Geometry Triangles Quadrilaterals Congruence Questions for CTET PAPER II

Triangle Basics and Angle Properties

A triangle has three sides and three angles. The most fundamental property is that the sum of all interior angles of any triangle is always 180 degrees. An exterior angle of a triangle equals the sum of the two non-adjacent interior angles — this is called the Exterior Angle Theorem. Triangles are classified by sides (scalene, isosceles, equilateral) and by angles (acute, right, obtuse).

• Angle sum of triangle = 180 degrees
• Exterior angle = sum of two opposite interior angles
• Equilateral triangle: all sides equal, each angle = 60 degrees
• Isosceles triangle: two equal sides, base angles are equal
• In a right triangle: the side opposite 90 degrees is the hypotenuse (longest side)
• Triangle inequality: sum of any two sides must be greater than the third side

Congruence of Triangles

Two triangles are congruent if they are exactly the same in shape and size — every side and every angle matches. Congruence means you can place one triangle exactly over the other. The order of vertices matters: if triangle ABC is congruent to triangle PQR, then AB = PQ, BC = QR, AC = PR, and corresponding angles match. CTET heavily tests which criterion applies in a given figure.

• SSS: all three sides of one triangle equal all three sides of another
• SAS: two sides and the included angle (angle between those sides) are equal
• ASA: two angles and the included side are equal
• AAS: two angles and a non-included side are equal
• RHS: right angle, hypotenuse, and one side are equal — only for right triangles
• AAA is NOT a congruence criterion — it only proves similarity

Properties of Special Triangles

CTET Class 6-8 syllabus includes Pythagoras theorem for right triangles. The square of the hypotenuse equals the sum of squares of the other two sides. Common Pythagorean triplets must be memorised. Also remember: in an isosceles triangle, the perpendicular from the apex bisects the base (and vice versa).

• Pythagoras: in right triangle, c squared = a squared + b squared where c is hypotenuse
• Common triplets: 3-4-5, 5-12-13, 8-15-17, 7-24-25
• Median: line from vertex to midpoint of opposite side; a triangle has 3 medians that meet at centroid
• Centroid divides each median in ratio 2:1 from vertex
• Altitude: perpendicular from vertex to opposite side; all three altitudes meet at orthocenter

Quadrilaterals and Their Properties

A quadrilateral has four sides and four angles. The angle sum of any quadrilateral is always 360 degrees. CTET tests properties of special quadrilaterals very frequently — know each type distinctly. The hierarchy is: all squares are rectangles, all rectangles are parallelograms, all parallelograms are quadrilaterals.

• Parallelogram: opposite sides parallel and equal, opposite angles equal, diagonals bisect each other
• Rectangle: parallelogram with all right angles, diagonals are equal in length
• Rhombus: parallelogram with all sides equal, diagonals bisect each other at 90 degrees
• Square: all sides equal AND all angles 90 degrees, diagonals equal and bisect at 90 degrees
• Trapezium: exactly one pair of parallel sides
• Kite: two pairs of adjacent equal sides, one diagonal bisects the other at 90 degrees

Congruence vs Similarity — Key Distinction

This is the most common confusion point in CTET. Congruent figures are identical in size and shape. Similar figures have the same shape but can be different sizes. For triangles, AAA proves similarity (not congruence). SSS, SAS, ASA, AAS, RHS prove congruence. The symbol for congruence is the equals sign with a tilde on top; similarity uses the tilde alone.

• Congruence: same shape + same size
• Similarity: same shape + proportional sides
• AAA is a similarity criterion, NOT congruence
• SSS similarity: all sides in proportion (not equal)
• Every congruent pair is also similar, but not every similar pair is congruent

• Treating AAA as a congruence criterion — it is only a similarity criterion. CTET uses this trap directly in MCQs.
• In SAS, the angle must be the INCLUDED angle (between the two given sides). Using a non-included angle is called SSA which is not a valid criterion.
• Confusing properties of rhombus and rectangle: rhombus has perpendicular diagonals (not equal), rectangle has equal diagonals (not perpendicular) — only square has both.
• Forgetting that exterior angle = sum of two remote interior angles, not all three interior angles.
• Assuming all parallelograms have equal diagonals — only rectangles and squares among parallelograms have equal diagonals.

• Congruence criteria mnemonic: Some Sailors Are Always Right — SSS, SAS, ASA, AAS, RHS
• Quadrilateral hierarchy ladder (bottom to top): Quadrilateral → Trapezium → Parallelogram → Rectangle/Rhombus → Square. Each step adds a property.
• DAISY rule for diagonals: D=equal diagonals (Rectangle, Square), A=diagonals bisect at 90 degrees (Rhombus, Square), I=diagonals bisect each other (all parallelograms). Square satisfies all three.
• For Pythagoras triplets remember 3-4-5 and just multiply: 6-8-10, 9-12-15. For 5-12-13, think of it as the unlucky triplet — must be memorised separately.

• CTET Paper II consistently gives 2-3 questions on congruence criteria. The typical format: a figure or description is given, and you must identify which criterion (SSS, SAS, etc.) proves congruence. Focus on SAS vs SSA trap.
• Quadrilateral property questions appear as true/false style MCQs: which statement is correct about a rhombus/rectangle/square. The diagonal property differences between these three are tested almost every year.
• Angle-based questions (exterior angle theorem, angle sum) are straightforward and can be solved in under 30 seconds — do these first to save time for harder Qs.
• Pedagogical angle: CTET also asks how to TEACH geometry at Class 6-8 level — expect 1-2 questions on using geoboards, dot paper, or paper folding to demonstrate congruence or properties. These come from the pedagogy sub-section.
• Pythagoras questions in CTET are typically one-step application. If numbers look messy, check for standard triplets first before computing — it saves time.