Sec 2 · Geometry · Deep Study
Understanding
Shape & Space
This guide builds your intuition for geometry from the ground up — why angles behave the way they do, where area formulas come from, and how the Pythagorean theorem really works. Work through each section in order, try every checkpoint question, then reveal the answer to check yourself.
📐 6 sections
🔍 Worked examples
✅ Checkpoint questions
💡 Intuition-first
1
Angle Relationships
How angles relate when lines meet
💭
Why do angles on a straight line sum to 180°?
A full rotation is 360°. A straight line cuts that rotation exactly in half — you can only go from one side to the other, which is half a turn. Half of 360° is 180°. So any angles that sit on one side of a straight line must together complete that half-rotation, giving a total of 180°.
A full rotation is 360°. A straight line cuts that rotation exactly in half — you can only go from one side to the other, which is half a turn. Half of 360° is 180°. So any angles that sit on one side of a straight line must together complete that half-rotation, giving a total of 180°.
Angle pairs
| Pair | Definition | Sum |
|---|---|---|
| Complementary | Two angles that together form a right angle | 90° |
| Supplementary | Two angles that together form a straight line | 180° |
| Vertical angles | Opposite angles formed when two lines cross | Equal to each other |
| Angles on a line | All angles on one side of a straight line | 180° |
| Angles at a point | All angles around a single point | 360° |
Complementary
∠A + ∠B = 90°
Supplementary
∠A + ∠B = 180°
Full rotation
all angles at a point = 360°
★ Easy
Show solution
Missing angle on a straight line
Two angles sit on a straight line. One angle is 53°. Find the other.
1
Identify the rule
Angles on a straight line sum to 180°.
2
Subtract
missing angle = 180° − 53° = 127°
Answer: 127°
★★ Medium
Show solution
Three unknowns using vertical and supplementary angles
Two straight lines cross. One of the four angles is 40°. Find all three remaining angles.
1
Label the angles
Call the known angle A = 40°. The four angles are A, B, C, D going around the crossing point.
2
Vertical angles are equal
C = A = 40° (C is opposite A)
3
Supplementary angles sum to 180°
B = 180° − 40° = 140°
4
D is opposite B (vertical angles)
D = B = 140°
Answer: 40°, 140°, 40°, 140°
✅ Checkpoint 1
a) Two complementary angles are in a ratio of 2:3. Find both angles.
b) An angle and its supplement differ by 50°. Find both angles.
c) Three angles meet at a point. Two of them are 115° and 130°. Find the third.
b) An angle and its supplement differ by 50°. Find both angles.
c) Three angles meet at a point. Two of them are 115° and 130°. Find the third.
a) Total = 90°. Parts: 2x + 3x = 90° → 5x = 90° → x = 18°. Angles: 36° and 54°.
b) Let the angle be x. Then x + (180° − x) aren't the same — set up: x − (180° − x) = 50° → 2x − 180° = 50° → 2x = 230° → x = 115°. Supplement = 65°. Angles: 115° and 65°.
c) All angles at a point = 360°. Third angle = 360° − 115° − 130° = 115°.
2
Parallel Lines & Transversal
The F, Z, C shapes and what they mean
💭
Why do parallel lines create predictable angle pairs?
Parallel lines are like two identical train tracks — they point in exactly the same direction. When a transversal (crossing line) cuts through them, it makes the same angle with each track. The lines are "copies" of each other, so the angle patterns repeat. This is why corresponding angles (same position at each intersection) are always equal, and co-interior angles always sum to exactly 180°.
Parallel lines are like two identical train tracks — they point in exactly the same direction. When a transversal (crossing line) cuts through them, it makes the same angle with each track. The lines are "copies" of each other, so the angle patterns repeat. This is why corresponding angles (same position at each intersection) are always equal, and co-interior angles always sum to exactly 180°.
Angle pairs with parallel lines
| Pair | Position | Relationship |
|---|---|---|
| Corresponding angles | Same position at each intersection (F-shape) | Equal |
| Alternate interior angles | Between the lines, opposite sides (Z-shape) | Equal |
| Co-interior angles | Between the lines, same side (C-shape) | Sum = 180° |
🔑
Memory trick: F = equal, Z = equal, C = 180°
Sketch the letter shape: F-angles are at corresponding positions, Z-angles alternate sides (both equal to each other), C-angles are on the same side and trapped between the parallel lines (they add to 180°).
Sketch the letter shape: F-angles are at corresponding positions, Z-angles alternate sides (both equal to each other), C-angles are on the same side and trapped between the parallel lines (they add to 180°).
