Grade 9 Mathematics
Grade 9 · Term 3Mathematics

Geometry of 2D Shapes

We study the properties of triangles and quadrilaterals, prove congruence and similarity of triangles, and use these properties to solve geometric problems with formal reasons.

5.1 Properties of Triangles5.2 Properties of Quadrilaterals5.3 Congruent and Similar Triangles
Week 1

5.1 Properties of Triangles

  • Recall and apply the angle sum of a triangle (180°)
  • Identify types of triangles: equilateral, isosceles, scalene, right-angled
  • Apply the exterior angle theorem
  • Solve problems using triangle properties
🌍

Real-World Connection

Triangles are the strongest geometric shape — that's why engineers use triangular trusses in bridges and roof structures. A triangle cannot be deformed without changing its side lengths, unlike a rectangle which can be pushed into a parallelogram. Every triangle, no matter how oddly shaped, always has interior angles summing to exactly 180°.

Property / Rule

Angle Sum of a Triangle

The interior angles of any triangle add up to 180°.

A^+B^+C^=180°(∠ sum of △)\hat{A} + \hat{B} + \hat{C} = 180° \quad (\angle\text{ sum of } \triangle)

Property / Rule

Exterior Angle Theorem

The exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.

ext A^=B^+C^(ext ∠ of △)\text{ext } \hat{A} = \hat{B} + \hat{C} \quad (\text{ext } \angle \text{ of } \triangle)

Property / Rule

Isosceles Triangle

In an isosceles triangle, the base angles (angles opposite the equal sides) are equal.

AB=AC⇒B^=C^(∠s opp. equal sides)AB = AC \Rightarrow \hat{B} = \hat{C} \quad (\angle\text{s opp. equal sides})

💡 Tip

Always write a reason for every statement in geometry. Use standard abbreviations: ∠ sum of △, ext ∠ of △, ∠s opp. equal sides, isosceles △.

Worked Examples

Worked Example

Using triangle properties to find angles

Problem

In △ABC, AB = AC, and A^=40°\hat{A} = 40°. Find B^\hat{B} and C^\hat{C}. Also find the exterior angle at C.

Worked Example

Exterior angle theorem

Problem

In △XYZ, the exterior angle at Z is 128°128°. X^=(2k)°\hat{X} = (2k)° and Y^=(3k+3)°\hat{Y} = (3k+3)°. Find k and all angles of the triangle.

Worked Example

Types of triangles and properties

Problem

A triangle has sides 5 cm, 5 cm, and 8 cm. (a) What type is it? (b) Find the angle between the two equal sides, given the base angles are each 36.9°.
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge2 marks
In △PQR, P^=65°\hat{P} = 65° and Q^=48°\hat{Q} = 48°. Find R^\hat{R}.
2
L1 · Knowledge1 mark
An equilateral triangle has all three sides equal. What is each interior angle?
3
L2 · Routine Procedures3 marks
In △ABC, the exterior angle at A is 125°125°. B^=3x\hat{B} = 3x and C^=2x\hat{C} = 2x. Find xx.
4
L2 · Routine Procedures4 marks
Triangle DEF has DE = EF. If D^=2x+10°\hat{D} = 2x + 10° and F^=3x−5°\hat{F} = 3x − 5°, find all angles.
5
L3 · Complex Procedures5 marks
In △PQR, PQ∥ST where S is on PR and T is on QR. If P^=55°\hat{P} = 55° and Q^=70°\hat{Q} = 70°, find ∠PST\angle PST and ∠STR\angle STR, giving reasons.
6
L3 · Complex Procedures4 marks
The angles of a triangle are in ratio 2:3:42:3:4. Find each angle and classify the triangle.
7
L4 · Problem Solving5 marks
Prove that the sum of exterior angles of a triangle (one at each vertex) is 360°.
8
L4 · Problem Solving6 marks
In △ABC, D is on BC such that AD ⊥ BC. If AB = 13, BD = 5 and AC = 15, find (a) AD, (b) DC, (c) the total length BC. (d) Verify your answers by calculating the area of △ABC two ways.
Week 2

5.2 Properties of Quadrilaterals

  • Identify and describe the properties of quadrilaterals: parallelogram, rectangle, rhombus, square, trapezium, kite
  • Solve problems using the properties of quadrilaterals
  • Show that the sum of interior angles of a quadrilateral is 360°
🌍

Real-World Connection

Quadrilaterals are the most common shape in construction. Floors are rectangles. Rooftops are trapeziums. Diamonds in fabric patterns are rhombuses. Kite shapes appear in actual kites and some architectural windows. Each type has special properties that engineers exploit — a rhombus's diagonals bisect each other at right angles, making it useful for creating perfectly perpendicular structures.

Property / Rule

Angle Sum of a Quadrilateral

The sum of interior angles of any quadrilateral is 360°. This follows because any quadrilateral can be divided into two triangles.

A^+B^+C^+D^=360°\hat{A} + \hat{B} + \hat{C} + \hat{D} = 360°

Properties of Quadrilaterals

Shape

Key Properties

Parallelogram

Both pairs of opposite sides parallel

Opp sides equal, opp angles equal, diagonals bisect each other

Rectangle

Parallelogram with all right angles

Diagonals equal and bisect each other

Rhombus

Parallelogram with all sides equal

Diagonals bisect each other at 90°, diagonals bisect vertex angles

Square

Rectangle AND rhombus

All properties of both

Definition

Trapezium

A quadrilateral with exactly ONE pair of parallel sides.

