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Geometry

Practice MCQs

Geometry is the branch of mathematics that studies shapes, sizes, positions, angles, lines, surfaces, and figures.

Elementary Mathematics Geometry Plane Geometry Competitive Exams

Geometry is the branch of mathematics that studies points, lines, angles, plane figures, triangles, quadrilaterals, circles, tangents, normals, and loci. In competitive examinations, Geometry questions mainly test theorem application, diagram reading, angle relations, congruency, similarity, and properties of standard figures.

What is Geometry?

Geometry deals with shapes and their properties. It explains how lines, angles, sides, diagonals, circles, and curves behave under fixed rules. Unlike arithmetic, Geometry often depends more on recognizing the figure and applying the correct theorem.

The important parts of elementary Geometry are lines and angles, plane figures, triangles, parallelograms, rectangles, squares, circles, tangents, normals, and loci. These concepts are widely used in school mathematics, aptitude tests, entrance examinations, and government exams.

Quick idea: Draw the diagram, mark the given angles and equal sides, identify the figure, and then apply the correct theorem.
Area What It Covers Exam Focus
Lines and Angles Angles at a point, linear pair, vertically opposite angles. Angle calculation.
Parallel Lines Corresponding, alternate, and co-interior angles. Transversal-based questions.
Triangles Sides, angles, congruency, similarity, medians, altitudes. Triangle theorems.
Quadrilaterals Parallelogram, rectangle, square and diagonal properties. Side, angle, diagonal relations.
Circles Radius, chord, tangent, normal, cyclic quadrilateral. Circle theorem questions.
Loci Path traced by a point satisfying a fixed condition. Basic locus identification.

“Geometry becomes easy when the figure is read correctly and the theorem is applied logically.”

Geometry Tip
Key points
  • Understand point, line, ray, segment, angle and plane.
  • Use angle properties at a point and on a straight line.
  • Apply parallel line angle rules correctly.
  • Know triangle angle and side properties.
  • Use congruency and similarity criteria.
  • Understand centroid and orthocentre.
  • Know quadrilateral diagonal properties.
  • Apply circle, tangent and normal rules.
  • Identify simple loci from conditions.
angles triangles similarity circles tangents loci

Basic Geometry Terms

Before studying theorems, students should clearly understand the basic language of Geometry.

Term Meaning Example / Note
Point An exact position with no length, breadth or thickness. Usually denoted by A, B, C.
Line A straight path extending endlessly in both directions. A line has no fixed endpoint.
Ray A part of a line with one endpoint and extending endlessly in one direction. \(\overrightarrow{AB}\)
Line Segment A part of a line with two fixed endpoints. \(\overline{AB}\)
Angle Figure formed by two rays with a common endpoint. The common endpoint is called vertex.
Plane A flat surface extending endlessly in all directions. Plane figures lie on a plane.
Plane Figure A two-dimensional figure lying on a plane. Triangle, square, rectangle, circle.
Important: A neat diagram is often half the solution in Geometry.
Theorem Bank
Angles at a Point
Sum of angles around a point \(=360^\circ\).
Angles on a straight line \(=180^\circ\).
Vertically Opposite Angles
When two lines intersect, vertically opposite angles are equal.
Parallel Lines
Corresponding angles are equal.
Alternate interior angles are equal.
Co-interior angles are supplementary.
Triangle Angle Sum
In any triangle:
\(\angle A+\angle B+\angle C=180^\circ\)
Exterior Angle
Exterior angle of a triangle equals the sum of the two opposite interior angles.
Triangle Inequality
Sum of any two sides of a triangle is greater than the third side.
Circle Tangent
Radius drawn to the point of contact is perpendicular to the tangent.
Cyclic Quadrilateral
Opposite angles of a cyclic quadrilateral are supplementary.

Tip: In Geometry, first identify the figure and then recall the theorem related to that figure.

Geometry Theorem Selection Guide

Geometry questions become easier when the student first identifies the figure and then selects the correct theorem. The table below helps choose the right property quickly.

