Notes
SSC CHSL Exam
Qunatiative Aptitude
Geometery

Geometery

1. Fundamental Concepts of Geometry

1.1 Points, Lines, Segments, and Rays

  • Point: An exact location in space with no size. Example: “.” (a dot on paper)
  • Line Segment: The shortest path joining two points, with definite length. Example: Segment $AB$ (see diagram below)
  • Ray: A line segment extended infinitely in one direction. Example: Ray $AB$ starts at $A$ and passes through $B$, continuing forever.
  • Visual Aid:

1.2 Types of Lines

TypeDescriptionExample/Diagram
Intersecting LinesMeet at a point (point of intersection)“X” shape
Concurrent LinesThree or more lines intersect at the same pointStar-like pattern
Parallel LinesNever meet, always same distance apart"
TransversalA line that cuts two or more lines at distinct pointsSee below
  • Visual Aid:

1.3 Angles and Their Types

Angle TypeMeasure (Degrees)Example/Diagram
Right$90^\circ$Corner of a square
Acute$< 90^\circ$$45^\circ, 60^\circ$
Obtuse$90^\circ < x < 180^\circ$$120^\circ$
Straight$180^\circ$A straight line
Reflex$180^\circ < x < 360^\circ$$270^\circ$
  • Example: An angle of $60^\circ$ is acute; $135^\circ$ is obtuse.
  • Visual Aid:

1.4 Angle Relationships

RelationshipDescriptionExample
ComplementarySum to $90^\circ$$30^\circ$ and $60^\circ$
SupplementarySum to $180^\circ$$130^\circ$ and $50^\circ$
Vertically Opposite AnglesOpposite angles formed by intersecting lines (equal)See diagram
Angle BisectorDivides an angle into two equal partsSee diagram
  • Visual Aid:

Practice Questions: Fundamental Concepts

  1. Name the type of angle for $110^\circ$.
  2. If two angles are complementary and one is $35^\circ$, what is the other?
  3. Draw two parallel lines and a transversal. Mark and name a pair of corresponding angles.

Answers:

  1. Obtuse angle
  2. $55^\circ$
  3. (Student drawing; corresponding angles are in matching corners at each intersection.)

2. Triangles

2.1 Types of Triangles

TypeSides/Angles CriteriaExample Diagram
EquilateralAll sides and angles equal ($60^\circ$)
IsoscelesTwo sides and two angles equal
ScaleneAll sides and angles different
Right-angledOne angle is $90^\circ$

2.2 Properties of Triangles

  • Sum of Angles:
$$ \angle A + \angle B + \angle C = 180^\circ $$
  • Triangle Inequality: The sum of any two sides is greater than the third:
$$ a + b > c,\quad a + c > b,\quad b + c > a $$
  • Exterior Angle Theorem: The exterior angle equals the sum of the two opposite interior angles.

Example: If two angles are $50^\circ$ and $60^\circ$, third angle = $180^\circ - (50^\circ + 60^\circ) = 70^\circ$.

2.3 Area of Triangle

FormulaWhen to Use
$\frac{1}{2} \times \text{base} \times \text{height}$Base and height known
Heron’s: $\sqrt{s(s-a)(s-b)(s-c)}$All sides known ($s = \frac{a+b+c}{2}$)
$\frac{1}{2}ab\sin C$Two sides and included angle

Example: Base = 8 cm, Height = 5 cm Area = $\frac{1}{2} \times 8 \times 5 = 20$ cm²

2.4 Special Lines in Triangles

LineDescriptionDiagram Example
MedianVertex to midpoint of opposite side
AltitudeVertex perpendicular to opposite side
Angle BisectorDivides angle into two equal parts
Perpendicular BisectorPerpendicular at midpoint of side

Practice Questions: Triangles

  1. Can a triangle have sides 3 cm, 4 cm, and 8 cm?
  2. Find the area of a triangle with sides 7 cm, 8 cm, and 9 cm.
  3. In a triangle, two angles are $45^\circ$ and $75^\circ$. Find the third angle.

Answers:

  1. No ($3+4=7<8$), so not possible.
  2. Use Heron’s formula: $s = 12$, Area $= \sqrt{12 \times 5 \times 4 \times 3} = \sqrt{720} \approx 26.83$ cm²
  3. $180^\circ - (45^\circ + 75^\circ) = 60^\circ$

3. Quadrilaterals and Polygons

3.1 Quadrilaterals

ShapePropertiesArea Formula
Square4 equal sides, all $90^\circ$ angles$a^2$
RectangleOpposite sides equal, all $90^\circ$ angles$l \times b$
ParallelogramOpposite sides parallel and equalbase $\times$ height
RhombusAll sides equal, diagonals bisect at right angles$\frac{1}{2} d_1 d_2$
TrapeziumOne pair parallel sides$\frac{1}{2}(a+b)h$

Example: Rectangle: Length = 6 cm, Breadth = 4 cm Area = $6 \times 4 = 24$ cm²; Perimeter = $2(6+4) = 20$ cm

3.2 Regular Polygons

  • Definition: All sides and angles are equal.
  • Sum of Interior Angles:
$$ (n-2) \times 180^\circ $$
  • Each Angle (Regular Polygon):
$$ \frac{(n-2) \times 180^\circ}{n} $$

Example: Octagon ($n=8$): Each angle = $\frac{6 \times 180}{8} = 135^\circ$

Practice Questions: Quadrilaterals and Polygons

  1. Find the area and perimeter of a square with side 5 cm.
  2. What is the sum of interior angles of a pentagon?
  3. Find each interior angle of a regular hexagon.

