
SideAngleSide (SAS) Congruence Postulate
If two sides (CA and CB) and the included angle ( BCA ) of a triangle are congruent to the corresponding two sides (C'A' and C'B') and the included angle (B'C'A') in another triangle, then the two triangles are congruent.
Example 1: Let ABCD be a parallelogram and AC be one of its diagonals. What can you say about triangles ABC and CDA? Explain your answer.
Solution to Example 1:
 In a parallelogram, opposite sides are congruent. Hence sides
BC and AD are congruent, and also sides AB and CD are congruent.
 In a parallelogram opposite angles are congruent. Hence angles
ABC and CDA are congruent.
 Two sides and an included angle of triangle ABC are congruent to two corresponding sides and an included angle in triangle CDA. According to the above postulate the two triangles ABC and CDA are congruent.
SideSideSide (SSS) Congruence Postulate
If the three sides (AB, BC and CA) of a triangle are congruent to the corresponding three sides (A'B', B'C' and C'A') in another triangle, then the two triangles are congruent.
Example 2: Let ABCD be a square and AC be one of its diagonals. What can you say about triangles ABC and CDA? Explain your answer.
Solution to Example 2:
 In a square, all four sides are congruent. Hence sides
AB and CD are congruent, and also sides BC and DA are congruent.
 The two triangles also have a common side: AC. Triangles ABC has three sides congruent to the corresponding three sides in triangle CDA. According to the above postulate the two triangles are congruent. The triangles are also right triangles and isosceles.
AngleSideAngle (ASA) Congruence Postulate
If two angles (ACB, ABC) and the included side (BC) of a triangle are congruent to the corresponding two angles (A'C'B', A'B'C') and included side (B'C') in another triangle, then the two triangles are congruent.
Example 3: ABC is an isosceles triangle. BB' is the angle bisector. Show that triangles ABB' and CBB' are congruent.
Solution to Example 3:
 Since ABC is an isosceles triangle its sides AB and BC are congruent and also its angles BAB' and BCB' are congruent. Since BB' is an angle bisector, angles ABB' and CBB' are congruent.
Two angles and an included side in triangles ABB' are congruent to two corresponding angles and one included side in triangle CBB'. According to the above postulate triangles ABB' and CBB' are congruent.
AngleAngleSide (AAS) Congruence Theorem
If two angles (BAC, ACB) and a side opposite one of these two angles (AB) of a triangle are congruent to the corresponding two angles (B'A'C', A'C'B') and side (A'B') in another triangle, then the two triangles are congruent.
Example 4: What can you say about triangles ABC and QPR shown below.
Solution to Example 4:
 In triangle ABC, the third angle ABC may be calculated using the theorem that the sum of all three angles in a triangle is equal to 180 derees. Hence
angle ABC = 180  (25 + 125) = 30 degrees
 The two triangles have two congruent corresponding angles and one congruent side.
angles ABC and QPR are congruent. Also angles BAC and PQR are congruent. Sides BC and PR are congruent.
 Two angles and one side in triangle ABC are congruent to two corresponding angles and one side in triangle PQR. According to the above theorem they are congruent.
Right Triangle Congruence Theorem
If the hypotenuse (BC) and a leg (BA) of a right triangle are congruent to the corresponding hypotenuse (B'C') and leg (B'A') in another right triangle, then the two triangles are congruent.
Example 5: Show that the two right triangles shown below are congruent.
Solution to Example 5:
 We first use Pythagora's theorem to find the length of side AB in triangle ABC.
length of AB = sqrt [5^{ 2}  3^{ 2}] = 4
 One leg and the hypotenuse in triangle ABC are congruent to a corresponding leg and hypotenuse in the right triangle A'B'C'. According to the above theorem, triangles ABC and B'A'C' are congruent.
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