Congruent Triangles
Chapter Summary
Two triangles are said to be congruent ($\cong$) if they are identical in both shape and size, meaning their corresponding sides and angles are equal. In board exams, proving triangle congruence is a fundamental skill. To prove congruence without having to measure all six parts (three sides and three angles), you can use standard Postulates and Theorems: SAS (Side-Angle-Side), SSS (Side-Side-Side), ASA (Angle-Side-Angle), and AAS (Angle-Angle-Side). For right-angled triangles, the special H.S (Hypotenuse-Side) postulate is frequently applied and tested in both short and long questions.
A major portion of this chapter is dedicated to important theoretical theorems that often appear as long-form questions (usually worth 8 marks in the geometry section). Key theorems include proving that in any triangle, the measure of an exterior angle is greater than either of its corresponding non-adjacent interior angles, and that if two sides of a triangle are unequal, the longer side has a greater angle opposite to it. You must also master the properties of isosceles triangles, specifically that the angles opposite to congruent sides are equal, and its converse.
To score full marks in board exams, strict adherence to the proof format is essential. Every geometric proof must be neatly divided into Given, To Prove, Construction (if applicable), and Proof (comprising Statements and Reasons). Pay special attention to identifying common sides, vertically opposite angles, or alternate interior angles in overlapping or adjacent triangles, as these are frequently the hidden clues needed to complete your congruence proofs successfully.
Important Questions
- Prove that if in any two triangles, two sides and their included angle of one triangle are congruent to the corresponding two sides and their included angle of the other, then the triangles are congruent. (SAS Postulate)8 marks
- Prove that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.5 marks
- Prove that if two angles of a triangle are congruent, then the sides opposite to them are also congruent.5 marks
- Prove that in any right-angled triangle, the square of the length of hypotenuse is equal to the sum of the squares of the lengths of the other two sides. (Pythagoras Theorem)8 marks
- Prove that the bisectors of the angles of a triangle are concurrent.5 marks
- Prove that the perpendicular bisectors of the sides of a triangle are concurrent.5 marks
- Prove that to a greater angle of a triangle, the side opposite to it is longer than the side opposite to the smaller angle.5 marks