Geometry:
Shapes, Lines & Space
Five geometry topics, explained simply. Master the rules, understand the reasons, and tackle any exam question with confidence.
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Book Tutoring →Think of a straight road as a straight line (180 degrees). When another road crosses it, the angles formed on either side of the intersection always follow specific rules. Those rules are what we study here.
All geometry angle problems are solved by applying a small set of rules. Once you know them, you can find any unknown angle.
Angles on a Straight Line
All angles on one side of a straight line add up to 180 degrees.
Angles Around a Point
All angles around a single point add up to 360 degrees.
Vertically Opposite
When two straight lines cross, the angles directly opposite each other are equal.
Right Angle
A right angle is exactly 90 degrees. Two lines that form a right angle are perpendicular.
Complementary and Supplementary Angles
| Type | Sum | Example | Find the unknown |
|---|---|---|---|
| Complementary | = 90° | 35° and x | x = 90° − 35° = 55° |
| Supplementary | = 180° | 112° and x | x = 180° − 112° = 68° |
| Adj supplementary | = 180° | x and 2x on a line | 3x = 180°, x = 60° |
Parallel Lines cut by a Transversal
In every geometry answer you must write a reason for every angle you calculate. "Corresp ∠s; AB ∥ CD" is a reason. "Because I know it" is not. You will lose marks without reasons.
These rules are used in almost every other geometry topic. Triangles, proofs, similarity - they all rely on these angle relationships. This is the grammar of geometry.
A triangle is the strongest shape in construction - you see it in bridges, rooftops and towers. That strength comes from fixed relationships between its sides and angles. Those relationships are what this topic is about.
The three interior angles of ANY triangle always add up to exactly 180 degrees. No exceptions. Ever. This is the most useful rule in all of triangle geometry.
Equilateral
All 3 sides equal. All 3 angles equal.
Isosceles
2 sides equal. The 2 base angles (opposite the equal sides) are equal.
Scalene
All 3 sides different lengths. All 3 angles different sizes.
Right-angled
One angle is exactly 90 degrees. The side opposite it is the longest side (hypotenuse).
Exterior Angle Theorem
| Triangle Type | Sides | Angles | Symbol |
|---|---|---|---|
| Equilateral | All equal | All 60° | 3 tick marks on sides |
| Isosceles | 2 equal | 2 base angles equal | 2 tick marks + arc marks |
| Scalene | All different | All different | No marks |
| Right-angled | Any | One = 90° | Small square at right angle |
If a triangle has two equal sides marked, immediately write down that the base angles are equal. This is almost always how you unlock the rest of the calculation. Reason: "isosceles △, equal sides."
Every shape can be broken into triangles. Triangles are the foundation of 2D geometry. Master this topic and proofs, congruency and similarity all become manageable.
Imagine a factory stamping out identical metal parts. Every part has the same shape and the same size - just maybe facing a different direction. That is congruency. Two shapes are congruent if one can be flipped, rotated or slid to fit exactly on top of the other.
Two triangles are congruent if all their corresponding sides and all their corresponding angles are equal. We write this as △ABC ≡ △DEF. The order of the letters matters - it tells you which vertices correspond.
The 4 Conditions for Congruency (learn all four)
SSS
3 sides equal
SAS
2 sides + included angle
AAS
2 angles + any side
RHS
Right angle + hypotenuse + side
When writing a congruency proof, state each pair of equal parts with a reason. Then state which condition (SSS / SAS / AAS / RHS) applies. End with: "therefore △ABC ≡ △DEF (SAS)". The condition in brackets is non-negotiable.
Congruent = same shape AND same size. Similar = same shape, different size. If triangles are congruent, they are also similar - but not the other way around. Think of it as congruency being the stricter version.
A photo and an enlarged print of the same photo are similar. The shape is identical, the proportions are the same, but the sizes differ. A map and the actual landscape it represents are similar figures - the angles are all the same, the distances are just scaled up by the same factor.
Two triangles are similar if their corresponding angles are all equal AND their corresponding sides are all in the same ratio (proportion). We write △ABC ||| △DEF. The symbol ||| means similar.
Conditions for Similarity
Working with the Scale Factor
AB/DE = BC/EF → 6/4 = 9/EF → EF = 9 x 4/6 = 6.
| Property | Congruent Triangles | Similar Triangles |
|---|---|---|
| Corresponding angles | Equal | Equal |
| Corresponding sides | Equal (same length) | In proportion (same ratio) |
| Size | Identical | May differ |
| Symbol | ≡ | ||| |
Always write your proportion with corresponding sides in the SAME position (numerator/denominator). Mixing them up is the most common similarity error. Check: small triangle sides on top, large on bottom (or vice versa - but stay consistent).
Similarity is how we calculate heights and distances we cannot measure directly. Surveyors, architects and engineers use similar triangles to find the height of a building, the width of a river, or the distance to a distant object.
