CBSE Class 10 Maths Chapter 6 Triangles Notes

Key Theorems and Concepts to Remember

• Basic Proportionality Theorem (BPT) / Thales Theorem: If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides proportionally.
• Converse of BPT: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
• Similarity Criteria for Triangles: Two triangles are similar if their corresponding angles are equal (AAA/AA similarity) OR their corresponding sides are in the same ratio (SSS similarity) OR one angle is equal and the sides including it are proportional (SAS similarity).
• Area of Similar Triangles Theorem: The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
• Pythagoras Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (hypotenuse² = base² + perpendicular²).
• Converse of Pythagoras Theorem: If in a triangle, the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.
• Altitude to Hypotenuse: If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then the triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

Essential Definitions

Similar Figures

Two figures are said to be similar if they have the same shape but not necessarily the same size. For polygons, corresponding angles must be equal and corresponding sides must be proportional.

Congruent Figures

Two figures are congruent if they have exactly the same shape and the same size. All corresponding parts (sides and angles) are equal.

Basic Proportionality Theorem (BPT)

Also known as Thales Theorem. States that if a line parallel to one side of a triangle intersects the other two sides at distinct points, then it divides the two sides in the same ratio.

Corresponding Sides

Sides of two figures that are in the same relative position and are opposite to corresponding angles.

Corresponding Angles

Angles of two figures that are in the same relative position and are opposite to corresponding sides.

Hypotenuse

The longest side of a right-angled triangle, opposite the right angle.

Altitude

A line segment from a vertex of a triangle perpendicular to the opposite side.

Understanding Similarity of Triangles

The concept of similarity is central to Chapter 6. Two triangles are similar if their corresponding angles are equal and their corresponding sides are in the same ratio (proportional). This means one triangle is essentially an enlarged or reduced version of the other, maintaining the same shape. There are three primary criteria to prove that two triangles are similar:

1. AAA (Angle-Angle-Angle) Similarity Criterion: If in two triangles, corresponding angles are equal, then their corresponding sides are proportional, and hence the triangles are similar. A common shortcut is AA Similarity: If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. This is because the third angle will automatically be equal (sum of angles in a triangle is 180°).

1. SSS (Side-Side-Side) Similarity Criterion: If the corresponding sides of two triangles are proportional, then their corresponding angles are equal, and hence the two triangles are similar. This means if AB/DE = BC/EF = CA/FD, then ΔABC ~ ΔDEF.

1. SAS (Side-Angle-Side) Similarity Criterion: If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar. For example, if ∠A = ∠D and AB/DE = AC/DF, then ΔABC ~ ΔDEF. It's crucial that the equal angle is included between the proportional sides. Mastering these criteria is key to solving most problems related to similar triangles, including those involving heights, distances, and areas. Remember that similarity is a fundamental concept for understanding geometric transformations and scaling.

Congruence vs. Similarity: Key Differences

Quick Solved Examples

• {"title":"Example 1: Basic Proportionality Theorem","bodyMarkdown":"In ΔABC, DE || BC. If AD = 3 cm, DB = 6 cm, and AE = 4 cm, find EC.\n\nSolution: By BPT, AD/DB = AE/EC.\n3/6 = 4/EC\n1/2 = 4/EC\nEC = 8 cm."}
• {"title":"Example 2: Pythagoras Theorem","bodyMarkdown":"A ladder 10 m long reaches a window 8 m above the ground. Find the distance of the foot of the ladder from the base of the wall.\n\nSolution: Let the ladder be the hypotenuse (c=10m), the height of the window be one leg (a=8m), and the distance from the wall be the other leg (b).\nBy Pythagoras Theorem: a² + b² = c²\n8² + b² = 10²\n64 + b² = 100\nb² = 36\nb = 6 m. The distance is 6 m."}

Exam Tip: Avoiding Common Mistakes

When proving similarity or applying theorems, always ensure you state the correct criterion (AAA, SSS, SAS for similarity) or theorem (BPT, Pythagoras). For BPT, clearly identify the parallel line and the sides it intersects. For Pythagoras, ensure the triangle is right-angled. When writing proofs, maintain a clear, step-by-step logical flow, justifying each step with a theorem or axiom. Be careful with corresponding parts; ensure you match corresponding vertices, sides, and angles correctly. A common error is mixing up ratios of sides when using the Area of Similar Triangles Theorem – it's the square of the ratio of sides!

Quick Revision Checks

• Q: What are the conditions for two triangles to be similar by SAS criterion? A: One angle of a triangle must be equal to one angle of the other triangle, AND the sides including these angles must be proportional.
• Q: If ΔABC ~ ΔPQR and AB/PQ = 2/3, what is the ratio of their areas? A: The ratio of their areas is the square of the ratio of their corresponding sides. So, Area(ΔABC) / Area(ΔPQR) = (AB/PQ)² = (2/3)² = 4/9.
• Q: State the Converse of Basic Proportionality Theorem. A: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
• Q: In a right-angled triangle, if the legs are 5 cm and 12 cm, what is the length of the hypotenuse? A: Using Pythagoras Theorem: hypotenuse² = 5² + 12² = 25 + 144 = 169. So, hypotenuse = √169 = 13 cm.

Frequently Asked Questions