Chapter 4 — Geometry

Basic geometric objects

A point shows a position in the plane and has no size. We usually mark a point with a dot and name it by a capital letter, for example A or P. A line is a straight path extending in both directions without end. It is shown with arrows at both ends and named by two letters, such as AB, or by a small letter. A line segment is part of a line with two endpoints; it has definite length and is written as AB where A and B are endpoints. A ray starts at an endpoint and continues forever in one direction; it is written as AB with the arrow shown only at B's side, meaning it starts at A and goes through B and beyond.

When drawing, use a ruler for straightness. For naming: the order of letters matters for rays but not for segments. Understanding these objects allows us to build other shapes by joining segments and intersecting lines. Intersections of lines and points where segments meet are important for constructing polygons.

Key practical skill: given two points, you can draw the segment joining them. Given a point and a direction, you can draw a ray. Given two points and arrows, you can show a line. These are the building blocks for all later constructions and proofs in geometry.

What is an angle?

An angle is formed when two rays share the same endpoint. The shared endpoint is called the vertex. The rays are called the arms of the angle. An angle measures the amount of turn from one arm to the other. We measure angles in degrees (°). A full circle is 360°, a straight line is 180°, a right angle is 90°. Learning to visualise and measure angles precisely helps in drawing shapes accurately and solving many geometric problems.

How to measure an angle

Use a protractor for measurement. Place the centre hole of the protractor on the vertex, align the baseline (zero mark) with one arm, and read the number where the other arm crosses the scale. Many protractors have inner and outer scales; choose the scale that starts at the aligned zero. When the arm falls near the outer scale use that value. Practice placing the protractor gently and reading to the nearest degree or half degree as needed. Label the measured angle with its degree measure.

Types of angles by size

• Acute angle: greater than 0° and less than 90°.
• Right angle: exactly 90°.
• Obtuse angle: greater than 90° but less than 180°.
• Straight angle: exactly 180° (arms opposite in a straight line).
• Reflex angle: greater than 180° and less than 360°.

Angle notation and addition

Angles are named using three letters with the vertex in the middle, for example ∠ABC where B is the vertex. If a point lies inside an angle, we can split the angle: for example if point D is inside ∠ABC, then ∠ABD + ∠DBC = ∠ABC. Angle addition helps break complex diagrams into smaller parts and set up equations to find unknown measures. Practise by drawing angles of different types, measuring them and recording their names and sizes. Accurate measurement and clear labelling are essential skills in geometry.

Angle relationships you will use often

Complementary and supplementary angles are simple but powerful ideas. Complementary angles are two angles whose measures add up to 90°. For example, a 30° angle and a 60° angle are complementary. These pairs often appear when a right angle is subdivided or when perpendicular lines form angles. Supplementary angles add up to 180°; for example, a 110° angle and a 70° angle are supplementary. These occur when two angles form a straight line or when adjacent angles around a point on a line sum to a straight angle.

How they appear in diagrams

If a line is drawn across two other lines, or if rays share a common endpoint on a straight line, complementary and supplementary relationships frequently help find unknown angles. For instance, if one angle adjacent to another along a straight line is given, subtracting from 180° finds the other. Similarly, if you know one part of a right angle, the other part is its complement.

Vertically opposite angles

When two straight lines cross, they form four angles. The angles that are opposite each other (across the intersection) are called vertically opposite angles and are equal. This equality is very useful: if one angle at intersection is known, you immediately know the opposite angle. Adjacent angles around the intersection are supplementary because they lie on a straight line. For problem solving, label angles clearly (for example ∠1, ∠2) and use equations such as ∠1 + ∠2 = 180° for straight lines or ∠1 = ∠3 for vertical opposites.

Using algebra with these ideas

Often a diagram gives angle expressions like (3x + 10)° and (2x + 20)°. If they are complementary set sum = 90, if supplementary set sum = 180, and solve for x. Always check your final angles are positive and of the correct type (acute/obtuse) for the situation. Practice with diagrams where multiple relationships combine: vertical opposites, supplementary pairs and complements can together determine unknowns quickly when labelled methodically.

