Congruence of Triangles
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Definition: If triangle ABC is congruent to triangle DEF, then all corresponding angles are congruent,
and all corresponding sides are congruent. In this example:
Angle A is congruent to Angle D
Angle B is congruent to Angle E
Angle C is congruent to Angle F
Segment AB is congruent to Segment DE
Segment AC is congruent to Segment DF
Segment BC is congruent to Segment EF
A
B
D
C
E
F
Congruence Properties of Triangles:
1) Side-Angle-Side (SAS) Congruence: Given triangles ABC and DEF, if two sides and the angle
formed by the two sides (called the included angle) of triangle ABC are congruent to the
corresponding two sides and the included angle on triangle DEF, then the triangles are congruent
2) Angle-Side-Angle (ASA) Congruence: Given triangles ABC and DEF, if two angles and the
included side of triangle ABC are congruent to the corresponding two angles and the included
side of triangle DEF, then the triangles are congruent.
3) Side-Side-Side (SSS) Congruence: Given triangles ABC and DEF, if the three sides of triangle ABC
are congruent to their corresponding sides of triangle DEF, then the triangles are congruent.
4) Angle-Angle-Side (AAS) Congruence: Given triangles ABC and DEF, if two of the angles and a
non-included side of triangle ABC are congruent to two angles and the corresponding nonincluded side of triangle DEF, then the triangles are congruent.
© Dr Brian Beaudrie and Dr Barbara Boschmans
Congruence of Triangles
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Theorem: Opposite sides and opposite angles of a parallelogram are congruent.
Theorem: The Triangle Inequality: the sum of any two sides of a triangle must be greater than the
length of its third side.
Theorem: Angles opposite the congruent sides of an isosceles triangle are congruent.
Definition: If triangle ABC is congruent to triangle DEF, then all corresponding angles are congruent,
and all corresponding sides are congruent.
A
D
Corresponding
Corresponding
Angles:
sides:
C
F
A  D
B  E
C  F
AB  DE
AC  DF
BC  EF
B
E
Note: There are certain conditions that, if met, imply that all corresponding angles and sides are
congruent of two triangles.
Examples
1. Given that ABC  XYZ , complete the following statements.
a. CBA   _______
c. CAB   _______
b. ACB   _______
d. BAC   _______
2. Suppose you know that for ABC and JKL , B  K .
a. To show the two triangles are congruent by SAS, what more would you need to know?
b. To show the two triangles are congruent by ASA, what more would you need to know (two
different solutions…list both)?
3. If possible, draw two non-congruent triangles that satisfy the following conditions. If not possible,
explain why not.
a) Three pairs of corresponding parts are congruent.
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Congruence of Triangles
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b) Four pairs of corresponding parts are congruent.
c) Five pairs of corresponding parts are congruent.
4. Could any of the congruence properties (ASA, SAS, SSS) be used to show if the following pairs of
triangles are congruent? If so, state which property can be used. If not, state “not congruent”.
a.
90o
b.
70o
40o
5
5
5
5
c.
7
65o
65o
65o
65o
7
d.
e)
5
8
5
8
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Congruence of Triangles
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5. From Point A, draw a line exactly 2 inches long that forms exactly a 55o angle. Label the point at
the end of your new line segment point C.
Connect point B and point C by drawing a line segment.
Q: What is the measure of angle B?
A
B
Q: What is the measure of angle C?
Q: What is the length of segment BC?
Compare your answers with those around you.
What do you notice?
6. From Point A, draw a line (longer than 3 inches) that forms exactly a 50o angle with segment AB.
From Point B, draw a line that forms exactly a 60o angle with segment AB and make sure it
intersects the other segment you just drew. Label the point of intersection point C.
Q: What is the length of segment AC?
A
B
Q: What is the length of segment BC?
Q: What is the measure of angle C?
Compare your answers with those around you.
What do you notice?
© Dr Brian Beaudrie and Dr Barbara Boschmans