★ Easy
Show solution
Finding a missing angle using parallel line rules
Two parallel lines are cut by a transversal. One corresponding angle is 68°. Find the other corresponding angle and the co-interior angle at the same intersection.
1
Corresponding angles (F-shape)
corresponding angle = 68° (F = equal)
2
Co-interior angle (C-shape)
co-interior angle = 180° − 68° = 112° (C = 180°)
Answer: Corresponding = 68°, Co-interior = 112°
★★ Medium
Show solution
All angles from two parallel lines and a transversal
A transversal crosses two parallel lines. At the first intersection, one angle is 55°. Name and find all 8 angles at both intersections.
1
First intersection — 4 angles
A = 55° (given)
B = 180° − 55° = 125° (supplementary to A on the line)
C = 55° (vertical to A)
D = 125° (vertical to B)
B = 180° − 55° = 125° (supplementary to A on the line)
C = 55° (vertical to A)
D = 125° (vertical to B)
2
Second intersection — use parallel line rules
E = 55° (corresponding to A, F-shape)
F = 125° (corresponding to B)
G = 55° (vertical to E)
H = 125° (vertical to F)
F = 125° (corresponding to B)
G = 55° (vertical to E)
H = 125° (vertical to F)
3
Verify co-interior angles
B + E = 125° + 55° = 180° ✓ (C-shape check)
Answer: Four 55° angles and four 125° angles
✅ Checkpoint 2
a) Two parallel lines are cut by a transversal. An alternate interior angle is 73°. What is its alternate interior pair? What is the co-interior angle on the same side?
b) A transversal forms a co-interior angle of 102° with two parallel lines. Find the corresponding angle on the other side.
c) True or false: Corresponding angles between parallel lines are supplementary. Explain.
b) A transversal forms a co-interior angle of 102° with two parallel lines. Find the corresponding angle on the other side.
c) True or false: Corresponding angles between parallel lines are supplementary. Explain.
a) Alternate interior pair = 73° (Z = equal). Co-interior = 180° − 73° = 107° (C = 180°).
b) Co-interior pair = 180° − 102° = 78°. Corresponding angle = 78° (F = equal, so same as the alternate interior).
c) False. Corresponding angles between parallel lines are equal, not supplementary. They would only be supplementary if the transversal were perpendicular (making them both 90°).
3
Triangles
Why all triangles have the same angle sum
💭
Proof intuition — why do triangle angles sum to 180°?
Draw any triangle. Through the top vertex (apex), draw a line parallel to the base. The angle at the apex now has three parts: the left part equals the bottom-left angle of the triangle (alternate interior angles, Z-shape), the right part equals the bottom-right angle (also alternate interior angles), and the middle part is the apex angle itself. Together, these three parts lie along the straight line through the apex — so they sum to 180°.
Draw any triangle. Through the top vertex (apex), draw a line parallel to the base. The angle at the apex now has three parts: the left part equals the bottom-left angle of the triangle (alternate interior angles, Z-shape), the right part equals the bottom-right angle (also alternate interior angles), and the middle part is the apex angle itself. Together, these three parts lie along the straight line through the apex — so they sum to 180°.
Triangle angle sum
∠A + ∠B + ∠C = 180°
True for every triangle, no exceptions
Types by sides
| Type | Sides | Angle properties |
|---|---|---|
| Equilateral | All 3 sides equal | All 3 angles = 60° |
| Isosceles | 2 sides equal | The 2 base angles (opposite the equal sides) are equal |
| Scalene | No sides equal | No angles equal |
Types by angles
| Type | Largest angle |
|---|---|
| Acute | All angles < 90° |
| Right | One angle = exactly 90° |
| Obtuse | One angle > 90° |
Exterior angle theorem
Exterior angle theorem
exterior angle = sum of the two non-adjacent interior angles
The exterior angle is always larger than either of the two remote interior angles
★ Easy
Show solution
Finding the missing angle in a triangle
A triangle has angles 47° and 85°. Find the third angle.
1
Use the angle sum property
∠A + ∠B + ∠C = 180°
47° + 85° + ∠C = 180°
47° + 85° + ∠C = 180°
2
Solve
∠C = 180° − 47° − 85° = 48°
Answer: 48°
★★ Medium
Show solution
Isosceles triangle — find all angles
An isosceles triangle has a vertex angle (the angle between the two equal sides) of 36°. Find the two base angles.
