showing that they are also congruent and thus, corresponding
sides.
• Finally, as you can see, BC ∼
= EF because they each have three tic marks.
Each of these pairs corresponds because they are congruent to each other.
When two triangles are congruent, the three pairs of corresponding angles are also congruent. Notice the tic marks
in the triangles below.
In congruent triangles, all three pairs of corresponding angles are ____________.
We use arcs inside the angle to show congruence in angles just as tic marks show congruence in sides. You can see
that 1, 2, or 3 arcs inside each angle show which angles are congruent and corresponding. From the markings in the
angles we can see:
6 A∼
and 6 C ∼
= 6 D, 6 B ∼
= 6 F,
= 6 E.
Which angle is congruent to 6 F? ___________________
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Which angle is congruent to 6 E? ___________________
Which angle is congruent to 6 D? ___________________
Which side is congruent to AC? ___________________
Which side is congruent to DF? ___________________
Which side is congruent to BC? ___________________
A term used to describe sides and angles is partparts.
Angles and sides are also called ___________________________ of a triangle.
By definition, if two triangles are congruent, then you know that all pairs of corresponding sides are congruent and
all pairs of corresponding angles are congruent. We can therefore say that the corresponding parts (sides and angles)
of congruent triangles are congruent. This is often called CPCTC.
CPCTC
Corresponding Parts of Congruent Triangles are Congruent.
Reading Check:
1. Give 2 examples of apart of a triangle.
2. Fill in the blanks: If two triangles are congruent, this means that all of its corresponding _________ and _________ are congruent.
3. What do the lettersCPCTC stand for?
Example 1
Are the two triangles below congruent?
Begin by examining the sides:
• AC and RI both have one tic mark, so they are congruent.
• AB and T I both have two tic marks, so they are congruent as well.
• BC and RT have three tic marks each.
So each pair of sides is congruent.
Next you must check each angle:
• 6 I and 6 A both have one arc, so they are congruent.
• 6 T∼
= 6 B because they each have two arcs.
• Finally, 6 R ∼
= 6 C because they have three arcs.
We can check that each angle in the first triangle matches with its corresponding angle in the second triangle by
examining the sides. 6 B corresponds with 6 T because they are formed by the sides with two and three tic marks.
Since all pairs of corresponding sides and angles are congruent in these two triangles, we conclude that YES, the
two triangles are congruent.
Creating Congruence Statements
We have already been using the congruence sign ∼
= when talking about congruent sides and congruent angles.
When writing congruence statements involving angles or triangles, use other symbols:
• The symbol BC means “segment BC”
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Concept 1. Corresponding Parts (CPCTC) and Identifying Minimal Conditions
• The symbol 6 B means “angle B”
• Similarly, the symbol ∆ABC means “triangle ABC”
In words, the symbol ∼
= means _________________________________.
In words, the symbol ∆LMN means _________________________________.
In words, the symbol 6 N means _________________________________.
In words, the symbol LM means _________________________________.
When you are creating a congruence statement of two triangles, the order of the letters is very importantCorresponding
parts must be written in order. This means that the angle at the first letter of the first triangle corresponds with the
angle at the first letter of the second triangle, the angles at the second letter correspond, and the angles at the third
letter correspond. If the angles are not matched up between the triangles, the parts will not correspond.
When writing congruence statements, __________________________________ parts must be written in order.
You can use either
For instance, in the congruence statement ∆XY Z ∼
= ∆LMN, the letters that match up tell you which angles in the
triangles are congruent:
You can see that angles X and L match up, so 6 X ∼
= 6 L angles Y and M match up, so 6 Y ∼
= 6 M and angles Z and N
∼6
match up, so 6
=
Likewise, in the picture of the triangles below, you can match up the marked angles (or sides) to see what parts
correspond:
∼ ∆PQR because the order of the letters
If you are writing a congruence statement, you could NOT say that ∆BCD =
does not match up to corresponding congruent angles.
