Learning Objectives
- Understand congruence of triangles and criteria for congruence
- Apply SAS, ASA, AAS, SSS, and RHS congruence rules
- Learn properties of isosceles and equilateral triangles
- Understand inequalities in triangles
Key Concepts
Congruence of Triangles
Two triangles are congruent if they are exact copies of each other โ all corresponding sides and angles are equal. The symbol for congruence is โ .
Criteria for Congruence
SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of the other triangle, the triangles are congruent.
ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to two angles and the included side of the other triangle, the triangles are congruent.
AAS (Angle-Angle-Side): If two angles and one non-included side of one triangle are equal to the corresponding parts of the other, the triangles are congruent.
SSS (Side-Side-Side): If three sides of one triangle are equal to the three sides of the other triangle, the triangles are congruent.
RHS (Right angle-Hypotenuse-Side): If the hypotenuse and one side of a right triangle are equal to the hypotenuse and one side of another right triangle, they are congruent.
Properties of Isosceles Triangle
Theorem: Angles opposite to equal sides of an isosceles triangle are equal.
Converse: If two angles of a triangle are equal, then the sides opposite to them are also equal.
Inequalities in Triangles
Theorem: In a triangle, the side opposite to the larger angle is longer.
Converse: In a triangle, the angle opposite to the longer side is larger.
Triangle Inequality: The sum of any two sides of a triangle is always greater than the third side.
Summary
Triangles can be proved congruent using five criteria: SAS, ASA, AAS, SSS, and RHS. In an isosceles triangle, equal sides have equal opposite angles and vice versa. In any triangle, the longer side is opposite the larger angle, and the sum of any two sides exceeds the third side.
Important Terms
- Congruent Triangles: Triangles with equal corresponding sides and angles
- Isosceles Triangle: A triangle with at least two equal sides
- Equilateral Triangle: A triangle with all three sides equal (all angles = 60ยฐ)
- CPCT: Corresponding Parts of Congruent Triangles (are equal)
Quick Revision
- Five congruence criteria: SAS, ASA, AAS, SSS, RHS
- AAA is NOT a congruence criterion (it shows similarity, not congruence)
- Equal sides โ equal opposite angles (isosceles triangle property)
- The sum of two sides > third side (triangle inequality)
- The side opposite the greater angle is the longer side