Geometry Guide & Practice

---

1. Fundamentals & Definitions

Point, Line, and Plane

• Point: A geometrical location with no dimensions (length, width, or height), represented by a fine dot.
• Line: A straight path of points that extends infinitely in both opposite directions. A line is uniquely determined by two points. Symbol: $AB$ or $k$.
• Line Segment: A bounded portion of a straight line between two distinct endpoints. It has a definite, measurable length. Symbol: $\overline{AB}$ or $AB$.
• Midpoint: A point on a line segment that is equidistant from both endpoints, dividing the segment into two equal halves ($AP = PB$).
• Ray: A part of a line that starts at one endpoint and extends infinitely far in one direction. Symbol: $\overrightarrow{AC}$.
• Plane: A flat, two-dimensional surface that extends infinitely in all directions.

Angles & Angle Relationships

• Angle: The union of two non-collinear rays (called arms or sides) sharing a common initial point (called the vertex). Symbol: $\angle BAC$ or $\angle A$.
• Acute Angle: An angle whose measure is strictly less than $90^\circ$ but greater than $0^\circ$.
• Right Angle: An angle whose measure is exactly $90^\circ$. It is formed when two lines or rays are perpendicular.
• Obtuse Angle: An angle whose measure is greater than $90^\circ$ but strictly less than $180^\circ$.
• Straight Angle: An angle whose sides lie along a straight line, measuring exactly $180^\circ$.
• Reflex Angle: An angle whose measure is greater than $180^\circ$ but strictly less than $360^\circ$.
• Adjacent Angles: Two angles that share a common vertex and a common side but do not overlap (have no common interior points).
• Vertical (Vertically Opposite) Angles: Opposite angles formed by the intersection of two straight lines. Vertical angles are always equal in measure ($\angle a = \angle c$ and $\angle b = \angle d$).
• Linear Pair of Angles: Two adjacent angles whose non-common arms form opposite rays. They are always supplementary ($\angle a + \angle b = 180^\circ$).
• Supplementary Angles: Any two angles whose measures sum to exactly $180^\circ$. Each is the supplement of the other.
• Complementary Angles: Any two angles whose measures sum to exactly $90^\circ$. Each is the complement of the other.

Parallel and Perpendicular Lines

• Parallel Lines: Lines lying in the same plane that never meet or intersect, regardless of how far they are extended. Symbol: $\parallel$ (e.g., $a \parallel b$).
• Perpendicular Lines: Lines that intersect each other at a right angle ($90^\circ$). Symbol: $\perp$ (e.g., $a \perp b$).
• Transversal: A straight line that intersects two or more other lines at distinct points.
• Interior Angles: The four angles formed between the two lines cut by a transversal.
• Exterior Angles: The four angles formed outside the two lines cut by a transversal.
• Alternate Angles: Pairs of interior or exterior angles situated on opposite sides of the transversal.
• Corresponding Angles: Pairs of angles situated in the same relative position at each intersection where a transversal cuts two lines.

Polygons

• Polygon: A closed two-dimensional plane figure bounded by three or more straight line segments (sides).
• Convex Polygon: A polygon where no line segment between any two vertices goes outside the boundary, meaning the line containing any side has all other vertices situated on the same side of the line.
• Regular Polygon: A polygon that is both equilateral (all sides are equal in length) and equiangular (all interior angles are equal in measure).
• Apothem: The line segment drawn from the center of a regular polygon perpendicular to one of its sides. It also corresponds to the radius of the inscribed circle.
• Diagonals: Line segments connecting any two non-consecutive vertices of a polygon.