1
Remaining angle sum
base angles together = 180° − 36° = 144°
2
Isosceles — base angles are equal, so split equally
each base angle = 144° ÷ 2 = 72°
Answer: Both base angles = 72°
★★★ Hard
Show solution
Exterior angle theorem + isosceles in a complex figure
Triangle ABC is isosceles with AB = AC. ∠BAC = 50°. Side BC is extended to point D. Find ∠ACD (the exterior angle at C).
1
Find the base angles using isosceles property
AB = AC, so ∠ABC = ∠ACB
∠ABC + ∠ACB = 180° − 50° = 130°
∠ABC = ∠ACB = 65°
∠ABC + ∠ACB = 180° − 50° = 130°
∠ABC = ∠ACB = 65°
2
Apply the exterior angle theorem at vertex C
∠ACD is the exterior angle at C
∠ACD = ∠BAC + ∠ABC (sum of the two non-adjacent interior angles)
∠ACD = 50° + 65° = 115°
∠ACD = ∠BAC + ∠ABC (sum of the two non-adjacent interior angles)
∠ACD = 50° + 65° = 115°
3
Verify: exterior angle + interior angle = 180°
∠ACB + ∠ACD = 65° + 115° = 180° ✓ (straight line)
Answer: ∠ACD = 115°
✅ Checkpoint 3
a) A right triangle has one acute angle of 34°. Find the other acute angle.
b) An isosceles triangle has base angles of 52°. Find the vertex angle.
c) Using the exterior angle theorem: a triangle has interior angles 40°, 65°, and 75°. Find all three exterior angles.
b) An isosceles triangle has base angles of 52°. Find the vertex angle.
c) Using the exterior angle theorem: a triangle has interior angles 40°, 65°, and 75°. Find all three exterior angles.
a) Right triangle: 90° + 34° + x = 180° → x = 56°.
b) Vertex angle = 180° − 52° − 52° = 76°.
c) Each exterior angle = sum of the other two interior angles:
Exterior at 40° vertex = 65° + 75° = 140°
Exterior at 65° vertex = 40° + 75° = 115°
Exterior at 75° vertex = 40° + 65° = 105°
4
Quadrilaterals
Four sides, 360 degrees, many families
💭
Why do quadrilateral angles sum to 360°?
Draw a diagonal across any quadrilateral — it splits the shape into two triangles. Each triangle has angles summing to 180°. Two triangles = 2 × 180° = 360°. This works for any quadrilateral, no matter how irregular.
Draw a diagonal across any quadrilateral — it splits the shape into two triangles. Each triangle has angles summing to 180°. Two triangles = 2 × 180° = 360°. This works for any quadrilateral, no matter how irregular.
Quadrilateral angle sum
∠A + ∠B + ∠C + ∠D = 360°
Quadrilateral families
| Shape | Key properties |
|---|---|
| Square | 4 equal sides · 4 right angles (90° each) · diagonals bisect at 90° |
| Rectangle | Opposite sides equal · 4 right angles · diagonals equal in length |
| Parallelogram | Opposite sides parallel & equal · opposite angles equal · consecutive angles supplementary |
| Rhombus | 4 equal sides · opposite angles equal · diagonals bisect each other at 90° |
| Trapezoid | Exactly 1 pair of parallel sides |
★ Easy
Show solution
Finding the missing angle in a quadrilateral
A quadrilateral has angles 95°, 110°, and 75°. Find the fourth angle.
1
Angle sum = 360°
95° + 110° + 75° + x = 360°
280° + x = 360°
280° + x = 360°
2
Solve
x = 360° − 280° = 80°
Answer: 80°
★★ Medium
Show solution
Parallelogram — find all angles from one
A parallelogram has one angle of 62°. Find all four angles.
1
Opposite angles are equal in a parallelogram
∠A = 62° → ∠C = 62° (opposite)
2
Consecutive angles are supplementary (co-interior, parallel sides)
∠B = 180° − 62° = 118°
3
∠D is opposite ∠B
∠D = 118°
4
Check
62° + 118° + 62° + 118° = 360° ✓
Answer: 62°, 118°, 62°, 118°
✅ Checkpoint 4
a) A quadrilateral has three angles of 90°, 85°, and 105°. Find the fourth.
b) A rhombus has one angle of 70°. Find all four angles.
c) Why are all four angles in a rectangle 90°? Explain using the properties of a parallelogram.
b) A rhombus has one angle of 70°. Find all four angles.
c) Why are all four angles in a rectangle 90°? Explain using the properties of a parallelogram.
a) x = 360° − 90° − 85° − 105° = 80°.
b) A rhombus is a parallelogram: opposite angles equal, consecutive angles supplementary.
