Geometry
Lines and Angles, Triangles, Quadrilaterals, Circles
Overview
Geometry forms the backbone of the Mathematics content section in PSTET Paper II. Questions typically test your ability to apply properties of lines, angles, triangles, quadrilaterals and circles to solve numerical problems or identify correct statements. This topic carries significant weightage because it integrates visual reasoning with algebraic calculation—skills essential for upper-primary mathematics teaching.
For PSTET, you must master angle relationships (complementary, supplementary, vertically opposite, angles on parallel lines), triangle properties (angle sum, congruence, similarity, Pythagoras), quadrilateral properties (parallelogram family, angle sums), and circle basics (chord-tangent-arc relationships). Most questions are direct application problems requiring one or two steps of reasoning. Diagrams are often provided, so practice interpreting figures quickly and accurately.
Key Concepts
- Basic angle pairs: Complementary angles sum to 90°; supplementary angles sum to 180°; vertically opposite angles are always equal.
- Parallel lines cut by a transversal produce four key angle pairs—corresponding angles (equal), alternate interior angles (equal), alternate exterior angles (equal), and co-interior/consecutive interior angles (supplementary, sum = 180°).
- Triangle angle-sum property: Interior angles of any triangle add to 180°. Exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
- Congruence criteria for triangles: SSS, SAS, ASA, AAS and RHS (right-angle-hypotenuse-side). Two triangles satisfying any one criterion are congruent (identical in shape and size).
- Similarity criteria: AA (angle-angle), SSS (sides in proportion), SAS (included angle equal, adjacent sides in proportion). Similar triangles have equal corresponding angles and proportional sides.
- Pythagoras theorem: In a right triangle, hypotenuse² = base² + perpendicular² (applies only when one angle is 90°).
- Quadrilateral angle-sum: Interior angles of any quadrilateral add to 360°. Special quadrilaterals (parallelogram, rectangle, rhombus, square, trapezium) have additional properties regarding sides, angles and diagonals.
- Circle fundamentals: All radii are equal; diameter = 2 × radius; a chord is a line segment with both endpoints on the circle; a tangent touches the circle at exactly one point and is perpendicular to the radius at that point.
Formulas / Key Facts
| Concept | Formula / Fact |
|---|---|
| Sum of angles in a triangle | 180° |
| Sum of angles in a quadrilateral | 360° |
| Sum of interior angles of n-gon | (n − 2) × 180° |
| Each interior angle of regular n-gon | (n − 2) × 180° / n |
| Exterior angle of triangle | Sum of two remote interior angles |
| Pythagoras theorem | c² = a² + b² (c = hypotenuse) |
| Area of triangle | ½ × base × height |
| Area of equilateral triangle (side a) | (√3 / 4) × a² |
| Parallelogram properties | Opposite sides equal and parallel; opposite angles equal; diagonals bisect each other |
| Rectangle diagonals | Equal in length; bisect each other |
| Rhombus diagonals | Bisect each other at 90°; bisect the vertex angles |
| Square | All properties of rectangle + rhombus |
| Circumference of circle | 2πr |
| Area of circle | πr² |
| Tangent-radius angle | 90° (tangent ⊥ radius at point of contact) |
| Angle in a semicircle | 90° |
Worked Examples
Example 1 — Angles on parallel lines
Two parallel lines are cut by a transversal. One of the interior angles on the same side of the transversal is 65°. Find the other.
Solution: Co-interior angles are supplementary. Other angle = 180° − 65° = 115°
Example 2 — Triangle angle and exterior angle
In triangle PQR, angle P = 50° and angle Q = 70°. Find angle R and the exterior angle at R.
Solution: Angle R = 180° − 50° − 70° = 60° Exterior angle at R = angle P + angle Q = 50° + 70° = 120°
Example 3 — Pythagoras theorem
A ladder 13 m long leans against a wall. Its foot is 5 m from the wall. How high up the wall does it reach?
Solution: Let height = h. h² + 5² = 13² h² = 169 − 25 = 144 h = 12 m
Example 4 — Quadrilateral angle sum
Three angles of a quadrilateral are 90°, 85° and 95°. Find the fourth angle.
Solution: Fourth angle = 360° − (90° + 85° + 95°) = 360° − 270° = 90°
Example 5 — Circle tangent property
A tangent PA is drawn to a circle with centre O. If angle OPA = 90° and OP = 10 cm, OA = 6 cm, find PA.
Solution: Tangent is perpendicular to radius at point of contact, so triangle OAP is right-angled at A. PA² = OP² − OA² = 100 − 36 = 64 PA = 8 cm
Common Mistakes
- Confusing co-interior with alternate angles → Co-interior angles are supplementary (sum 180°); alternate angles are equal. Always check whether angles are on the same side or opposite sides of the transversal.
- Applying Pythagoras to non-right triangles → Pythagoras theorem works only when one angle is exactly 90°. Verify the right angle before using c² = a² + b².
- Mixing congruence and similarity → Congruent triangles are identical (equal sides and angles). Similar triangles have equal angles but sides only in proportion. SSS for congruence needs equal sides; SSS for similarity needs sides in the same ratio.
- Forgetting that diagonals of a parallelogram only bisect each other, not the angles → Angle bisection occurs in a rhombus, not in a general parallelogram or rectangle.
- Assuming all quadrilaterals have equal diagonals → Only rectangles and squares have equal diagonals. In a parallelogram or rhombus, diagonals are generally unequal.
- Ignoring the tangent-radius perpendicularity → When a tangent is involved, immediately mark the 90° angle at the point of contact; this often creates a right triangle to solve.
Quick Reference
- Triangle interior angles = 180°; quadrilateral = 360°; n-gon = (n − 2) × 180°.
- Alternate angles equal; co-interior angles supplementary.
- Congruence: SSS, SAS, ASA, AAS, RHS. Similarity: AA, SSS (ratio), SAS (ratio).
- Pythagoras: hypotenuse² = sum of squares of other two sides (right triangle only).
- Parallelogram diagonals bisect each other; rhombus diagonals bisect at 90°; rectangle diagonals are equal.
- Tangent ⊥ radius at contact point; angle in semicircle = 90°.