Triangles & Geometrical Centers

• Triangle: A three-sided polygon. The sum of its three interior angles is always $180^\circ$. Symbol: $\Delta$.
• Equilateral Triangle: A triangle in which all three sides are equal, and all three interior angles measure $60^\circ$.
• Isosceles Triangle: A triangle with at least two equal sides. The angles opposite these equal sides are also equal (base angles).
• Scalene Triangle: A triangle with all three sides of different lengths and all angles of different measures.
• Cevian: Any line segment extending from a vertex of a triangle to the opposite side (or its extension). Examples include medians, altitudes, and angle bisectors.
• Median: A cevian that connects a vertex to the midpoint of the opposite side.
• Altitude: A perpendicular cevian drawn from a vertex to the opposite side (base).
• Angle Bisector: A cevian that divides an interior angle of a triangle into two equal angles.
• Perpendicular Bisector: A line that passes through the midpoint of a triangle's side at a right angle ($90^\circ$).
• Centroid ($G$): The point of concurrency where the three medians of a triangle intersect.
• Orthocenter ($H$): The point of concurrency where the three altitudes of a triangle intersect.
• Circumcenter ($O$): The point of concurrency where the three perpendicular bisectors of a triangle's sides intersect. It is the center of the circumcircle.
• Incenter ($I$): The point of concurrency where the three interior angle bisectors of a triangle intersect. It is the center of the incircle (inscribed circle).
• Excenter: The point of concurrency of one interior angle bisector and the two opposite exterior angle bisectors. A triangle has exactly three excenters.
• Circumcircle (Circumscribed Circle): The circle passing through all three vertices of a triangle.
• Circumradius ($R$): The radius of the circumcircle.
• Incircle (Inscribed Circle): The circle that lies inside a triangle and touches all three of its sides.
• Inradius ($r$): The radius of the incircle.
• Congruent Triangles: Triangles that have the exact same shape and size. Corresponding sides and angles are equal. Symbol: $\cong$.
• Similar Triangles: Triangles that have the same shape but may differ in size. Their corresponding angles are equal, and their corresponding sides are proportional. Symbol: $\sim$.

Circles & Parts of a Circle

• Circle: The set of all points in a plane that are at a constant distance (radius) from a fixed point (center).
• Circumference: The perimeter or boundary length of a circle.
• Radius ($r$): The line segment from the center of the circle to any point on its circumference.
• Chord: A straight line segment connecting any two distinct points on a circle.
• Diameter ($d$): A chord that passes through the center of the circle. It is the longest possible chord and is equal to twice the radius ($d = 2r$).
• Secant: A straight line that cuts through a circle, intersecting it at two distinct points.
• Tangent: A straight line that touches a circle at exactly one point (point of contact or tangency) and never crosses the interior.
• Arc: A portion of the circumference of a circle.
• Sector: The region of a circle bounded by two radii and their intercepted arc.
• Segment of a Circle: The region of a circle bounded by a chord and its corresponding arc.
• Cyclic Quadrilateral: A quadrilateral whose four vertices all lie on the circumference of a single circle.
• Direct Common Tangent (DCT): A common tangent to two circles where both circles lie on the same side of the tangent line.
• Transverse Common Tangent (TCT): A common tangent to two circles where the circles lie on opposite sides of the tangent line.

---

2. Core Concepts & Formulas

2.1 Angles, Lines & Transversals

When a transversal intersects two parallel lines $a$ and $b$, the following angle relationships hold:

• Corresponding Angles are Equal: $\angle y = \angle q$, $\angle z = \angle r$, $\angle x = \angle p$, and $\angle w = \angle s$.
• Alternate Interior Angles are Equal: $\angle z = \angle p$ and $\angle w = \angle q$.
• Alternate Exterior Angles are Equal: $\angle y = \angle s$ and $\angle x = \angle r$.
• Consecutive Interior Angles are Supplementary: $\angle z + \angle q = 180^\circ$ and $\angle p + \angle w = 180^\circ$.

Note on textual layout: Imagine two horizontal parallel lines cut by a diagonal transversal. The top intersection contains angles $y, x$ (above line $a$, left and right) and $z, w$ (below line $a$, left and right). The bottom intersection contains angles $q, p$ (above line $b$, left and right) and $r, s$ (below line $b$, left and right).