∠A = 70°, ∠C = 70°, ∠B = 180° − 70° = 110°, ∠D = 110°. Angles: 70°, 110°, 70°, 110°.
c) A rectangle is a parallelogram, so consecutive angles are supplementary (sum to 180°). If one angle is defined as 90°, then its consecutive angle = 180° − 90° = 90°. By the opposite-angle property, all four angles equal 90°.
5
Perimeter & Area
Measuring the boundary and the surface
💭
Perimeter vs area — what they actually measure
Perimeter is the total length of the boundary — like the amount of fencing needed to go around a yard. It's measured in metres, centimetres, etc. Area is the amount of surface inside the boundary — like the amount of grass in the yard. It's measured in square units (m², cm²). A long thin rectangle and a square can have the same perimeter but very different areas. Never confuse the two.
Perimeter is the total length of the boundary — like the amount of fencing needed to go around a yard. It's measured in metres, centimetres, etc. Area is the amount of surface inside the boundary — like the amount of grass in the yard. It's measured in square units (m², cm²). A long thin rectangle and a square can have the same perimeter but very different areas. Never confuse the two.
Perimeter formulas
Square
P = 4s
Rectangle
P = 2(l + w)
Triangle
P = a + b + c
Circle (circumference)
C = 2πr = πd
Area formulas
Area formulas
Square: A = s²
Rectangle: A = l × w
Triangle: A = (b × h) ÷ 2
Parallelogram: A = b × h
Trapezoid: A = ((b₁ + b₂) ÷ 2) × h
Circle: A = πr²
Rectangle: A = l × w
Triangle: A = (b × h) ÷ 2
Parallelogram: A = b × h
Trapezoid: A = ((b₁ + b₂) ÷ 2) × h
Circle: A = πr²
⚠️
The height must be perpendicular to the base — not the slant side.
For a parallelogram or triangle, the height is the vertical distance between the base and the opposite side (or vertex). If you use the slant edge instead, you will overestimate the area. Always draw in the perpendicular height before calculating.
For a parallelogram or triangle, the height is the vertical distance between the base and the opposite side (or vertex). If you use the slant edge instead, you will overestimate the area. Always draw in the perpendicular height before calculating.
★ Easy
Show solution
Area and perimeter of a rectangle
A rectangle has length 12 cm and width 7 cm. Find its area and perimeter.
1
Perimeter
P = 2(l + w) = 2(12 + 7) = 2 × 19 = 38 cm
2
Area
A = l × w = 12 × 7 = 84 cm²
Answer: Perimeter = 38 cm, Area = 84 cm²
★★ Medium
Show solution
Composite shape — rectangle minus a triangle
A rectangle is 10 cm wide and 8 cm tall. A right triangle with base 4 cm and height 5 cm is cut from one corner. Find the remaining area.
1
Area of the full rectangle
A_rect = 10 × 8 = 80 cm²
2
Area of the triangle cut out
A_tri = (b × h) ÷ 2 = (4 × 5) ÷ 2 = 10 cm²
3
Remaining area
A_remaining = 80 − 10 = 70 cm²
Answer: 70 cm²
✅ Checkpoint 5
a) A square has perimeter 36 cm. Find its area.
b) A trapezoid has parallel sides of 8 cm and 14 cm, and a perpendicular height of 6 cm. Find its area.
c) A circle has radius 5 cm. Find its circumference and area. Use π ≈ 3.14.
b) A trapezoid has parallel sides of 8 cm and 14 cm, and a perpendicular height of 6 cm. Find its area.
c) A circle has radius 5 cm. Find its circumference and area. Use π ≈ 3.14.
a) Side = 36 ÷ 4 = 9 cm. Area = 9² = 81 cm².
b) A = ((8 + 14) ÷ 2) × 6 = (22 ÷ 2) × 6 = 11 × 6 = 66 cm².
c) C = 2πr = 2 × 3.14 × 5 = 31.4 cm. A = πr² = 3.14 × 25 = 78.5 cm².
6
Volume, Surface Area & Pythagorean Theorem
3D shapes and the most famous theorem in geometry
💭
Why does a² + b² = c²? Geometric proof intuition
Draw a right triangle, then build a square on each of its three sides. The area of the square on the hypotenuse exactly equals the combined area of the squares on the two legs. This isn't just algebra — you can physically cut up the two smaller squares and rearrange them to perfectly fill the large square. That's why the relationship holds for every right triangle.
