Geometry and Trigonometry
About 6 min · Mathematics · English
Time worked out from the lesson's 399 words at 150 words a minute, plus a minute for each question.
This topic tests your ability to visualize spatial relationships, apply angle theorems, and solve practical height-distance problems using trigonometric ratios.
Key ideas
- Angle relationships: Vertically opposite angles are equal; linear pair sums to 180°; angles around a point sum to 360°. Parallel lines cut by a transversal create equal corresponding angles, equal alternate interior angles, and co-interior angles summing to 180°.
- Triangle properties: Sum of interior angles is 180°; exterior angle equals sum of two opposite interior angles. In isosceles triangles, angles opposite equal sides are equal. The area of a triangle equals ½ × base × height.
- Congruence and similarity: Triangles are congruent if SSS (all sides equal), SAS (two sides and included angle), ASA (two angles and included side), or RHS (right angle, hypotenuse, and one side). Similar triangles have proportional sides and equal corresponding angles.
- Circle theorems: Angle in a semicircle is 90°; angles subtended by the same arc at the circumference are equal; angle at the centre is twice the angle at the circumference; opposite angles in a cyclic quadrilateral sum to 180°.
- Trigonometric ratios: In a right triangle with angle θ, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. The reciprocal ratios are cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
Quick reference
- Triangle angle sum = 180°; exterior angle = sum of opposite interior angles
- Pythagoras: a² + b² = c² in right triangle
- sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ
- Complementary angles: sin(90° − θ) = cos θ; tan(90° − θ) = cot θ
- tan θ = sin θ / cos θ; tan θ = opposite/adjacent in right triangle
Check yourself: 3 questions
1. In a right-angled triangle, one of the acute angles is 30°. If the side opposite to this angle is 7 cm, what is the length of the hypotenuse?
2. Two angles of a triangle are 45° and 65°. What is the measure of the third angle?
3. A ladder 15 m long reaches a window which is 9 m above the ground on one side of a street. Keeping its foot at the same point, the ladder is turned to the other side of the street to reach a window 12 m high. What is the width of the street?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 18 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
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