Geometry — Triangles, Quadrilaterals, Congruence and Similarity

Overview

Geometry forms a substantial portion of the Mathematics section in Bihar TET Paper II. This topic tests both conceptual understanding and problem-solving ability — expect questions on properties of triangles and quadrilaterals, congruence criteria, similarity theorems, and area relationships. Mastery here directly helps in mensuration problems as well.

For Bihar TET, focus on the standard results: angle-sum properties, congruence rules (SSS, SAS, ASA, AAS, RHS), similarity criteria (AA, SSS, SAS), and the Basic Proportionality Theorem (Thales theorem). Questions typically involve finding unknown angles, proving triangles congruent or similar, and applying properties to calculate lengths or areas. A clear grasp of definitions and theorems, combined with practice on standard problem types, ensures scoring in this section.

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Key Concepts

**Triangle angle-sum property**: The sum of interior angles of any triangle is 180°. The exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

**Types of triangles**: Classified by sides (scalene, isosceles, equilateral) and by angles (acute, right, obtuse). An equilateral triangle has all angles equal to 60°.

**Quadrilateral angle-sum property**: The sum of interior angles of any quadrilateral is 360°.

**Special quadrilaterals hierarchy**: Square → Rectangle → Parallelogram → Quadrilateral; Square → Rhombus → Parallelogram. A square has all properties of both rectangle and rhombus.

**Congruence of triangles**: Two triangles are congruent if they have exactly the same shape and size — all corresponding sides and angles are equal.

**Similarity of triangles**: Two triangles are similar if they have the same shape but not necessarily the same size — corresponding angles are equal and corresponding sides are in proportion.

**Basic Proportionality Theorem (BPT / Thales theorem)**: If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.

**Area relationship in similar triangles**: If two triangles are similar with sides in ratio k, then their areas are in ratio k².

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Formulas / Key Facts

**Triangle Properties**

Sum of angles = 180°
Exterior angle = Sum of two remote interior angles
In a right triangle: hypotenuse² = base² + perpendicular² (Pythagoras theorem)

**Quadrilateral Properties**

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Q1 · Geometry · EASY
In triangle ABC, AB = 5 cm, BC = 12 cm and AC = 13 cm. What type of triangle is ABC?
Q2 · Geometry · MEDIUM
Two triangles ABC and PQR are congruent by SAS criterion. If AB = 6 cm, AC = 8 cm, angle A = 60°, and PQ = 6 cm, PR = 8 cm, then what is the measure of angle P?
Q3 · Geometry · MEDIUM
In a quadrilateral ABCD, the diagonals AC and BD intersect at point O. If AO = 3 cm, OC = 3 cm, BO = 4 cm, OD = 4 cm, and angle AOB = angle COD, then what type of quadrilateral is ABCD?
Q4 · Geometry · HARD
Two similar triangles ABC and PQR have their corresponding sides in the ratio 3:5. If the area of triangle ABC is 27 square cm, what is the area of triangle PQR in square cm?
Q5 · Geometry · MEDIUM
In a triangle ABC, if ∠A = 60° and ∠B = 70°, then ∠C is:

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Notes generated on 27 Jun 2026

Geometry

Geometry — Triangles, Quadrilaterals, Congruence and Similarity

Overview

Key Concepts

Formulas / Key Facts

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