Triangles — Class 10 (Advanced Notes)

Congruence • Similarity • Pythagoras • Area relations • Inequalities • Practice
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1. Definitions & Types

A triangle is a three-sided polygon formed by joining three non-collinear points. Types (by sides):

  • Equilateral (all sides equal)
  • Isosceles (two sides equal)
  • Scalene (all sides different)

By angles: acute, right, obtuse.

2. Congruence of Triangles

Two triangles are congruent if all corresponding sides and angles are equal. Common congruence criteria:

  • SSS: three sides equal
  • SAS: two sides and included angle equal
  • ASA: two angles and included side equal
  • AAS: two angles and a non-included side equal
  • RHS (right triangle): hypotenuse and one side equal

Use congruence to prove base angles of an isosceles triangle are equal (SAS on halves after drawing altitude).

3. Similarity of Triangles

Triangles are similar if their corresponding angles are equal and corresponding sides are in proportion. Common tests:

  • AA: two angles equal ⇒ triangles similar
  • SSS: corresponding sides in same ratio ⇒ similar
  • SAS: two sides proportional and included angle equal

Important application: corresponding sides ratio = scale factor; use for indirect measurement problems.

4. Pythagoras Theorem & Applications

In a right triangle with legs a, b and hypotenuse c:

a² + b² = c²

Use Pythagoras for finding missing sides, proving right angles, and in converse form: if a² + b² = c², triangle is right-angled.

Find hypotenuse of right triangle with legs 6 and 8: c = √(36+64) = √100 = 10.

5. Important Properties & Theorems

  • Basic Proportionality Theorem (Thales): If a line parallel to one side of a triangle intersects the other two sides, it divides them proportionally.
  • Triangle Inequality: Sum of any two sides > third side.
  • Midpoint Theorem: Segment joining midpoints of two sides is parallel to third side and half its length.
  • Exterior Angle Theorem: Exterior angle = sum of opposite interior angles.

6. Area Relations

Area of triangle = (1/2) × base × height. Other useful relations:

  • Area using two sides and included angle: (1/2)ab sin C.
  • Triangles on same base and between same parallels have equal area.
  • Area ratios follow square of similarity ratio for similar triangles.

7. Solved Examples

Example 1. Prove that in triangle ABC, if AD is median and AB = AC, then AD ⟂ BC.

Sketch: AB = AC ⇒ triangle is isosceles. Median to base is also altitude and perpendicular bisector. Use congruence of ABD and ACD (SAS) to conclude AD ⟂ BC.

Example 2. A line through midpoints of two sides of triangle ABC is parallel to third side — prove and compute length relation.

Use midpoint theorem: length equals half of third side.

8. Practice Questions

  1. In ΔABC, AB = AC and AD is altitude. Prove BD = DC.
  2. Prove that the sum of angles in a triangle = 180° using parallel lines.
  3. A line parallel to BC intersects AB at D and AC at E such that AD/DB = AE/EC = 2. If BC = 9 cm, find DE.
  4. Find area of Δ with sides 13, 14, 15.

9. Exam Tips & Strategy

  • Label diagrams carefully; write known equalities on the figure.
  • Use congruence for exact equalities and similarity for proportional reasoning.
  • For area and length problems, look for similar triangles — they simplify ratios.
  • Practice classic theorems (Thales, midpoint, exterior angle) for quick recall under exam pressure.