Understanding triangles fully

A triangle is the simplest polygon with three sides, three vertices and three interior angles. A key fact you must remember is that the sum of the three interior angles of any triangle is 180°. This gives a powerful tool: when two angles are known, the third can be found by subtraction. Triangles are classified in two helpful ways: by their sides and by their angles. Use both views together to understand properties and to solve problems.

Classification by sides

An equilateral triangle has all three sides equal; because the angles must sum to 180°, each interior angle is 60°. An isosceles triangle has two equal sides; the base is the unequal side and the two base angles opposite equal sides are equal. A scalene triangle has all sides different and therefore all interior angles different. Knowing which sides are equal tells you which angles are equal and vice versa, and this is often used to find unknown measures.

Classification by angles

Based on angle sizes, a triangle can be acute (all three angles less than 90°), right-angled (one angle exactly 90°) or obtuse (one angle greater than 90°). For a right-angled triangle, the side opposite the right angle is called the hypotenuse and is the longest side. The relationship between side lengths and opposite angles is important: the largest side is opposite the largest angle, and the smallest side is opposite the smallest angle. This ordering helps in comparing sides or angles when only partial information is given.

Using properties in problems

Apply the angle sum rule and equal-angle rules in isosceles triangles. For example if two angles are given you can detect whether a triangle is acute or obtuse. Use geometric drawing and labelling: always mark equal sides with ticks and equal angles with arcs when showing a solution. These visual marks make reasoning clearer and reduce mistakes when applying angle rules or congruence later.

What does congruence mean?

Congruent figures have the same shape and same size. For triangles, congruence means that every corresponding side and every corresponding angle of one triangle equals that of the other. When triangles are congruent, we can say corresponding parts are equal without further measurement. This allows us to transfer known lengths or angles from one triangle to another and to prove other properties about a figure.

Common congruence criteria

There are several easy criteria you will use to decide when two triangles are congruent. SSS (side-side-side): if all three sides of one triangle equal the three sides of another, the triangles are congruent. SAS (side-angle-side): if two sides and the included angle of one triangle equal the two sides and included angle of another, they are congruent. ASA (angle-side-angle): if two angles and the included side of one equal two angles and the included side of another, triangles are congruent. These rules come from rigid geometry: once the matching parts are fixed, the triangle cannot change shape.

Careful matching

When using these criteria, match corresponding parts carefully. For SAS the angle must be between the two matched sides; for ASA the side must be between the two matched angles. Do not confuse ASA with AAS (two angles and a non-included side) which also leads to congruence at this level because two angles determine the third; but practise with ASA and SAS first. Note that AA (just two angles) alone does not give congruence because triangles could be similar but of different sizes.

How to present a congruence proof

State the matching parts and which rule applies (for example, AB = DE, BC = EF, AC = DF, hence SSS so ΔABC ≅ ΔDEF). Then write the consequences: corresponding angles or sides are equal. Use tick marks on diagrams to show equal sides and arcs to show equal angles; this makes your reasoning clear and follows ICSE style for geometric proofs.

Perimeter and area — what they mean

Perimeter measures the distance around a shape while area measures the space inside it. For rectangles and squares these measures are simple to compute because of their straight sides and right angles. Understanding the formulas and the units used is essential for solving mensuration problems correctly and for checking answers.

Perimeter of rectangles and squares

For a rectangle with length l and breadth b, opposite sides are equal so perimeter P = l + b + l + b = 2(l + b). This counts the total outer boundary. For a square with side s, all four sides equal so P = s + s + s + s = 4s. Always write the unit with your answer, for example cm or m. When given measurements in different units convert them first so units match before computing perimeter.

Area of rectangles and squares

Area of a rectangle is the product of its length and breadth, A = l × b. This follows because a rectangle can be tiled by unit squares: length times breadth gives the number of unit squares inside. For a square, since l = b = s, the area is A = s × s = s2. The units for area become square units, for example cm2 or m2. Be careful to square the unit when writing the final answer.

Worked approach and checking

When solving problems, draw and label the figure first. Substitute numerical values into the formula and perform arithmetic carefully. If dimensions are given in mixed units convert to the same unit before using formulas. Check if the answer is reasonable: area should be positive and perimeter should be longer than any single side. For composite shapes built from rectangles and squares, divide the figure into parts, find each area, and add or subtract as required.