AB∥DC, but AD is NOT parallel to BC\text{AB} \parallel \text{DC, but AD is NOT parallel to BC}

Definition

Kite

A quadrilateral with two pairs of adjacent sides equal. One diagonal is the axis of symmetry. One pair of opposite angles are equal.

AB=AD and CB=CDAB = AD \text{ and } CB = CD
Worked Examples

Worked Example

Solving angles in a parallelogram

Problem

ABCD is a parallelogram. A^=3x+10°\hat{A} = 3x + 10° and C^=2x+40°\hat{C} = 2x + 40°. Find all angles.

Worked Example

Properties of a rhombus

Problem

PQRS is a rhombus with P^=110°\hat{P} = 110°. Find all angles and prove that the diagonals bisect at right angles.

Worked Example

Sum of interior angles of a quadrilateral

Problem

In quadrilateral ABCD, A^=x\hat{A} = x, B^=2x−10°\hat{B} = 2x - 10°, C^=x+30°\hat{C} = x + 30°, D^=3x−20°\hat{D} = 3x - 20°. Find all angles.
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge2 marks
Three interior angles of a quadrilateral are 85°, 95°, and 110°. Find the fourth angle.
2
L1 · Knowledge1 mark
Name the quadrilateral that has all sides equal AND all angles equal to 90°.
3
L2 · Routine Procedures4 marks
In rhombus PQRS, ∠PQR=68°\angle PQR = 68°. Find all other angles.
4
L2 · Routine Procedures4 marks
A kite ABCD has AB=AD=5 cm and CB=CD=8 cm. The diagonal AC = 6 cm. Find the area of the kite.
5
L3 · Complex Procedures4 marks
ABCD is a rectangle. Its diagonals AC and BD intersect at O. If AC = 10 cm, find OB and explain why.
6
L3 · Complex Procedures5 marks
PQRS is a parallelogram. Prove that the diagonals bisect each other. (Use the properties you know.)
7
L4 · Problem Solving5 marks
A quadrilateral has angles xx, 2x2x, 3x3x, and 4x4x. Find all angles and identify the quadrilateral if the largest two angles are adjacent.
8
L4 · Problem Solving5 marks
In square ABCD, E is the midpoint of AB. Prove that DE = CE.
Week 3

5.3 Congruent and Similar Triangles

  • Identify and apply criteria for congruent triangles: SSS, SAS, AAS, RHS
  • Identify and apply criteria for similar triangles: AA, SSS (proportional), SAS (proportional)
  • Use congruence and similarity to solve geometric problems
🌍

Real-World Connection

Congruence and similarity are used in map-making (scale drawings), architecture (scaled models), and manufacturing (identical parts). When a company makes thousands of identical components, each is congruent to the original. When an architect creates a scale model, all distances are proportional — the model is similar to the real building.

Conditions for Congruent Triangles

  • SSS: Three sides of one triangle equal three sides of the other.
  • SAS: Two sides and the INCLUDED angle are equal.
  • AAS: Two angles and any one corresponding side are equal.
  • RHS: Right angle, Hypotenuse, and one Side are equal.

Conditions for Similar Triangles

  • AA: Two angles of one triangle equal two angles of the other (third angle follows automatically).
  • SSS: All three pairs of corresponding sides are in the same ratio.
  • SAS: Two sides in proportion AND the included angle equal.

Proportional Sides in Similar Triangles

ABDE=BCEF=ACDF=k(scale factor)\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k \quad (\text{scale factor})

k = scale factor; k > 1 means enlargement, k < 1 means reduction

Worked Examples

Worked Example

Proving triangles similar and finding missing sides

Problem

In △ABC and △DEF, A^=D^=50°\hat{A} = \hat{D} = 50° and B^=E^=70°\hat{B} = \hat{E} = 70°. If AB = 6, BC = 8, and DE = 9, find EF.

Worked Example

Proving congruence using SSS

Problem

In △ABC and △DEF: AB=DE=5cm, BC=EF=7cm, CA=FD=9cm. Prove they are congruent.

Worked Example

Using similarity to find unknown lengths

Problem

A 1.8 m tall person casts a 2.4 m shadow. At the same time, a tree casts a 16 m shadow. How tall is the tree?
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge1 mark
Name the congruence condition: Two triangles have two sides equal and the included angle equal.
2
L1 · Knowledge2 marks
△PQR and △XYZ have all three sides equal. Are they congruent? Give the condition.
3
L2 · Routine Procedures4 marks
In △ABC, AB=5, BC=7, AC=9. In △DEF, DE=10, EF=14, DF=18. Are they similar? Find the scale factor.
4
L2 · Routine Procedures4 marks
△ABD ≅ △CBD with BD as a common side. If AB=8, A^=35°\hat{A}=35°, and ABD^=55°\hat{ABD}=55°. Find CB and C^\hat{C}.
5
L3 · Complex Procedures4 marks
A 2 m vertical stick casts a shadow of 3 m. At the same time, a tree casts a shadow of 18 m. How tall is the tree?
6
L3 · Complex Procedures5 marks
In the diagram, DE∥BC in △ABC. AD=4, DB=6, AE=5. Find EC and show △ADE ∼ △ABC.
7
L4 · Problem Solving5 marks
Prove that if the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.
8
L4 · Problem Solving5 marks
△ABC has AB=c, BC=a, AC=b. A point D on BC divides it so that BD:DC=2:3. If similar △ADE∼△ABC (with E on AC), find DE in terms of a.

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Area, Perimeter & Pythagoras

Geometry of 2D Shapes Grade 9 Maths CAPS Notes & Examples | MathSciBuddy