A builder wants to check that a wall is perfectly vertical (at 90 degrees to the floor). They measure 3 metres along the floor from the wall and 4 metres up the wall. If the diagonal distance between those two points is exactly 5 metres, the corner is a perfect right angle. That is Pythagoras in action, used on building sites every day.
The Three Types of Pythagoras Question
| Find | Formula | Example | Answer |
|---|---|---|---|
| Hypotenuse (c) | c = √(a² + b²) | a=3, b=4 | c = √(9+16) = √25 = 5 |
| Shorter side (a) | a = √(c² - b²) | c=13, b=5 | a = √(169-25) = √144 = 12 |
| Is it right-angled? | Check: a²+b² = c² | 5, 12, 13 | 25+144=169 ✓ Yes |
Always identify the hypotenuse FIRST before writing the formula. The hypotenuse is opposite the right angle (marked with a small square). It is always the longest side. Mixing up which side is c is the single biggest Pythagoras mistake.
Pythagoras connects all of geometry. It is used in coordinate geometry (distance formula), trigonometry, 3D problems, and real-world calculations from navigation to construction. It is also one of the most tested theorems in the entire Grade 9 syllabus.
Quadrilateral questions appear in almost every geometry paper. You will be given a shape with some information and asked to find unknown angles or side lengths. The only way to answer those questions is to know the properties of each shape cold. Properties of sides, angles and diagonals are all fair game.
A quadrilateral is any closed shape with exactly FOUR sides. The sum of interior angles in any quadrilateral is always 360°. The six quadrilaterals you must know for Grade 9 are: square, rectangle, rhombus, parallelogram, trapezium, and kite.
Properties of Each Quadrilateral
Square
Sides: All 4 sides equal. Opposite sides parallel.
Angles: All 4 angles = 90°.
Diagonals: Equal in length. Bisect each other at 90°. Bisect corner angles (each = 45°).
Rectangle
Sides: Opposite sides equal and parallel.
Angles: All 4 angles = 90°.
Diagonals: Equal in length. Bisect each other (but NOT at 90°).
Rhombus
Sides: All 4 sides equal. Opposite sides parallel.
Angles: Opposite angles equal. Co-interior angles supplementary. Angles are NOT necessarily 90°.
Diagonals: Bisect each other at 90°. Bisect corner angles. NOT necessarily equal in length.
Parallelogram
Sides: Opposite sides equal and parallel.
Angles: Opposite angles equal. Co-interior angles supplementary (add to 180°).
Diagonals: Bisect each other. NOT necessarily equal or at 90°.
Trapezium
Sides: ONE pair of parallel sides (called the parallel sides or bases). Other two sides are NOT parallel.
Angles: Co-interior angles between parallel sides are supplementary.
Diagonals: No special properties (unless it is an isosceles trapezium).
Kite
Sides: Two pairs of adjacent (touching) sides are equal. No sides are parallel.
Angles: One pair of opposite angles equal (the angles between the unequal sides).
Diagonals: One diagonal bisects the other at 90°. The longer diagonal bisects the two angles at its ends.
Quick Comparison Table
| Shape | All sides = | All angles 90° | Opp sides ∥ | Diag bisect @90° | Diag equal |
|---|---|---|---|---|---|
| Square | ✓ | ✓ | ✓ | ✓ | ✓ |
| Rectangle | ✗ | ✓ | ✓ | ✗ | ✓ |
| Rhombus | ✓ | ✗ | ✓ | ✓ | ✗ |
| Parallelogram | ✗ | ✗ | ✓ | ✗ | ✗ |
| Trapezium | ✗ | ✗ | One pair only | ✗ | ✗ |
| Kite | ✗ | ✗ | ✗ | One diag only | ✗ |
Using Properties to Find Unknown Angles and Sides
In geometry, every step in your working must be justified with a reason in brackets. For example: x = 60° (opposite angles of a parallelogram are equal). Without the reason you lose the method mark even if the value is correct. This applies to all six quadrilaterals.
A rhombus has ALL FOUR SIDES EQUAL - just like a square. But a rhombus does NOT have 90° angles (unless it is a square). Do not assume angles are right angles just because all sides are equal. A square is a special rhombus where all angles are also 90°. The diagonals of a rhombus cross at 90°, but the diagonal lengths are NOT equal.
Quick Reference
Six Rules to Always Know
Straight Lines
Angles on a line = 180° (supplementary). Around a point = 360°. Complementary = 90°. Vert opp equal.
Triangles
Angles sum to 180°. Ext angle = sum of 2 non-adjacent interior angles.
Congruency
SSS, SAS, AAS, RHS. Same shape AND same size. Symbol: ≡
Similarity
Equal angles + sides in proportion. Same shape, different size. Symbol: |||
Pythagoras
a² + b² = c². Hypotenuse is opposite the right angle. Always longest side.
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