Area formula applies to triangles

For any triangle the area equals half the product of a base and its corresponding height: A = 1/2 × base × height. The height is the length of the perpendicular dropped from the opposite vertex to the chosen base. In a right-angled triangle the two legs are perpendicular, so one leg can be treated as the base and the other as the height. This makes area calculation straightforward for right triangles.

Perimeter and the hypotenuse

The perimeter of a triangle is the sum of its three sides. For a right-angled triangle with legs a and b and hypotenuse c, the perimeter P = a + b + c. Often you are given the legs and must find the hypotenuse to compute the perimeter. The hypotenuse can be found using the Pythagorean relation: c2 = a2 + b2, so c = √(a2 + b2). At Class 7 you should be comfortable applying this to whole-number examples such as 3-4-5 triangles, and to find missing sides in simple problems.

Using area in algebraic problems

Sometimes problems provide the area and one dimension and ask for another. Rearrange the area formula: height = (2 × area) / base. Use this when a right triangle’s area is known and you need a leg length. Always keep units consistent. Sketch the triangle, label legs and hypotenuse, and mark the right angle clearly. When working with integer values check if the hypotenuse is a whole number; if not, leave it as a square root and give the perimeter in exact form unless decimal approximation is requested.

Practical tips

Use the Pythagorean relationship only for right-angled triangles. For area, choose the base where the height is easy to find. In diagrams practice drawing the altitude if it is not one of the sides. Always write the final answers with proper units and simplify square root expressions when possible for clarity.

Understanding four-sided figures

A quadrilateral is any polygon with four sides and four vertices. Because there is a wide variety, it is useful to study special types that have extra properties. These special quadrilaterals — square, rectangle, parallelogram, rhombus, trapezium (trapezoid), and kite — appear often in problems and real life. Learn their defining features and diagonal behaviour to answer most class problems confidently.

Square and rectangle

A square has four equal sides and four right angles. Its diagonals are equal, they bisect each other and are at right angles only in square they are equal and perpendicular. A rectangle has opposite sides equal and all angles right. Diagonals are equal in length and bisect each other, but in a rectangle they are not generally perpendicular unless it is a square.

Parallelogram and rhombus

A parallelogram has both pairs of opposite sides parallel and equal. Opposite angles are equal, and adjacent angles are supplementary. Its diagonals bisect each other but are not necessarily equal. A rhombus has all sides equal (like a slanted square) and opposite sides parallel; diagonals of a rhombus bisect the angles and meet at right angles, and they bisect each other as well. These diagonal properties help in many constructions and proofs.

Trapezium and kite

A trapezium has at least one pair of parallel sides; the parallel sides are called bases. Many angle relationships around transversals apply here. A kite has two pairs of adjacent equal sides; one diagonal often bisects the other at right angles. Recognising these patterns in diagrams lets you write equations to find missing angles or sides. For example, in a parallelogram if one angle is 70° then opposite is 70° and adjacent are 110°.

Problem approach

When given a quadrilateral, label sides and mark equal lengths and equal angles with ticks and arcs. Use parallelism to relate angles (alternate and corresponding). Check diagonal properties depending on the type. Sketches and clear labels make reasoning straightforward and reduce errors in calculation or proof.

Basic parts of a circle

A circle is the set of all points that are at the same distance from a fixed point called the centre. The fixed distance is the radius. The diameter is a special chord passing through the centre; its length equals twice the radius. A chord is any segment joining two points on the circle. The circumference is the complete round boundary of the circle. An arc is part of the circumference between two points. A sector is the region bounded by two radii and the included arc; a segment is the region between a chord and the arc it cuts off.

Relationships and simple facts

Important simple facts you should know: all radii in a circle are equal; a diameter is the longest chord; a perpendicular drawn from the centre to a chord bisects the chord and the corresponding arcs; conversely, the line joining centre to the midpoint of a chord is perpendicular to the chord. If a diameter meets the circumference at points A and B, the arc AB is a semicircle and the angle subtended by the semicircle at any point on the circumference is a right angle (Thales' idea in simple form).