2.2 General & Regular Polygons

For any convex polygon with $n$ sides:

• Sum of Interior Angles: $$\text{Sum} = (n - 2) \times 180^\circ$$
• Sum of Exterior Angles: Always $360^\circ$.
• Number of Diagonals: $$\text{Diagonals} = \frac{n(n - 3)}{2}$$

For a Regular Polygon with $n$ sides and side length $s$:

• Measure of Each Interior Angle ($i$): $$i = \frac{(n - 2) \times 180^\circ}{n}$$
• Measure of Each Exterior Angle ($e$): $$e = \frac{360^\circ}{n} = 180^\circ - i$$
• Area of Regular Polygon: $$\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} = \frac{1}{2} n s r$$ (where $r$ is the apothem/inradius).
• Regular Hexagon Specifics ($n = 6$):
  • Circumradius: $R = s$
  • Apothem (Inradius): $r = \frac{s\sqrt{3}}{2}$
  • Area: $$\text{Area} = 6 \times \left(\frac{s^2\sqrt{3}}{4}\right) = \frac{3\sqrt{3}}{2}s^2$$

2.3 Triangles — Metric Relations & Classification

Let $\Delta ABC$ have side lengths $a, b, c$ opposite to vertices $A, B, C$.

• Triangle Inequality: The sum of any two sides must be strictly greater than the third side: $$a + b > c, \quad b + c > a, \quad c + a > b$$ The absolute difference of any two sides must be strictly less than the third side: $$|a - b| < c, \quad |b - c| < a, \quad |c - a| < b$$
• Angle-Side Relationship: The largest side is opposite the largest angle, and the smallest side is opposite the smallest angle: $$a > b > c \iff \angle A > \angle B > \angle C$$
• Classification by Sides (using the largest side $a$):
  • Acute-angled triangle: $a^2 < b^2 + c^2$
  • Right-angled triangle: $a^2 = b^2 + c^2$
  • Obtuse-angled triangle: $a^2 > b^2 + c^2$
• Area Formulas:
  • Heron's Formula: $$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}, \quad \text{where } s = \frac{a+b+c}{2} \text{ (semi-perimeter)}$$
  • Base-Height Formula: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} b h_b$$
  • Trigonometric Area Formula: $$\text{Area} = \frac{1}{2} a b \sin C = \frac{1}{2} b c \sin A = \frac{1}{2} c a \sin B$$
  • Cosine Rule: $$\cos A = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos B = \frac{c^2 + a^2 - b^2}{2ca}, \quad \cos C = \frac{a^2 + b^2 - c^2}{2ab}$$
  • Projection Formula: $$a = b\cos C + c\cos B, \quad b = c\cos A + a\cos C, \quad c = a\cos B + b\cos A$$

2.4 Congruence & Similarity Criteria

• Congruence Criteria: Two triangles are congruent ($\cong$) if they satisfy:
  • SSS: Three corresponding sides are equal.
  • SAS: Two corresponding sides and the included angle are equal.
  • ASA / AAS: Two corresponding angles and the adjacent (or corresponding non-included) side are equal.
  • RHS: In two right-angled triangles, the hypotenuses and one pair of corresponding legs are equal.
• Similarity Criteria: Two triangles are similar ($\sim$) if they satisfy:
  • AA / AAA: Two corresponding angles are equal.
  • SSS: All three pairs of corresponding sides are in the same ratio.
  • SAS: Two pairs of corresponding sides are proportional and their included angles are equal.
• Properties of Similar Triangles: If $\Delta ABC \sim \Delta DEF$ with a scale factor $k = \frac{AB}{DE}$:
  • The ratio of sides, heights ($h$), medians ($m$), angle bisectors, perimeters ($P$), inradii ($r$), and circumradii ($R$) are all equal to $k$: $$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = \frac{h_1}{h_2} = \frac{m_1}{m_2} = \frac{P_1}{P_2} = \frac{r_1}{r_2} = \frac{R_1}{R_2} = k$$
  • The ratio of their areas is the square of the scale factor $k$: $$\frac{\text{Area}(\Delta ABC)}{\text{Area}(\Delta DEF)} = k^2 = \left(\frac{AB}{DE}\right)^2$$

2.5 Triangle Centers & Associated Circles

The primary properties and coordinate geometry integrations of the four centers are summarized below:

• Excenter Properties: The radius of the ex-circle opposite to vertex $A$ is: $$r_a = \frac{\Delta}{s - a}$$
• Special Configurations:
  • Equilateral Triangle of side $a$:
    • Altitude: $h = \frac{a\sqrt{3}}{2}$
    • Area: $\Delta = \frac{a^2\sqrt{3}}{4}$
    • Inradius: $r = \frac{a}{2\sqrt{3}}$
    • Circumradius: $R = \frac{a}{\sqrt{3}} = 2r$
    • Centroid, Orthocenter, Circumcenter, and Incenter all coincide at the exact same point.
  • Right-Angled Triangle (legs $a, b$, hypotenuse $c$):
    • Inradius: $r = \frac{a + b - c}{2}$
    • Circumradius: $R = \frac{c}{2}$
    • Right Triangle Perpendicular Altitudes: If the right angle is at $A$, and $AD \perp BC$:
      • $\Delta ABD \sim \Delta CAD \sim \Delta CBA$
      • $AD^2 = BD \cdot DC$
      • $AB^2 = BD \cdot BC$
      • $AC^2 = CD \cdot BC$
      • $AD = \frac{ab}{c}$
      • $\frac{1}{AD^2} = \frac{1}{a^2} + \frac{1}{b^2}$

2.6 Advanced Cevian & Intersection Theorems

• Midpoint Theorem: If $D$ and $E$ are the midpoints of $AB$ and $AC$ in $\Delta ABC$, then $DE \parallel BC$ and: $$DE = \frac{1}{2}BC$$
• Basic Proportionality Theorem (Thales' Theorem): If $DE \parallel BC$, then: $$\frac{AD}{DB} = \frac{AE}{EC} \implies \frac{AD}{AB} = \frac{AE}{AC}$$
• Angle Bisector Theorem: If $AD$ is the internal bisector of $\angle A$, then: $$\frac{BD}{DC} = \frac{AB}{AC}$$ The length of the internal angle bisector segment satisfies: $$AD^2 = AB \cdot AC - BD \cdot DC$$
• Ceva's Theorem: Cevians $AD, BE, CF$ intersect at a single concurrent point in the interior of $\Delta ABC$ if and only if: $$\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB} = 1$$
• Menelaus' Theorem: Three points $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ are collinear if and only if: $$\frac{BD}{DC} \cdot \frac{CE}{EA} \cdot \frac{AF}{FB} = 1 \quad \text{(using absolute segment lengths)}$$
• Stewart's Theorem: If $AD$ is a cevian of length $d$ from vertex $A$ to side $BC$ that divides $BC$ into segments $BD = m$ and $DC = n$ (where $m + n = a$): $$b^2 m + c^2 n = a(d^2 + mn)$$
• Apollonius' Theorem: If $AD$ is a median ($BD = CD$): $$AB^2 + AC^2 = 2(AD^2 + BD^2)$$

2.7 Quadrilaterals & Metric Relations

• Angle Sum: $\angle A + \angle B + \angle C + \angle D = 360^\circ$
• Parallelogram:
  • Area: $\text{Area} = \text{base} \times \text{height}$
  • Adjacent angles sum to $180^\circ$ (e.g., $\angle A + \angle B = 180^\circ$).
  • Parallelogram Law (Diagonals and Sides relationship): $$2(AB^2 + BC^2) = AC^2 + BD^2$$
• Rectangle:
  • Area: $\text{Area} = \text{length} \times \text{breadth} = l \cdot b$
  • Diagonal length: $d = \sqrt{l^2 + b^2}$
  • Diagonals are equal in length ($AC = BD$) and bisect each other.
• Rhombus:
  • All sides are equal in length ($AB = BC = CD = DA$).
  • Diagonals $d_1$ and $d_2$ are perpendicular bisectors of each other ($\angle APD = 90^\circ$).
  • Side length: $s = \frac{1}{2}\sqrt{d_1^2 + d_2^2}$
  • Area: $$\text{Area} = \frac{1}{2} d_1 d_2$$
  • Perimeter: $P = 4s = 2\sqrt{d_1^2 + d_2^2}$
• Square:
  • Area: $\text{Area} = s^2 = \frac{1}{2} d^2$
  • Diagonal length: $d = s\sqrt{2}$
• Trapezium:
  • Area: $$\text{Area} = \frac{1}{2} (a + b) h, \quad \text{where } a, b \text{ are the parallel sides and } h \text{ is the height.}$$
  • Median of a Trapezium (line segment joining midpoints of non-parallel sides): $$\text{Median} = \frac{a+b}{2} \implies \text{Area} = \text{Median} \times h$$