Draw a right triangle, then build a square on each of its three sides. The area of the square on the hypotenuse exactly equals the combined area of the squares on the two legs. This isn't just algebra — you can physically cut up the two smaller squares and rearrange them to perfectly fill the large square. That's why the relationship holds for every right triangle.
Pythagorean theorem
a² + b² = c²
c is the hypotenuse (longest side, opposite the right angle)
Common Pythagorean triples
Triple 1
3 – 4 – 5
Triple 2
5 – 12 – 13
Triple 3
8 – 15 – 17
Volume formulas
Volume
Rectangular prism: V = l × w × h
Cylinder: V = πr²h
Pyramid / Cone: V = (1/3) × base area × h
Sphere: V = (4/3)πr³
Cylinder: V = πr²h
Pyramid / Cone: V = (1/3) × base area × h
Sphere: V = (4/3)πr³
Surface area formulas
Surface Area
Rectangular prism: SA = 2(lw + lh + wh)
Cylinder: SA = 2πr² + 2πrh
Sphere: SA = 4πr²
Cylinder: SA = 2πr² + 2πrh
Sphere: SA = 4πr²
⚠️
Units matter — don't confuse surface area and volume.
Surface area is measured in square units (cm², m²) — it's a 2D measurement of the outside. Volume is measured in cubic units (cm³, m³) — it's a 3D measurement of what's inside. A common mistake is writing cm³ for an area answer or cm² for a volume answer.
Surface area is measured in square units (cm², m²) — it's a 2D measurement of the outside. Volume is measured in cubic units (cm³, m³) — it's a 3D measurement of what's inside. A common mistake is writing cm³ for an area answer or cm² for a volume answer.
★ Easy
Show solution
Find the hypotenuse
A right triangle has legs of length 6 cm and 8 cm. Find the hypotenuse.
1
Apply a² + b² = c²
6² + 8² = c²
36 + 64 = c²
100 = c²
36 + 64 = c²
100 = c²
2
Take the square root
c = √100 = 10 cm
Answer: c = 10 cm (recognise the 6–8–10 triple, a scaled 3–4–5)
★★ Medium
Show solution
Find a missing leg; verify a right triangle
Part A: hypotenuse = 26 cm, one leg = 10 cm. Find the missing leg. Part B: Is a triangle with sides 7, 24, 25 a right triangle?
1
Part A — rearrange for the missing leg
a² + 10² = 26²
a² + 100 = 676
a² = 576
a = √576 = 24 cm
a² + 100 = 676
a² = 576
a = √576 = 24 cm
2
Part B — check if 7² + 24² = 25²
7² + 24² = 49 + 576 = 625
25² = 625
625 = 625 ✓
25² = 625
625 = 625 ✓
Answer: Part A: missing leg = 24 cm | Part B: Yes, it is a right triangle
★★★ Hard
Show solution
Composite 3D volume problem
A solid is made by placing a square pyramid (base 6 cm × 6 cm, height 4 cm) on top of a rectangular prism (6 cm × 6 cm × 10 cm). Find the total volume.
1
Volume of the rectangular prism
V_prism = l × w × h = 6 × 6 × 10 = 360 cm³
2
Volume of the pyramid
Base area = 6 × 6 = 36 cm²
V_pyramid = (1/3) × 36 × 4 = 48 cm³
V_pyramid = (1/3) × 36 × 4 = 48 cm³
3
Total volume
V_total = 360 + 48 = 408 cm³
Answer: 408 cm³
✅ Checkpoint 6
a) A right triangle has legs 9 cm and 12 cm. Find the hypotenuse.
b) A cylinder has radius 3 cm and height 8 cm. Find its volume and surface area. Use π ≈ 3.14.
c) A rectangular prism measures 5 cm × 4 cm × 3 cm. Find its surface area.
b) A cylinder has radius 3 cm and height 8 cm. Find its volume and surface area. Use π ≈ 3.14.
c) A rectangular prism measures 5 cm × 4 cm × 3 cm. Find its surface area.
a) c² = 9² + 12² = 81 + 144 = 225 → c = √225 = 15 cm (3–4–5 triple × 3).
b) V = πr²h = 3.14 × 9 × 8 = 226.08 cm³.
SA = 2πr² + 2πrh = 2(3.14)(9) + 2(3.14)(3)(8) = 56.52 + 150.72 = 207.24 cm².
c) SA = 2(lw + lh + wh) = 2(5×4 + 5×3 + 4×3) = 2(20 + 15 + 12) = 2 × 47 = 94 cm².