Using these properties

These properties help solve construction and reasoning problems: for example, to find the centre of a given circle you can construct perpendicular bisectors of two chords; their intersection is the centre. To check if a line is a tangent (introduced later) note that radius to point of contact is perpendicular to the tangent. In Class 7 you mainly identify parts, use d = 2r, and apply the chord and perpendicular facts. While formulas for circumference and area involve π and are usually covered later, you should be comfortable with naming and drawing sectors, chords and diameters and using basic perpendicular bisector facts in constructions and proofs.

Why constructions are useful

Geometric constructions with compass and ruler teach precise methods to create lines, angles and points that satisfy given conditions without measuring. Two of the most used constructions are the perpendicular bisector of a segment and the angle bisector. These help locate midpoints, centres of circles, and divide angles into equal parts. Follow the steps carefully and keep the compass width constant where required.

Perpendicular bisector of a segment

To construct the perpendicular bisector of segment AB, first set the compass to a radius greater than half of AB. With the compass point at A draw arcs above and below the segment. Without changing the compass width, draw similar arcs from B so that the arcs from A and B intersect above and below the segment. Join the two intersection points of the arcs with a straightedge; this line will cross AB at its midpoint and will be perpendicular to AB. The reason this works is that the intersection points are equidistant from A and B, so the line joining them is the locus of points equidistant from A and B and therefore the perpendicular bisector.

Angle bisector

To bisect ∠X formed by rays XA and XB, set the compass at X and draw an arc that meets both arms at points P and Q. With the same compass radius draw arcs from P and Q that intersect at point R on the interior of the angle. Draw the ray XR. This ray XR divides ∠AXB into two equal angles. The construction works because R is equidistant from the arms, giving equal arcs and therefore equal angles.

Practice and verification

After construction measure the produced angles or use folding to check they are equal. For perpendicular bisector check that the intersection point on the segment is equidistant from the endpoints by measuring or by constructing congruent triangles. Learn to describe each step clearly in answers and to justify why the construction gives the claimed property — this builds logical explanation skills useful in geometry.

What are parallel lines?

Parallel lines are lines in a plane that never meet no matter how far they are extended. They keep a constant separation. When another line, called a transversal, crosses two parallel lines, many useful angle relationships are created. Recognising and using these relationships helps in finding unknown angles and in proving lines are parallel in reverse.

Angle pairs formed by a transversal

There are several named pairs of angles that appear. Corresponding angles are in matching positions relative to the two parallel lines and the transversal; if the lines are parallel these are equal. Alternate interior angles are on opposite sides of the transversal and between the two lines; they are equal when lines are parallel. Alternate exterior angles are on opposite sides of the transversal and outside the two lines; they are also equal if the lines are parallel. Consecutive interior (or co-interior) angles lie on the same side of the transversal and between the lines; when the lines are parallel their measures add to 180°.

Using the relationships in problems

Label the diagram clearly with the angles and identify which pair you can use. For example, if one angle at the top left is 70° then the corresponding angle at the bottom left is 70°, and the adjacent interior angle on the same side is 110° because they are supplementary. These facts allow you to set up equations when angles are given as algebraic expressions. For instance if one corresponding angle is (3x+10)° and its match is (2x+40)°, set them equal and solve for x, then substitute back to find specific angles.

Converse statements

The converse facts are just as useful: if corresponding angles are equal then the lines are parallel; if alternate interior angles are equal then the lines are parallel. Use these converses to prove a pair of lines are parallel given angle conditions. Practise many diagrams with different transversal positions to become quick at spotting which rule applies. Clear labelling and step-by-step reasoning keep answers neat and convincing.

Understanding symmetry visually

Symmetry describes balance and repetition in figures. It helps in recognising patterns and simplifying geometric reasoning. Two main types are line (mirror) symmetry and rotational symmetry. Identifying symmetry gives quick information about equal parts of a figure and is often used in design, art and mathematics. Practice by folding shapes on paper or by imagining rotations to see how they match themselves.