Question Type What to Use Typical Clue
Angles at a point Sum of angles is \(360^\circ\) Several angles meet at one point
Angles on a straight line Linear pair rule Two adjacent angles form a straight line
Intersecting lines Vertically opposite angles are equal Two lines cross each other
Parallel lines Corresponding, alternate and co-interior angle rules Two parallel lines cut by a transversal
Triangle angle question Angle sum or exterior angle property Three angles, exterior angle, missing angle
Triangle side question Triangle inequality or side-angle relation Check possible sides or compare greater side and greater angle
Congruency question SSS, SAS, ASA, AAS or RHS Prove two triangles are equal in shape and size
Similarity question AA, SSS or SAS similarity Proportional sides or equal angles
Median question Centroid property Medians meet at one point; ratio \(2:1\)
Altitude question Orthocentre property Altitudes of a triangle are involved
Quadrilateral question Parallelogram, rectangle, square or rhombus properties Opposite sides, opposite angles or diagonals
Circle question Chord, angle, cyclic quadrilateral or tangent theorem Radius, chord, tangent, cyclic quadrilateral
Tangent question Radius is perpendicular to tangent at point of contact Circle touched by a line at one point
Locus question Identify the path satisfying the fixed condition Equidistant from point, line, two points or two lines
Exam shortcut: First draw or read the diagram carefully. Then mark equal sides, equal angles, parallel lines, tangents and known values. Most Geometry mistakes happen because the theorem is chosen before the figure is properly identified.

Lines and Angles

Lines and angles form the base of plane Geometry. Many problems are solved using angle properties at a point, on a straight line, and between intersecting lines.

Concept Property Use
Angles at a Point Sum is \(360^\circ\). Used when several angles meet at one point.
Linear Pair Sum is \(180^\circ\). Used when two adjacent angles form a straight line.
Vertically Opposite Angles They are equal. Used when two lines intersect.
Complementary Angles Sum is \(90^\circ\). Used in right angle problems.
Supplementary Angles Sum is \(180^\circ\). Used in straight line and cyclic quadrilateral problems.
Exam tip: Mark all known angles first, then use angle sum properties.

Parallel Lines and Transversals

When a line cuts two parallel lines, it is called a transversal. Several important angle pairs are formed.

Angle Pair Property When Lines Are Parallel Example Use
Corresponding Angles Equal If one is \(70^\circ\), corresponding angle is also \(70^\circ\).
Alternate Interior Angles Equal Used inside the two parallel lines.
Alternate Exterior Angles Equal Used outside the two parallel lines.
Co-interior Angles Supplementary Their sum is \(180^\circ\).
Rule: If corresponding angles are equal or alternate interior angles are equal, then the two lines are parallel.

Triangles: Sides and Angles

A triangle is a plane figure formed by three line segments. Triangle theorems are among the most frequently tested parts of Geometry.

Angle Sum Property

The sum of the three interior angles of a triangle is \(180^\circ\).

Exterior Angle Property

An exterior angle is equal to the sum of the two opposite interior angles.

Triangle Inequality

The sum of any two sides of a triangle is greater than the third side.

Side-Angle Relation

The side opposite the greater angle is longer. The angle opposite the longer side is greater.

Important: In a right triangle, Pythagoras theorem is used: \(Hypotenuse^2=Base^2+Height^2\).

Congruency of Triangles

Two triangles are congruent if they have the same shape and same size. Their corresponding sides and corresponding angles are equal.

Criterion Meaning Use
SSS Three sides of one triangle are equal to three sides of another triangle. Side-side-side congruency.
SAS Two sides and included angle are equal. Side-angle-side congruency.
ASA Two angles and included side are equal. Angle-side-angle congruency.
AAS Two angles and one corresponding side are equal. Angle-angle-side congruency.
RHS Right angle, hypotenuse and one side are equal. Used only for right-angled triangles.
Exam tip: After proving two triangles congruent, use CPCT: Corresponding Parts of Congruent Triangles are equal.

