Geometry
About 6 min · Quantitative Aptitude (Basic Arithmetic Skill) · English
Time worked out from the lesson's 385 words at 150 words a minute, plus a minute for each question.
This topic tests your understanding of lines, angles, triangles, circles, and the relationships between these shapes through congruence, similarity, and tangent properties. Unlike mensuration (which focuses on area and volume calculations), geometry questions require you to apply theorems, properties, and logical reasoning to find unknown angles, lengths, or prove relationships.
Key ideas
- Lines and Angles: When two lines intersect, vertically opposite angles are equal. When a transversal cuts two parallel lines, corresponding angles are equal, alternate interior angles are equal, and co-interior angles sum to 180°. These properties are the foundation for angle-chasing problems.
- Triangle Fundamentals: The sum of interior angles in any triangle is 180°. An exterior angle equals the sum of the two non-adjacent interior angles.
- Congruence vs Similarity: Congruent triangles have identical shape and size (all corresponding sides and angles equal). Similar triangles have the same shape but not necessarily the same size (corresponding angles equal, corresponding sides proportional). Know the criteria: SSS, SAS, ASA, AAS for congruence; AA, SSS, SAS for similarity.
- Circle Properties: A tangent is perpendicular to the radius at the point of contact. Two tangents drawn from an external point are equal in length. Angles subtended by the same arc at the circumference are equal; the angle at the centre is double the angle at the circumference.
- Pythagoras Theorem: In a right-angled triangle, (hypotenuse)² = (base)² + (perpendicular)². Memorise Pythagorean triplets: (3,4,5), (5,12,13), (8,15,17), (7,24,25) to save calculation time.
Quick reference
- Triangle angle sum = 180°; exterior angle = sum of opposite two interior angles.
- Pythagoras: c² = a² + b² (right triangle only). Know triplets (3,4,5), (5,12,13), (8,15,17).
- Tangent ⊥ radius at point of contact; two tangents from external point are equal.
- Angle at centre = 2 × angle at circumference (same arc). Angle in semicircle = 90°.
- Parallel lines + transversal: corresponding angles equal, alternate interior angles equal, co-interior angles sum to 180°.
Check yourself: 3 questions
1. In a triangle ABC, angle A = 50° and angle B = 60°. What is the measure of angle C?
2. Two parallel lines are cut by a transversal. If one of the alternate interior angles is 65°, what is the measure of the other alternate interior angle?
3. In triangle PQR, PQ = PR and angle Q = 65°. What is the measure of angle P?
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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