Answers:

  1. Area = $25$ cm², Perimeter = $20$ cm
  2. $(5-2) \times 180 = 540^\circ$
  3. $\frac{(6-2) \times 180}{6} = 120^\circ$

4. Circles

4.1 Key Terms and Properties

TermDescriptionFormula/Diagram
Radius ($r$)Center to any point on circle
Diameter ($d$)Longest chord, $d = 2r$
CircumferencePerimeter of the circle$2\pi r$
AreaSpace inside the circle$\pi r^2$
ChordLine joining two points on the circle
ArcPart of circumference
Sector‘Pie slice’ of the circle
  • Visual Aid:

4.2 Sector and Arc Calculations

QuantityFormula
Length of Arc$\frac{\theta}{360^\circ} \times 2\pi r$
Area of Sector$\frac{\theta}{360^\circ} \times \pi r^2$

Example: Find the length of a $60^\circ$ arc in a circle of radius 7 cm. Arc length = $\frac{60}{360} \times 2\pi \times 7 = \frac{1}{6} \times 44 = 7.33$ cm

4.3 Important Circle Rules

  • Equal chords are equidistant from the center.
  • The perpendicular from the center to a chord bisects the chord.
  • The angle in a semicircle is a right angle.
  • Angles in the same segment are equal.
  • The angle subtended by an arc at the center is twice that at the circumference.

Practice Questions: Circles

  1. Find the area and circumference of a circle with radius 10 cm.
  2. What is the area of a sector with angle $90^\circ$ in a circle of radius 4 cm?
  3. In a circle, a chord of length 8 cm is twice as far from the center as a chord of length 10 cm. Find the radius.

Answers:

  1. Area = $100\pi \approx 314$ cm²; Circumference = $20\pi \approx 62.8$ cm
  2. $\frac{90}{360} \times \pi \times 16 = \frac{1}{4} \times \pi \times 16 = 4\pi \approx 12.57$ cm²
  3. Use Pythagoras theorem for chords and distances (see worked example above).

5. Visual Aids and Diagram Tools

  • Draw diagrams for every problem.
  • Use online tools like GeoGebra for accurate, interactive diagrams12.
  • Label all points, sides, and angles clearly.
  • Maintain even spacing and alignment for clarity.

6. Math Fonts and Spacing

  • Use clear math fonts (e.g., Cambria Math) for all formulas3.
  • For LaTeX or digital notes, use proper spacing commands for readability45.

Summary Table: Key Formulas

Shape/ConceptArea FormulaPerimeter/Circumference
Triangle$\frac{1}{2} \times \text{base} \times \text{height}$sum of all sides
Rectangle$l \times b$$2(l+b)$
Square$a^2$$4a$
Parallelogrambase $\times$ height$2(a+b)$
Rhombus$\frac{1}{2} d_1 d_2$$4a$
Trapezium$\frac{1}{2}(a+b)h$sum of all sides
Circle$\pi r^2$$2\pi r$
Sector (circle)$\frac{\theta}{360^\circ} \pi r^2$$\frac{\theta}{360^\circ} 2\pi r$

Final Practice Set

  1. Draw a triangle with sides 5 cm, 6 cm, and 7 cm. Find its area.
  2. Calculate the perimeter of a parallelogram with sides 8 cm and 5 cm.
  3. If a circle has a diameter of 14 cm, what is its area?
  4. Find the sum of the interior angles of a decagon.
  5. In a square of side 4 cm, what is the length of the diagonal?

Answers:

  1. Use Heron’s formula: $s=9$, Area = $\sqrt{9 \times 4 \times 3 \times 2} = \sqrt{216} \approx 14.7$ cm²
  2. $2 \times (8+5) = 26$ cm
  3. Radius = 7 cm; Area = $\pi \times 49 \approx 154$ cm²
  4. $(10-2) \times 180 = 1440^\circ$
  5. Diagonal = $4\sqrt{2} \approx 5.66$ cm

Tip: Always visualize, draw, and label diagrams for every geometry problem. Practice using digital tools like GeoGebra for clean, accurate figures. Use clear math fonts and spacing for notes and answers.

This guide follows a consistent structure: concept, example, visual, then practice-ensuring clarity and mastery of geometry fundamentals.

⁂

  1. https://www.geogebra.org/geometry?lang=en ↩︎

  2. https://www.math10.com/en/geometry/geogebra/geogebra.html ↩︎

  3. https://learn.microsoft.com/en-us/typography/cleartype/pdfs/cambriamath.pdf ↩︎

  4. https://ecampusontario.pressbooks.pub/math1020/chapter/bold-math-and-spacing/ ↩︎

  5. https://ctan.math.illinois.edu/macros/latex/contrib/spacingtricks/spacingtricks.pdf ↩︎

Data Interpretation (DI)Interest