Similar Triangles

Two triangles are similar if they have the same shape but not necessarily the same size. Their corresponding angles are equal and corresponding sides are proportional.

Criterion Meaning Result
AA / AAA Similarity Corresponding angles are equal. Triangles are similar.
SSS Similarity Corresponding sides are proportional. Triangles are similar.
SAS Similarity Two sides are proportional and included angle is equal. Triangles are similar.
Important: Congruent triangles have the same shape and size. Similar triangles have the same shape but may have different sizes.

Concurrence of Medians and Altitudes

In a triangle, medians and altitudes meet at special points. These points are important in theorem-based Geometry questions.

Line / Point Definition Important Property
Median A line segment from a vertex to the midpoint of the opposite side. The three medians of a triangle are concurrent.
Centroid Point of concurrence of medians. Divides each median in the ratio \(2:1\).
Altitude A perpendicular drawn from a vertex to the opposite side or its extension. The three altitudes of a triangle are concurrent.
Orthocentre Point of concurrence of altitudes. May lie inside, outside or on the triangle depending on triangle type.
Remember: Centroid belongs to medians. Orthocentre belongs to altitudes.

Parallelogram, Rectangle and Square

A quadrilateral is a four-sided plane figure. Parallelogram, rectangle and square have important properties related to sides, angles and diagonals.

Figure Sides and Angles Diagonals
Parallelogram Opposite sides are equal and parallel. Opposite angles are equal. Diagonals bisect each other.
Rectangle Opposite sides are equal and parallel. Each angle is \(90^\circ\). Diagonals are equal and bisect each other.
Square All sides are equal. Each angle is \(90^\circ\). Diagonals are equal, perpendicular and bisect each other.
Rhombus All sides are equal. Opposite angles are equal. Diagonals are perpendicular and bisect each other.
Exam tip: A square has the properties of both a rectangle and a rhombus.

Circles, Tangents and Normals

A circle is the set of all points in a plane that are at a fixed distance from a fixed point. The fixed point is called the centre and the fixed distance is called the radius.

Concept Meaning / Property Use
Radius Line segment joining centre to any point on the circle. All radii of a circle are equal.
Diameter Chord passing through the centre. Diameter is twice the radius.
Chord Line segment joining two points on a circle. Equal chords subtend equal angles at the centre.
Angle in a Semicircle Angle in a semicircle is a right angle. Used to identify right triangles.
Cyclic Quadrilateral All four vertices lie on a circle. Opposite angles are supplementary.
Tangent A line touching the circle at exactly one point. Radius to point of contact is perpendicular to tangent.
Normal Line perpendicular to tangent at the point of contact. For a circle, normal passes through the centre.
Tangents from External Point Two tangents drawn from an external point are equal. If PA and PB are tangents, then \(PA=PB\).
Remember: In tangent questions, immediately draw the radius to the point of contact and mark it perpendicular to the tangent.

Loci

A locus is the path traced by a point that moves according to a fixed condition. Locus problems require converting a condition into a geometrical path.

Condition Locus Example
Point is at a fixed distance from a fixed point. Circle All points 5 cm from O form a circle with centre O and radius 5 cm.
Point is equidistant from two fixed points. Perpendicular bisector of the segment joining the two points. Locus of points equidistant from A and B.
Point is equidistant from two intersecting lines. Angle bisectors of the angles formed by the lines. Locus of points equidistant from two sides of an angle.
Point is at a fixed distance from a fixed line. A pair of lines parallel to the given line. All points 3 cm from a given line.
Important: Locus means path. Read the fixed condition carefully and then identify the path that satisfies it.

Step-by-Step Solving Method

Step Action Example Focus
Step 1 Draw or observe the figure carefully. Line, angle, triangle, quadrilateral, circle, tangent or locus.
Step 2 Mark the given information. Equal sides, equal angles, parallel lines, chords, tangents.
Step 3 Identify the theorem required. Angle property, triangle property, congruency, similarity, circle theorem.
Step 4 Apply the theorem step by step. Use angle relations, side relations, diagonal properties or tangent rules.
Step 5 Verify the result. Angles should total correctly and sides should satisfy the figure property.
Important: In Geometry, selecting the correct theorem is usually more important than lengthy calculation.

