Sec 1 · G1 (N(T))

Triangles, quadrilaterals and polygons

Practise triangles, quadrilaterals and polygons with step-by-step help. Free account.

1. Triangles

Triangle	Property	Reason to write
any	angles add up to $180^\circ$	$\angle$ sum of $\triangle$
isosceles	two equal sides; the base angles opposite them are equal	base $\angle$ s of isos. $\triangle$
equilateral	three equal sides; each angle is $60^\circ$	equil. $\triangle$
right-angled	one angle is $90^\circ$
exterior angle	equals the sum of the two interior opposite angles	ext. $\angle$ of $\triangle$

Left: triangle ABC with angles 52° at A and 67° at C; AB extended past B makes an exterior angle of 119°, which is 52° + 67°. Right: a hexagon with its sides extended, showing the exterior angle at each vertex.

Isosceles: apex 44°, find each base angle x

\displaystyle 2x + 44 = 180

e.g.

\displaystyle 2x + 44 = 180

\displaystyle 2x = 136

\displaystyle x = 68

Exterior angle at B

\displaystyle 52^\circ + 67^\circ

e.g.

\displaystyle 52 + 67

\displaystyle = 119

📖 Worked example

In triangle $ABC$ , $AB = AC$ . Find $x$ .

Step 1 of 2

Use AB = AC: find ∠ACB and give the reason

\angle ACB = 62^\circ

(base ∠s of isos. △)

📖 Worked example

In the diagram, $BC$ is produced to $D$ . Find $x$ .

Step 1 of 2

Which angle fact gives x?

112^\circ = 46^\circ + x^\circ

(ext. ∠ of △)

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2. Constructing triangles and quadrilaterals

Three sides (SSS), AB = 7 cm, AC = 5 cm, BC = 6 cm: 1. draw AB = 7 cm; 2. an arc of radius 5 cm about A; 3. an arc of radius 6 cm about B cuts it at C; join AC and BC.

Two sides and the angle between them (SAS), AB = 7 cm, angle ABC = 50°, BC = 5 cm: 1. draw AB; 2. a protractor at B, 50°, and a ray; 3. an arc of radius 5 cm about B cuts the ray at C; join AC.

Two angles and the side between them (ASA), AB = 7 cm, angle BAC = 50°, angle ABC = 60°: 1. draw AB; 2. a ray at 50° from A; 3. a ray at 60° from B; the rays cross at C.

Given	Tools	Fixes one triangle?
three sides (SSS)	ruler, compasses	yes, if the two shorter sides add up to more than the longest
two sides and the angle between (SAS)	ruler, protractor, compasses	yes
two angles and the side between (ASA)	ruler, protractor	yes, if the two angles add up to less than $180^\circ$
three angles		no: any size

📖 Worked example

Jun Jie is constructing triangle $ABC$ with $AB = 7$ cm, $AC = 5$ cm and $BC = 6$ cm. The diagram shows the construction so far. Which of the following is the next step?

Step 1 of 2

How many steps are done?

Done: draw AB, arc about A.

📖 Worked example

Construct triangle $ABC$ with $AB = 6$ cm, $\angle ABC = 115^\circ$ and $BC = 5$ cm. Measure the length of $AC$ , to the nearest $0.1$ cm.

Step 1 of 4

Draw the base AB with a ruler

Draw

AB = 6

cm.

📖 Worked example

Construct quadrilateral $ABCD$ with $AB = 4$ cm, $BC = 6$ cm, $CD = 5$ cm, $DA = 6$ cm and the diagonal $AC = 7$ cm. Measure the length of the diagonal $BD$ , to the nearest $0.1$ cm.

Step 1 of 4

Start with the diagonal: it splits ABCD into two triangles

Draw

AC = 7

cm.

Practise this →

Practise triangles, quadrilaterals and polygons with step-by-step help

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1. Triangles

Triangle	Property	Reason to write
any	angles add up to $180^\circ$	$\angle$ sum of $\triangle$
isosceles	two equal sides; the base angles opposite them are equal	base $\angle$ s of isos. $\triangle$
equilateral	three equal sides; each angle is $60^\circ$	equil. $\triangle$
right-angled	one angle is $90^\circ$
exterior angle	equals the sum of the two interior opposite angles	ext. $\angle$ of $\triangle$

Isosceles: apex 44°, find each base angle x

\displaystyle 2x + 44 = 180

e.g.

\displaystyle 2x + 44 = 180

\displaystyle 2x = 136

\displaystyle x = 68

Exterior angle at B

\displaystyle 52^\circ + 67^\circ

e.g.

\displaystyle 52 + 67

\displaystyle = 119

📖 Worked example

In triangle $ABC$ , $AB = AC$ . Find $x$ .

Step 1 of 2

Use AB = AC: find ∠ACB and give the reason

\angle ACB = 62^\circ

(base ∠s of isos. △)

📖 Worked example

In the diagram, $BC$ is produced to $D$ . Find $x$ .

Step 1 of 2

Which angle fact gives x?

112^\circ = 46^\circ + x^\circ

(ext. ