📏 Geometry & Congruence
Geometry questions test your knowledge of angle properties, triangle theorems, polygon rules, congruence, similarity, and transformations. The most important thing in geometry is to always state the property or theorem you are using — marks are awarded for reasoning, not just the final answer.
Geometry in O-Level E-Math is about knowing a toolkit of angle properties and applying them systematically. Most questions involve finding unknown angles by applying 2-3 properties in sequence. The key to success is recognising which property to use, applying it correctly, and stating it clearly in your working.
Angle Properties of Lines
These are the foundational angle properties. You must memorise them and be able to quote them precisely. Examiners look for the exact property name in your working, so do not just write "angles add up" — write the specific property such as "angles on a straight line sum to 180 degrees".
Essential Angle Properties
Angles on a straight line
Sum to 180 degrees. Two or more adjacent angles that together form a straight line always add up to 180 degrees.
Vertically opposite angles
Are equal. When two straight lines intersect, the opposite angles formed are always equal.
Angles at a point
Sum to 360 degrees. All angles around a single point add up to one complete revolution.
Alternate angles (Z-angles)
Equal when lines are parallel. Look for the Z-shape (or S-shape) between two parallel lines cut by a transversal.
Corresponding angles (F-angles)
Equal when lines are parallel. Look for the F-shape between two parallel lines cut by a transversal.
Co-interior angles (C/U-angles)
Sum to 180 degrees when lines are parallel. Same-side interior angles between parallel lines.
Angles on a straight line
Vertically opposite angles
Angles at a point
Alternate angles (Z-angles)
Corresponding angles (F-angles)
Co-interior angles (C-angles)
Triangle angle sum (180°) and exterior angle theorem
📐Angle Properties
Fundamental angle relationships for lines, triangles, and polygons.
📝Worked Example 1: Parallel Lines and Angles
Lines AB and CD are parallel. A transversal crosses them, making angle BAE = 65 degrees at the top line. Find angle DCE at the bottom if angle DCE and angle BAE are co-interior.
Parallel lines AB and CD with a transversal through E and F
📝Worked Example 2: Multi-Step Angle Problem
In the figure, PQ is parallel to RS. Angle QPT = 42 degrees and angle TRS = 75 degrees. Find angle PTR.
PQ is parallel to RS with point T between the lines
Polygon Angle Properties
A polygon is a closed shape with straight sides. The formulas below apply to any polygon. For regular polygons (all sides and angles equal), we can find each individual angle. The exterior angle and interior angle at any vertex are supplementary (add to 180 degrees).
Regular hexagon: interior angle = 120°, exterior angle = 60°
🔷Polygon Properties
Interior and exterior angle formulas for regular and irregular polygons.
📝Worked Example 3: Finding the Number of Sides
Each interior angle of a regular polygon is 156 degrees. How many sides does it have?
Regular hexagon: interior angle = 120°, exterior angle = 60°
📝Worked Example 4: Interior Angle Sum
Find the sum of interior angles of a decagon (10 sides). If the decagon is regular, find each interior angle.
Regular hexagon: interior angle = 120°, exterior angle = 60°
What is the sum of interior angles of a regular octagon (8 sides)?
Congruence & Similarity
Two figures are congruent if they are exactly the same shape AND size. Two figures are similar if they are the same shape but may be different sizes. For triangles, there are specific tests to prove congruence or similarity without checking every measurement.
🔄Congruence & Similarity
Tests for congruent triangles and properties of similar figures.
4 Congruence Tests for Triangles
SSS (Side-Side-Side)
All three sides of one triangle equal the corresponding three sides of another.
SAS (Side-Angle-Side)
Two sides and the INCLUDED angle (the angle between those two sides) are equal.
AAS or ASA
Two angles and a corresponding side are equal. (If two angles match, the third automatically matches too.)
RHS (Right angle-Hypotenuse-Side)
Both triangles have a right angle, equal hypotenuses, and one other equal side.
SSS
Side-Side-Side
SAS
Side-Angle-Side
AAS
Angle-Angle-Side
RHS
Right angle-Hypotenuse-Side
AAA Does NOT Prove Congruence!
Having all three angles equal (AAA) proves similarity, NOT congruence. Two triangles can have identical angles but be completely different sizes. This is a favourite trap in exam questions. AAA establishes similar shapes; you need at least one matching side for congruence.
📝Worked Example 5: Similarity and Area Ratio
Triangles ABC and DEF are similar with corresponding sides in the ratio 3 : 5. If the area of triangle ABC is 27 cm squared, find the area of triangle DEF.
Similar triangles BDE and BAC with right angles at D and A
📝Worked Example 6: Similarity and Volume Ratio
Two similar cylinders have heights 6 cm and 9 cm. The smaller cylinder has volume 48pi cm cubed. Find the volume of the larger cylinder.
Two similar triangles have sides in ratio 2:3. If the smaller triangle has area 16 cm, find the area of the larger.
⚠️Common Mistakes — Geometry
Why: Students mix up interior and exterior angle formulas. Interior + Exterior = 180 degrees for each vertex.
Why: AAA proves similarity only, not congruence. Congruence requires SSS, SAS, AAS, or RHS.
Why: SOH CAH TOA: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.
Similarity: Length, Area, Volume Ratios
When two shapes are similar with a length ratio of , the area ratio is and the volume ratio is . This applies to all similar shapes, not just triangles. Exam questions frequently test your ability to move between these ratios. A common mistake is confusing the length ratio with the area ratio.
| If Length Ratio Is | Area Ratio Is | Volume Ratio Is |
|---|---|---|
| 1 : 2 | 1 : 4 | 1 : 8 |
| 2 : 3 | 4 : 9 | 8 : 27 |
| 3 : 5 | 9 : 25 | 27 : 125 |
📝Worked Example 6b: Finding Length Ratio from Area Ratio
Two similar shapes have areas in the ratio 16 : 49. Find the ratio of their corresponding lengths and the ratio of their volumes.