Solved Examples

Question Method Answer
Two angles form a linear pair. One angle is \(65^\circ\). Find the other angle. Linear pair angles sum to \(180^\circ\).
Other angle \(=180^\circ-65^\circ=115^\circ\). 		\(115^\circ\)
In a triangle, two angles are \(50^\circ\) and \(60^\circ\). Find the third angle. Sum of angles in a triangle \(=180^\circ\).
Third angle \(=180^\circ-(50^\circ+60^\circ)=70^\circ\). 		\(70^\circ\)
If two parallel lines are cut by a transversal and one corresponding angle is \(75^\circ\), find the corresponding angle. Corresponding angles are equal when lines are parallel. \(75^\circ\)
Two tangents PA and PB are drawn from external point P to a circle. If PA = 9 cm, find PB. Tangents drawn from an external point to a circle are equal.
Therefore, \(PA=PB\). 		9 cm
In a parallelogram, one angle is \(80^\circ\). Find the opposite angle. Opposite angles of a parallelogram are equal. \(80^\circ\)
In a rectangle, one diagonal is 10 cm. Find the other diagonal. Diagonals of a rectangle are equal. 10 cm
What is the locus of points at a distance of 4 cm from a fixed point O? Points at a fixed distance from a fixed point form a circle. Circle with centre O and radius 4 cm
What is the point of concurrence of the medians of a triangle called? The three medians of a triangle meet at a point called centroid. Centroid

Note: In theorem-based questions, mention the theorem or property used. This makes the solution clearer and more exam-ready.

Common Traps and Shortcuts

Common Traps
  • Confusing complementary and supplementary angles.
  • Forgetting that angles on a straight line sum to \(180^\circ\).
  • Using parallel line rules when lines are not given parallel.
  • Confusing congruent triangles with similar triangles.
  • Forgetting CPCT after proving congruency.
  • Assuming all quadrilaterals have equal diagonals.
  • Confusing rectangle, rhombus and square diagonal properties.
  • Forgetting radius is perpendicular to tangent at point of contact.
  • Misidentifying the locus condition.
Useful Shortcuts
  • Mark equal angles and equal sides immediately.
  • For parallel lines, check corresponding and alternate angles first.
  • For triangles, use angle sum and exterior angle properties.
  • For congruency, identify SSS, SAS, ASA, AAS or RHS.
  • For similarity, check AA similarity first.
  • For parallelograms, remember diagonals bisect each other.
  • For rectangles, remember diagonals are equal.
  • For squares, remember diagonals are equal and perpendicular.
  • For tangent questions, draw radius to point of contact.
Exam approach: Identify whether the question is based on lines and angles, parallel lines, triangles, congruency, similarity, medians and altitudes, quadrilaterals, circles, tangents, or loci.

Practice

A) Multiple Choice Questions
  1. The sum of angles around a point is:
    \(90^\circ\) \(180^\circ\) \(270^\circ\) \(360^\circ\)
  2. The sum of the interior angles of a triangle is:
    \(90^\circ\) \(120^\circ\) \(180^\circ\) \(360^\circ\)
  3. Which congruency criterion applies only to right-angled triangles?
    SSS SAS ASA RHS
  4. Opposite angles of a cyclic quadrilateral are:
    Equal Supplementary Complementary Always \(90^\circ\)
  5. The point of concurrence of medians of a triangle is called:
    Orthocentre Centroid Incentre Circumcentre
B) Solve the Higher-Order Problems
  1. Two angles form a straight line. One angle is \(125^\circ\). Find the other angle. (Hint: Angles on a straight line sum to \(180^\circ\).)
  2. In a triangle, two angles are \(45^\circ\) and \(75^\circ\). Find the third angle. (Hint: Sum of angles in a triangle is \(180^\circ\).)
  3. A tangent is drawn to a circle at point A. What is the angle between the tangent and radius OA? (Hint: Radius to point of contact is perpendicular to tangent.)
  4. In a rectangle, one diagonal is 13 cm. Find the length of the other diagonal. (Hint: Diagonals of a rectangle are equal.)
  5. What is the locus of points equidistant from two fixed points A and B? (Hint: Think of the perpendicular bisector of AB.)
C) Match the Concept with the Correct Rule
Concept Correct Rule / Meaning
Linear Pair Two adjacent angles whose sum is \(180^\circ\)
Vertically Opposite Angles Angles formed by intersecting lines that are equal
Congruent Triangles Triangles having same shape and same size
Similar Triangles Triangles having same shape but not necessarily same size
Centroid Point of concurrence of medians
Orthocentre Point of concurrence of altitudes
Tangent Line touching a circle at exactly one point
Locus Path traced by a point satisfying a fixed condition
Geometry Reminder

Geometry is a theorem-based and visual branch of mathematics. It requires clear understanding of lines, angles, triangles, quadrilaterals, circles, tangents, normals and loci. A neat diagram, correct marking and proper theorem selection are the keys to solving Geometry questions accurately.

Task: Create five Geometry questions using one question each from lines and angles, triangles, quadrilaterals, circles and loci.

Show Suggested Answers
Multiple Choice
  1. \(360^\circ\)
    The sum of angles around a point is \(360^\circ\).
  2. \(180^\circ\)
    The sum of the interior angles of a triangle is \(180^\circ\).
  3. RHS
    RHS congruency applies only to right-angled triangles.
  4. Supplementary
    Opposite angles of a cyclic quadrilateral sum to \(180^\circ\).
  5. Centroid
    The point of concurrence of medians is called centroid.
Higher-Order Problems
  1. Angles on a straight line sum to \(180^\circ\).
    Other angle \(=180^\circ-125^\circ=55^\circ\).
    Answer = \(55^\circ\).
  2. Sum of angles in a triangle \(=180^\circ\).
    Third angle \(=180^\circ-(45^\circ+75^\circ)=60^\circ\).
    Answer = \(60^\circ\).
  3. Radius drawn to the point of contact is perpendicular to tangent.
    Therefore, angle between tangent and radius \(=90^\circ\).
    Answer = \(90^\circ\).
  4. Diagonals of a rectangle are equal.
    If one diagonal is 13 cm, the other diagonal is also 13 cm.
    Answer = 13 cm.
  5. The locus of points equidistant from two fixed points A and B is the perpendicular bisector of segment AB.
    Answer = Perpendicular bisector of AB.
Concept Matching
  1. Linear Pair → Two adjacent angles whose sum is \(180^\circ\)
  2. Vertically Opposite Angles → Angles formed by intersecting lines that are equal
  3. Congruent Triangles → Triangles having same shape and same size
  4. Similar Triangles → Triangles having same shape but not necessarily same size
  5. Centroid → Point of concurrence of medians
  6. Orthocentre → Point of concurrence of altitudes
  7. Tangent → Line touching a circle at exactly one point
  8. Locus → Path traced by a point satisfying a fixed condition
Clue Explanation

Geometry problems are solved by identifying the figure and applying the right theorem. Lines and angles require angle relationships, triangles require angle sum, congruency or similarity, quadrilaterals require side-angle-diagonal properties, circles require chord and tangent theorems, and loci require understanding of the fixed condition.

Exam tips
  • Draw a clean diagram before solving.
  • Mark given equal sides and angles.
  • Use angle sum properties first.
  • Check parallel line conditions carefully.
  • Use CPCT after proving congruency.
  • Use AA rule quickly for similarity questions.
  • Remember rectangle, square and parallelogram diagonal properties.
  • In circle tangent questions, draw radius to the point of contact.
  • For loci, identify the fixed condition first.
Practice MCQs