Geometry: Angles, Lines and Triangles

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Geometry is the study of shapes, sizes, positions, and properties of figures in space. In the NCEE, you need to understand angles, the relationships between lines, and the properties of triangles.

Types of Angles:

Angle Type	Size	Description
Acute	$0° < x < 90°$	Smaller than a right angle
Right	Exactly $90°$	Quarter turn; forms an L shape
Obtuse	$90° < x < 180°$	Between a right angle and a straight line
Straight	Exactly $180°$	A straight line
Reflex	$180° < x < 360°$	More than a straight line
Complementary	Sum = $90°$	Two angles that add to 90°
Supplementary	Sum = $180°$	Two angles that add to 180°

Angle Facts:

• Angles around a point add up to $360°$
• Angles on a straight line add up to $180°$
• Vertically opposite angles are equal (when two lines cross)
• Complementary angles: $x + y = 90°$
• Supplementary angles: $x + y = 180°$

Types of Lines:

Type	Description
Parallel lines	Never meet; always the same distance apart; marked with arrows
Perpendicular lines	Meet at a right angle ($90°$)
Intersecting lines	Cross at any angle
Transversal	A line that crosses two or more parallel lines

Angles formed by a Transversal crossing Parallel Lines:

Angle Relationship	Description
Corresponding angles	In the same relative position at each intersection
Alternate interior angles	On opposite sides of the transversal, inside the parallel lines
Alternate exterior angles	On opposite sides of the transversal, outside the parallel lines
Co-interior angles	On the same side of the transversal, inside the parallel lines

Key Rule: When a transversal crosses parallel lines:

• Corresponding angles are equal
• Alternate interior angles are equal
• Alternate exterior angles are equal

⚡ Exam Tip (NCEE): When two parallel lines are crossed by a transversal, corresponding angles look the same shape (F-shape) on the same side. Alternate angles are on opposite sides of the transversal (Z or N-shape). Use these shape patterns to identify equal angles quickly.

---

🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding of geometric reasoning.

Triangles — Types:

Triangle Type	Properties
Equilateral	All 3 sides equal; all 3 angles = $60°$
Isosceles	2 sides equal; 2 angles equal (base angles are equal)
Scalene	All sides different; all angles different
Right-angled	One angle = $90°$

Triangle Angle Sum:

The interior angles of ANY triangle add up to $180°$: $$x + y + z = 180°$$

Example: In a triangle, one angle is $50°$ and another is $60°$. Find the third angle. $50 + 60 + x = 180$ $110 + x = 180$ $x = 70°$

Exterior Angle Theorem:

An exterior angle of a triangle equals the sum of the two opposite interior angles.

Example: If an exterior angle is $110°$ and one opposite interior angle is $45°$, find the other opposite interior angle. $45° + x = 110°$ $x = 65°$

Congruent Triangles (Same Size and Shape):

Two triangles are congruent if they have exactly the same three sides and three angles. Test by:

• SSS: All three sides equal
• SAS: Two sides and the included angle equal
• ASA (or AAS): Two angles and a side equal
• RHS: Right-angled triangles with hypotenuse and one side equal

Similar Triangles (Same Shape, Different Size):

Two triangles are similar if:

• Corresponding angles are equal
• Corresponding sides are in the same ratio

If triangle ABC is similar to triangle DEF:

• $\angle A = \angle D, \angle B = \angle E, \angle C = \angle F$
• $AB/DE = BC/EF = AC/DF$

Pythagoras’ Theorem (Right-Angled Triangles):

In a right-angled triangle with sides $a$, $b$ (legs) and $c$ (hypotenuse, opposite the right angle): $$a^2 + b^2 = c^2$$

Example: Find the hypotenuse of a right triangle with legs 3 cm and 4 cm. $c^2 = 3^2 + 4^2 = 9 + 16 = 25$ $c = 5$ cm

⚡ Common NCEE Error: In Pythagoras’ theorem, $c^2 = a^2 + b^2$, where $c$ is the hypotenuse (longest side, opposite the right angle). Students sometimes put the wrong side as $c$. Always verify which side is opposite the right angle.

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Angle Bisectors and Perpendicular Bisectors:

• An angle bisector divides an angle into two equal angles
• A perpendicular bisector of a side: a line perpendicular to the side that passes through its midpoint
• The point where the three angle bisectors of a triangle meet is the incentre (centre of the incircle)
• The point where the three perpendicular bisectors meet is the circumcentre (centre of the circumcircle)

Altitude, Median, and Perimeter:

• An altitude of a triangle is a perpendicular from a vertex to the opposite side (or its extension)
• A median joins a vertex to the midpoint of the opposite side
• The perimeter of a triangle is the sum of its three sides

Special Triangles — Pythagorean Triples:

Common Pythagorean triples (integer solutions to $a^2 + b^2 = c^2$):

• $(3, 4, 5)$ and multiples: $(6, 8, 10), (9, 12, 15), …$
• $(5, 12, 13)$
• $(8, 15, 17)$
• $(7, 24, 25)$

Area of a Triangle:

$$Area = \frac{1}{2} \times base \times height$$

For any triangle with sides $a$, $b$, $c$ and semi-perimeter $s = (a+b+c)/2$ (Heron’s formula): $$Area = \sqrt{s(s-a)(s-b)(s-c)}$$

Example: Triangle with sides 3, 4, 5 cm: $s = (3+4+5)/2 = 6$ $Area = \sqrt{6(6-3)(6-4)(6-5)} = \sqrt{6 \times 3 \times 2 \times 1} = \sqrt{36} = 6$ cm² (This is a right triangle with legs 3 and 4, so area = $1/2 \times 3 \times 4 = 6$ cm² ✓)

Circle Geometry Basics:

• A circle has $360°$ all the way around
• The diameter = 2 × radius
• Circumference = $2\pi r$ or $\pi d$
• Area = $\pi r^2$
• $\pi \approx 3.142$ or $22/7$

Angles in a Circle:

• Angle subtended by an arc at the centre = twice the angle subtended at the circumference
• Angle in a semicircle = $90°$ (Thales’ theorem)

Angles of Elevation and Depression:

• Angle of elevation: The angle between the horizontal and the line of sight when looking UP at an object
• Angle of depression: The angle between the horizontal and the line of sight when looking DOWN at an object

In right-angled trigonometry problems involving angles of elevation/depression, the horizontal line is the reference.

⚡ Extended Tip — Working Backwards in Geometry: When asked to prove an angle equality in a complex diagram, start from what you know and work forward using angle rules (supplementary, complementary, angle sum of triangle = 180°, exterior angle = sum of opposite interior angles, etc.). Check each angle relationship systematically: vertically opposite, corresponding, alternate, co-interior, and angles around a point.

---

Content adapted based on your selected roadmap duration. Switch tiers using the pill selector above.