If you look at 6 B, it does not correspond to 6 P.
angles).
6
6
B corresponds to 6 Q instead (indicated by the two arcs in the
C corresponds to 6 ______ (three arcs), and 6 D corresponds to 6 ______ (one arc).
Remember, you must compose the congruent statement so that the vertices are lined up for congruence, which is
noted by the number of arcs inside the angles. The statement below is correct:
∆BCD ∼
= ∆QPR
Reading Check:
Use the congruence statement
6
∼
=6
∼
=6
6
∼
=6
6
Example 2
Compose a congruence statement for the two triangles below.
To write an accurate congruence statement, you must be able to identify the corresponding pairs in the triangles
above. Notice that:
•
6
R and 6 F each have one arc mark.
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• Similarly, 6 S and 6 E each have two arcs, and
• 6 T and 6 D have three arcs.
Additionally, you can see from the tic marks on each side that:
• RS = FE (or RS ∼
= FE),
• ST = ED, and
• RT = FD.
So, the two triangles are congruent, and to make the most accurate statement, this should be expressed by matching
corresponding vertices. You can spell the first triangle in alphabetical order, for example, and then align the second
triangle so its angles match with the angles in the first one:
∆RST ∼
= ∆FED
Notice in example 2 that you don’t need to write the angles in alphabetical order, as long as the corresponding parts
match.
There are six ways to name any triangle by its vertices. You can start at any of the three vertices and then name
the triangle’s other vertices by progressing clockwise or counter-clockwise around the diagram. This process would
give six different possible names for a triangle. For the diagram in Example 2, we could also express the congruence
statement as follows:
∆DEF ∼
= ∆T SR
Observe that this time we named the triangle on the left of the diagram first. The order does not matter. Both of
these congruence statements are accurate because corresponding sides and angles are aligned within the statement.
Reading Check:
1. In the diagram below, the two triangles are congruent. Create a congruence statement (using the geometry
symbols∆ and ∼
= ) for the diagram. Remember to be careful with corresponding angles and sides!
2. For the same diagram above, create
3. For the same diagram above, create a
The Third Angle Theorem
Previously, you studied the Triangle Sum Theorem, which states that the sum of the measures of the interior angles
in a triangle will always be equal to 180◦ . This information is useful when showing congruence.
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Concept 1. Corresponding Parts (CPCTC) and Identifying Minimal Conditions
The Triangle Sum Theorem says that the measures of all three angles inside a triangle add up to ____________.
As you practiced, if you know the measures of two angles within a triangle, there is only one possible measurement
of the third angle. Thus, if you can prove two corresponding angle pairs congruent, the third pair is also guaranteed
to be congruent.
Third Angle Theorem
If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles are also
congruent.
This means that once you know two congruent angle pairs, then the last angle pair is also _______________________________.
Example 3
Identify whether or not the missing angles in the triangles below are congruent.
One triangle looks bigger than the other. Does that mean all of its angles are bigger? To identify whether or not the
third angles are congruent, you must first find their measures.
Start with the triangle on the left. Since you know two of the angles in the triangle, you can use the triangle sum
theorem to find the missing angle. In ∆WV X we know:
m6 W + m6 V + m6 X = 180◦
80◦ + 35◦ + m6 X = 180◦
115◦ + m6 X = 180◦
m6 X = 65◦
The missing angle of the triangle on the left measures 65◦ . Repeat this process for the triangle on the right:
m6 C + m6 A + m6 T = 180◦
80◦ + 35◦ + m6 T = 180◦
115◦ + m6 T = 180◦
m6 T = 65◦
Since the measure of both angles is 65◦ , 6 X ∼
= 6 T.
Reading Check:
1. Fill in the blanks:
TheThird Angle ____________ says that if two angles in one triangle are ____________ to two angles in another
triangle, then the third pair of angles are also ____________.
2. In Example 3 (on the previous page), create
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