Geometry: Triangles, Quadrilaterals, Congruence and Similarity

Overview

Geometry forms a significant portion of the Mathematics section in CG TET Paper II, typically contributing 3–5 questions. This topic tests your understanding of shapes, their properties, and logical reasoning—skills essential for teaching upper primary students (Classes VI–VIII).

The scope covers two major plane figures (triangles and quadrilaterals) along with two fundamental concepts for comparing figures (congruence and similarity). Mastery here requires memorising properties and theorems, but more importantly, understanding *why* these properties hold and *how* to apply them in problem-solving.

For exam success, focus on: triangle classification and angle properties, quadrilateral properties (especially parallelograms), the five congruence criteria, and the distinction between congruence and similarity. These concepts also appear in EVS-integrated questions about measurement and spatial reasoning.

Key Concepts

**Triangle Angle Sum Property**: The sum of interior angles of any triangle equals 180°. This is the foundation for solving most triangle problems.

**Exterior Angle Theorem**: An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

**Triangle Inequality**: The sum of any two sides of a triangle must be greater than the third side. This determines whether three given lengths can form a triangle.

**Congruence means identical**: Two figures are congruent if they have exactly the same shape AND size—one can be superimposed on the other perfectly.

**Similarity means same shape, different size**: Similar figures have equal corresponding angles and proportional corresponding sides. All congruent figures are similar, but not all similar figures are congruent.

**Quadrilateral Angle Sum**: The sum of interior angles of any quadrilateral equals 360°.

**Parallelogram Properties**: Opposite sides are equal and parallel; opposite angles are equal; diagonals bisect each other.

**Special Quadrilaterals Hierarchy**: Square → Rectangle → Parallelogram → Quadrilateral (each inherits properties from the level above).

Formulas / Key Facts

**Triangle Formulas:**

Sum of angles = 180°
Exterior angle = Sum of two interior opposite angles
Area = (1/2) × base × height
Area using Heron's formula: √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2

**Quadrilateral Formulas:**

Sum of angles = 360°
Area of rectangle = length × breadth
Area of square = side²
Area of parallelogram = base × height
Area of rhombus = (1/2) × d₁ × d₂ (where d₁, d₂ are diagonals)
Area of trapezium = (1/2) × (sum of parallel sides) × height

**Congruence Criteria (5 rules):**

SSS (Side-Side-Side): Three sides equal
SAS (Side-Angle-Side): Two sides and included angle equal
ASA (Angle-Side-Angle): Two angles and included side equal
AAS (Angle-Angle-Side): Two angles and one side equal
RHS (Right angle-Hypotenuse-Side): For right triangles only

**Similarity Criteria (3 rules):**

AAA or AA: Two angles equal (third automatically equal)
SSS: All three sides in same ratio
SAS: Two sides in same ratio with included angle equal

**Similarity Ratio Property:** If two triangles are similar with sides in ratio k, then:

Corresponding altitudes, medians, angle bisectors are also in ratio k
Areas are in ratio k²

Worked Examples

**Example 1: Finding an unknown angle** In triangle ABC, angle A = 65° and angle B = 48°. Find angle C.

*Solution:* Sum of angles in a triangle = 180° Angle C = 180° − 65° − 48° = 67°

**Example 2: Proving congruence** In triangles PQR and XYZ: PQ = XY = 5 cm, QR = YZ = 7 cm, and angle Q = angle Y = 60°. Are the triangles congruent?

*Solution:* We have two sides equal (PQ = XY and QR = YZ) and the included angle equal (angle Q = angle Y). This satisfies the SAS criterion. Therefore, triangle PQR ≅ triangle XYZ.

**Example 3: Using similarity** Triangles ABC and DEF are similar. If AB = 4 cm, DE = 6 cm, and the area of triangle ABC = 24 cm², find the area of triangle DEF.

*Solution:* Ratio of corresponding sides = DE/AB = 6/4 = 3/2 Ratio of areas = (ratio of sides)² = (3/2)² = 9/4 Area of DEF = Area of ABC × (9/4) = 24 × (9/4) = 54 cm²

**Example 4: Quadrilateral property** ABCD is a parallelogram. If angle A = 70°, find all other angles.

*Solution:* In a parallelogram, opposite angles are equal and consecutive angles are supplementary. Angle A = 70°, so angle C = 70° (opposite) Angle B = 180° − 70° = 110° (consecutive angles supplementary) Angle D = 110° (opposite to B)

Common Mistakes

**Confusing congruence criteria**: Students often think AAA proves congruence. *Correction*: AAA only proves similarity, not congruence—two triangles can have the same angles but different sizes.

**Wrong order in congruence statements**: Writing triangle ABC ≅ triangle PQR means A↔P, B↔Q, C↔R. *Correction*: Always match corresponding vertices in the same order.

**Assuming all properties transfer from parallelogram to all quadrilaterals**: Not every quadrilateral has equal opposite sides. *Correction*: Check whether the figure is specifically a parallelogram before applying its properties.

**Forgetting the "included angle" requirement in SAS**: Two sides and ANY angle being equal doesn't guarantee congruence. *Correction*: The angle must be between the two given sides for SAS to apply.

**Confusing perimeter ratio with area ratio in similar figures**: If sides are in ratio 2:3, students assume areas are also 2:3. *Correction*: Areas are in the ratio of the square of the sides, so 4:9.

Quick Reference

Triangle angle sum = 180°; Quadrilateral angle sum = 360°
Congruence criteria: SSS, SAS, ASA, AAS, RHS (not AAA)
Similarity criteria: AA, SSS (proportional), SAS (proportional)
Congruent = same shape + same size; Similar = same shape only
In similar triangles: sides ratio = k, area ratio = k²
Parallelogram: opposite sides equal and parallel, diagonals bisect each other

Geometry