How To Test Congruent and Similar Triangles

Types of triangles:

Triangles can be classified according to the relative lengths of their sides:
• In an equilateral triangle, all sides are of equal length. An equilateral triangle is also an equiangular polygon, i.e. all its internal angles are equal—namely, 60°; it is a regular polygon
• In an isosceles triangle, two sides are of equal length. An isosceles triangle also has two congruent angles (namely, the angles opposite the congruent sides). An equilateral triangle is an isosceles triangle, but not all isosceles triangles are equilateral triangles.
• In a scalene triangle, all sides have different lengths. The internal angles in a scalene triangle are all different.

Triangles can also be classified according to the their internal angles, described below using degrees of arc.
• A right triangle (or right-angled triangle, has one 90° internal angle (a right angle). The side opposite to the right angle is the hypotenuse; it is the longest side in the right triangle. The other two sides are the legs or catheti (singular: cathetus) of the triangle.
• An obtuse triangle has one internal angle larger than 90° (an obtuse angle).
• An acute triangle has internal angles that are all smaller than 90° (three acute angles). An equilateral triangle is an acute triangle, but not all acute triangles are equilateral triangles.
• An oblique triangle has only angles that are smaller or larger than 90°. It is therefore any triangle that is not a right triangle.

Congruent and Similar Triangles:

Rules in Geometry to tests for congruent triangles:
a. SAS Test – Side-Angle-Side
b. SSS Test - Side-Side-Side
c. ASA Test - Angle-Side-Angle d. AAS Test - Angle-Angle-Side

a.) Side-Angle-Side
The rule states that if two sides and the included angle are congruent to two sides and the included angle of a second triangle, the two triangles are congruent. An included angle is an angle created by two sides of a triangle.

b.) Side-Side-Side
The rule states that if three sides of one triangle are congruent to three sides of a second triangle, the two triangles are congruent.

c.) Angle-Side-Angle
The rule states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. An included side is a side that is common to (between) two angles. For example, in the figure used in the problem below, segment AB is an included side to angles A and B.

d.) Angle-Angle-Side
The rule states that if two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, the two triangles are congruent.

CPCTC
When two triangles are congruent, all six pairs of corresponding parts (angles and sides) are congruent. This statement is usually simplified as corresponding parts of congruent triangles are congruent, or CPCTC for short.

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