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Math-2
Lesson 5-2 (Triangle Congruence)
Vocabulary: Congruence, corresponding angles, corresponding sides, CPCTC, included angle, included side, congruence
statement, angle congruence, segment congruence,
Lesson Objectives:
1. Segments are congruent if they have the same measure.
2. The symbol

means “congruent”
3. Segments are congruent if they have the same measure (length).
In the triangles above: AC  DF therefore A C  D F .
4. The “included side” is the side of the triangle that is common to two angles of the triangle. In the triangles above;
A C is the included side of A and C . Notice that the end points of the included side are the vertices of the angles
that include the side.
5. The “included angle” is the angle made of the two segments that are part of the sides of the angle. In the triangles
above; A is the included angle of side A C and A B . Notice that the common end points of the two sides is the
vertex of the included angle.
6. Two triangles are congruent if there are three pairs of congruent angles and three pairs of congruent sides between
the two triangles. RST  ZYX because: R  Z , S  Y , T  X , RS = ZY, ST = YX, and TR = XZ
7. The order of the letters in the triangles matters in this congruence statement RST  ZYX because
R corresponds to Z , S corresponds to Y , and T corresponds to X
8. If two triangles are congruent then we say “CPCTC” meaning “corresponding parts of congruent triangles are
congruent”.
9. If two triangles are congruent, then three pairs of angles are congruent and three pairs of sides are congruent
between the two triangles.
10. We can prove that two triangles are congruent by linking only a combination or three pairs of angle/sides if they are
arranged as:
a) Side-Side-Side (SSS): all three pairs of sides are congruent.
b) Side-Angle-Side (SAS): two pairs of sides and the included angles are congruent.
c) Angle-Side-Angle (ASA): two pairs of angles and the included side are congruent.
d) Angle-Angle-Side (AAS): two pairs of angles and the corresponding non-included side are congruent.
11. Angle-Angle-Angle (AAA) does NOT prove congruence because the triangles may not have the same size.
12. Side-Side-Angle (SSA) does NOT prove congruence.
13. We label triangles with “tic marks” through sides to indicate sides with the same measure. A “two tic mark side” in
one triangle has the same measure (and is therefore congruent) to a “two tic mark side” in another triangle.
14. We label angles with the same measure with an arc inside of each angle with the same number of “tic marks”
through the arcs.
15. Given two angle measures in a triangle, we can find the measure of the third angle by substituting the known angle
measures into the equation given in the Triangle Angle Sum Theorem.
Theorems:
1. The Triangle Angle Sum Theorem: If A , B , and C are the interior angles of a triangle, then
mA  mB  C  180
2. Side-Side-Side (SSS) Congruence Theorem: if there are three pairs on congruent sides between two
triangles then the triangles are congruent.
3. Side-Angle-Side (SAS) Congruence Theorem: if there are two pairs of congruent sides between two
triangles and the included angle is also congruent between the two triangles, then the two triangles are
congruent.
4. Angle-Side-Angle (ASA) Congruence Theorem: If there are two pairs of congruent angles between two
triangles, and the included side is also congruent between the two triangles, then the two triangles are
congruent.
5. Angle-Angle-Side (AAS Congruence Theorem: If there are two pairs of congruent angles between two
triangles and the corresponding non-included side is also congruent, then the two triangles are
congruent.