Line symmetry

A figure has line symmetry if there is a line (axis) such that folding the figure along that line makes the two halves coincide. The axis divides the figure into mirror-image halves. Simple examples: an isosceles triangle has one axis through the apex and midpoint of base; a rectangle has two axes through midlines; a circle has infinitely many axes through its centre. To test, fold paper on the proposed axis or draw perpendiculars to check corresponding points match. When a figure has line symmetry, corresponding points on either side are at equal distances from the axis.

Rotational symmetry

A figure has rotational symmetry if rotating it about a central point by some angle less than 360° maps the figure onto itself. The number of times it maps onto itself during a full 360° rotation is the order of rotational symmetry. For example, a square has order 4 because rotations by 90°, 180° and 270° bring it to the same appearance; an equilateral triangle has order 3 (rotations by 120° and 240°). The smallest positive angle giving the match is called the angle of rotation.

Connections and problem solving

Some shapes have both types: a regular polygon often has multiple axes and rotational symmetry equal to the number of sides. Use symmetry to deduce equal lengths or angles without calculation — for example symmetric halves imply equal areas. When asked to draw axes, place them through vertices and centres as required. State the order of rotational symmetry and sketch the positions the figure takes under rotation. Showing how symmetry reduces work is a good method in exam answers.

Composite figures and strategy

Composite figures are shapes made by joining or cutting away simple shapes like rectangles, squares and triangles. To find perimeter and area you should not try to memorize new formulas. Instead break the figure into parts whose areas or side lengths you can find, compute each part, then add or subtract as needed. Clear labelling and separate working for each part prevent mistakes and make your answer easy to follow.

Area method

For area, divide the composite figure into rectangles, squares and triangles so that their individual areas can be calculated by known formulas. If a region is missing (a hole), compute the area of the larger shape first and subtract the hole’s area. Keep track of units: area results are in square units. When dimensions are not given directly for a part, use subtraction of lengths to determine them from the whole; always sketch and mark those derived lengths on your diagram.

Perimeter method

Perimeter of a composite figure is the total length around the outer boundary only. Do not include internal dividing lines. To compute the perimeter, list the outer segments in order and add their lengths. If some outer sides are not given but can be found from other dimensions or using the Pythagorean theorem for right-angled parts, compute them before summing. Be careful with overlapping edges; ensure each outer edge is counted exactly once.

Worked approach and checks

Begin by drawing the figure, label known lengths, and separate the figure into parts with dotted lines if needed. Write formulas for each part and substitute values. After computing, check units and compare magnitude: the area should be reasonable given the outer dimensions and perimeter should be larger than any single side but not excessively large. Practice with L-shaped figures, figures with triangular cutouts, and combinations of rectangles and triangles to build confidence.

Why coordinates help

The coordinate plane gives a simple way to locate points precisely and to connect geometry with arithmetic. Two number lines, the horizontal x-axis and the vertical y-axis, meet at the origin (0,0). Every point on the plane can be described by an ordered pair (x, y) where x is the horizontal position (abscissa) and y is the vertical position (ordinate). Learning to plot and read points accurately opens the path to analytic geometry where algebra and geometry work together.

How to plot a point

To plot (x, y) start at the origin. Move horizontally x units: right if x is positive, left if negative. Then move vertically y units: up if y is positive, down if negative. Put a dot and label it. For example, to plot (3, 2) move 3 units to the right and 2 units up. To plot (−2, 4) move 2 units left and 4 units up. Practice with various quadrants so you are comfortable with positive and negative coordinates.

Shapes and simple distances

When you join plotted points you can draw shapes. If sides are horizontal or vertical, their lengths are differences of x-coordinates or y-coordinates respectively. For example, a rectangle with corners at (1,1), (4,1), (4,3), (1,3) has base length 3 and height 2, so area is 6. Distance between two points on the same vertical line is the difference of their y-values; on the same horizontal line use difference of x-values. These simple ideas let you compute perimeter or area for axis-aligned shapes quickly.

Checking and using the grid

Always label axes and scale, and mark units on grid paper. Ensure equal spacing so plotted positions are accurate. Coordinate geometry problems at this level focus on plotting, reading coordinates, drawing simple shapes and using coordinate differences to find lengths and areas. This introduction prepares you for distance and midpoint formulas studied later, and helps visualise algebraic solutions geometrically.