Quick Summary
The Geometry chapter in Class 8 covers triangle properties, congruence of triangles, and properties of special quadrilaterals. Congruence proofs are new at this level and require careful logical presentation. This chapter typically carries 10–15 marks in the annual exam.
Key Definitions
- Congruent Figures: Two figures are congruent if they have exactly the same shape and size.
- CPCTC: Corresponding Parts of Congruent Triangles are Congruent.
- Parallelogram: A quadrilateral with both pairs of opposite sides parallel.
- Rhombus: A parallelogram with all four sides equal.
- Rectangle: A parallelogram with all angles equal to 90°.
- Square: A rectangle with all sides equal.
Properties of Triangles
| Property | Statement |
|---|---|
| Angle sum | Sum of angles = 180° |
| Exterior angle | Exterior angle = sum of two non-adjacent interior angles |
| Triangle inequality | Sum of any two sides > third side |
| Isosceles triangle | Base angles are equal |
| Equilateral triangle | All angles = 60° |
Triangle Congruence Criteria
| Criterion | Condition |
|---|---|
| SSS | All three sides equal |
| SAS | Two sides and included angle equal |
| ASA | Two angles and included side equal |
| AAS | Two angles and non-included side equal |
| RHS | Right angle, hypotenuse, and one side equal |
Note: AAA is NOT a congruence criterion (it gives similarity, not congruence).
Properties of Quadrilaterals
Parallelogram
- Opposite sides are equal and parallel.
- Opposite angles are equal.
- Consecutive angles are supplementary.
- Diagonals bisect each other.
Rhombus
- All sides are equal.
- Diagonals bisect each other at right angles.
- Diagonals bisect the vertex angles.
Rectangle
- All angles = 90°.
- Diagonals are equal in length and bisect each other.
Square
- All sides equal; all angles = 90°.
- Diagonals are equal, bisect each other at right angles, and bisect the vertex angles.
Important Questions
One Mark
- State the angle sum property of a triangle.
- Which congruence criterion applies when two sides and the included angle are equal?
- The angles of a triangle are 45°, 65°, and x°. Find x.
- Name the quadrilateral whose diagonals bisect each other at right angles.
- In triangle ABC ≅ triangle PQR, which side corresponds to BC?
Short Answer (3 marks)
- In triangle ABC, AB = 5 cm, BC = 7 cm, and angle B = 60°. In triangle PQR, PQ = 5 cm, QR = 7 cm, and angle Q = 60°. Are the triangles congruent? State the criterion.
- List four properties of a parallelogram.
- Prove that the angles at the base of an isosceles triangle are equal.
Long Answer (5 marks)
- Two triangles ABC and DEF have AB = DE, BC = EF, and angle B = angle E. Prove they are congruent and state all corresponding equal parts.
- ABCD is a parallelogram. Prove that AB = CD and AD = BC using triangle congruence.
Common Mistakes
- Stating AAA as a congruence criterion — it is not.
- Not stating the congruence criterion and CPCTC when writing proofs.
- Confusing properties of rhombus and rectangle in answers.
Revision Checklist
- [ ] I know all five congruence criteria and can apply them.
- [ ] I can find missing angles using the angle sum and exterior angle properties.
- [ ] I know the properties of parallelogram, rhombus, rectangle, and square.
- [ ] I can write a congruence proof with proper steps.
- [ ] I have solved all textbook exercises and corrected mistakes.
Frequently Asked Questions
The four congruence criteria are: SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), and RHS (Right angle-Hypotenuse-Side for right triangles).