2.8 Circle Geometry (Chords, Angles & Arcs)

• Chord Properties:
  • A perpendicular drawn from the center of a circle to a chord bisects the chord.
  • Chords of equal length are equidistant from the center.
  • Chord length $L$ at distance $d$ from center in circle of radius $R$: $$L = 2\sqrt{R^2 - d^2}$$
• Angle Relations:
  • Angle at the Center Theorem: The angle subtended by an arc at the center is double the angle subtended by the same arc at any point on the remaining part of the circle: $$\angle AOB = 2 \angle ACB$$
  • Angles in Semicircle: An angle subtended by a diameter at the circumference is always a right angle ($90^\circ$).
  • Angles in Same Segment: Angles subtended by the same chord in the same segment of a circle are equal.
• Cyclic Quadrilaterals:
  • Opposite angles are supplementary: $\angle A + \angle C = 180^\circ$ and $\angle B + \angle D = 180^\circ$.
  • An exterior angle is equal to the interior opposite angle.
  • Brahmagupta's Formula (Area): $$\text{Area} = \sqrt{(s-a)(s-b)(s-c)(s-d)}, \quad \text{where } s = \frac{a+b+c+d}{2}$$
  • Ptolemy's Theorem: The product of the diagonals of a cyclic quadrilateral is equal to the sum of the products of its opposite sides: $$AC \cdot BD = AB \cdot CD + AD \cdot BC$$
• Arc & Sector Metric Formulas:
  • Length of an arc ($l$) subtending angle $\theta$ (in degrees) at the center: $$l = \frac{\theta}{360^\circ} \times 2\pi r$$
  • Area of a sector ($A_{sector}$) with angle $\theta$ (in degrees): $$A_{sector} = \frac{\theta}{360^\circ} \times \pi r^2 = \frac{1}{2} l r$$
  • Area of a segment ($A_{segment}$) cut by a chord subtending angle $\theta$ at the center: $$A_{segment} = \text{Area of Sector} - \text{Area of Triangle} = r^2 \left( \frac{\pi \theta}{360^\circ} - \frac{\sin\theta}{2} \right)$$

2.9 Tangent & Secant Theorems

• Radius-Tangent Perpendicularity: A tangent at any point on a circle is perpendicular to the radius drawn through the point of contact.
• Equal Tangent Segments: Tangents drawn to a circle from an external point are equal in length.
• Alternate Segment Theorem: The angle formed between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment.
• Power of a Point Theorems:
  • Intersecting Chords (Internal): If two chords $AB$ and $CD$ intersect inside a circle at point $P$: $$PA \cdot PB = PC \cdot PD$$
  • Intersecting Secants (External): If two chords $AB$ and $CD$ are extended to intersect outside the circle at point $P$: $$PA \cdot PB = PC \cdot PD$$
  • Tangent-Secant Theorem: If a tangent segment $PT$ and a secant $PAB$ are drawn from an external point $P$: $$PT^2 = PA \cdot PB$$

2.10 Common Tangent Formulas

Let $d$ be the distance between the centers of two circles with radii $R$ and $r$ ($R \ge r$).

---

Typical Exam Weightage

One of the highest-weightage SSC quant topics — circle theorems and triangle properties are tested most frequently.

Figures are typical ranges based on recent-year patterns, not a guarantee for any specific upcoming paper — always cross-check against the latest official syllabus and previous-year papers for Geometry.