📝Worked Example 6c: Mass and Similarity
Two similar statues are made from the same material. The smaller statue is 15 cm tall and weighs 2 kg. The larger statue is 25 cm tall. Find its weight.
📝Worked Example 6d: Finding a Missing Side Using Similarity
In triangle ABC, D is on AB and E is on AC such that DE is parallel to BC. If AD = 4 cm, DB = 6 cm, and BC = 15 cm, find DE.
Similar triangles BDE and BAC with right angles at D and A
Two similar solids have volumes in the ratio 8:27. What is the ratio of their surface areas?
Symmetry & Transformations
Transformations move a shape from one position to another. There are four types: translation (slide), reflection (flip), rotation (turn), and enlargement (resize). For each transformation, you need to be able to describe it fully and find image or object points.
🪞Symmetry & Transformations
Properties of line symmetry, rotational symmetry, and basic transformations.
📝Worked Example 7: Describing a Transformation
Triangle A has vertices (1,1), (3,1), (1,3). Triangle B has vertices (5,1), (7,1), (5,3). Describe the transformation from A to B.
Triangle A reflected across the y-axis to form Triangle B
📝Worked Example 8: Enlargement
Triangle PQR is enlarged by scale factor 2 with centre of enlargement at the origin. P = (1, 2), Q = (3, 1), R = (2, 4). Find the image triangle.
Triangle A reflected across the y-axis to form Triangle B
📝Worked Example 9: Map Scales
A map has scale 1:50000. Two towns are 6 cm apart on the map. Find the actual distance in km. If the actual area of a park is 2 km squared, find the area on the map in cm squared.
A shape has area 12 cm. It is enlarged by scale factor 3. What is the new area?
Geometry Exam Strategy
For geometry questions, always draw a large, clear diagram and label all given information. Mark parallel lines with arrows and equal lengths with tick marks. When stating a reason, use the exact property name from the syllabus. Partial marks are given for correct reasoning even if the final answer is wrong.
Proving Congruence in Exam Questions
Congruence proofs are a common structured question in Paper 2, typically worth 3-4 marks. You must identify the correct congruence test (SSS, SAS, AAS, or RHS), state each corresponding pair of equal sides or angles with reasons, and then conclude with the congruence statement. The congruence statement must list vertices in the correct corresponding order.
📝Worked Example 10: Proving Triangles Are Congruent
ABCD is a rectangle. M is the midpoint of AB. Prove that triangles AMD and BMC are congruent.
Rectangle ABCD with M as midpoint of AB (AM = MB, AD = BC)
📝Worked Example 11: Proving Similarity
In the figure, angle BAC = angle BDE = 90 degrees. AB = 6 cm, AC = 8 cm, BD = 3 cm. Show that triangles BAC and BDE are similar, and find DE.
Similar triangles BDE and BAC with right angles at D and A
Properties of Special Quadrilaterals
You should know the properties of all special quadrilaterals. These properties are often needed in geometry proofs and angle-finding questions. Each quadrilateral has specific properties about its sides, angles, and diagonals.
| Quadrilateral | Key Properties |
|---|---|
| Parallelogram | Opposite sides equal and parallel. Opposite angles equal. Diagonals bisect each other. |
| Rectangle | All angles 90 degrees. Diagonals equal and bisect each other. Opposite sides equal and parallel. |
| Rhombus | All sides equal. Diagonals bisect each other at 90 degrees. Opposite angles equal. |
| Square | All sides equal. All angles 90 degrees. Diagonals equal, bisect at 90 degrees. |
| Trapezium | One pair of parallel sides. Co-interior angles between parallel sides sum to 180 degrees. |
| Kite | Two pairs of adjacent equal sides. One pair of opposite angles equal. Diagonals perpendicular. |
Rectangle ABCD with M as midpoint of AB (AM = MB, AD = BC)
📝Worked Example 12: Properties of a Parallelogram
ABCD is a parallelogram. Angle A = (3x + 10) degrees and angle B = (2x + 20) degrees. Find x and the angles.
📝Worked Example 13: Midpoint Theorem
In triangle ABC, M is the midpoint of AB and N is the midpoint of AC. MN = 7 cm. Find BC.
M and N are midpoints of AB and AC respectively
Describing Transformations Fully
When describing a transformation in the exam, you must give all required information. Incomplete descriptions lose marks. Here is what you need for each type:
What to Include When Describing a Transformation
Translation
State the column vector (horizontal, vertical displacement). Example: "Translation by column vector (3, -2)".
Reflection
State the line of reflection (equation). Example: "Reflection in the line y = x" or "Reflection in the line x = -1".
Rotation
State THREE things: (1) centre of rotation, (2) angle of rotation, (3) direction (clockwise or anticlockwise). Example: "Rotation 90 degrees clockwise about (0, 0)".
Enlargement
State TWO things: (1) centre of enlargement, (2) scale factor. A negative scale factor means the image is on the opposite side of the centre. Example: "Enlargement scale factor 2, centre (1, 3)".
📝Worked Example 14: Reflection
Triangle A has vertices (1,1), (3,1), (1,4). Triangle B has vertices (-1,1), (-3,1), (-1,4). Describe the transformation from A to B.
Triangle A reflected across the y-axis to form Triangle B
📝Worked Example 15: Rotation
Triangle P has vertices (2,1), (4,1), (2,3). Triangle Q has vertices (-1,2), (-1,4), (-3,2). Describe the transformation from P to Q.
Triangle A reflected across the y-axis to form Triangle B
A rotation is described as '90 degrees about (2,1)'. What important detail is missing?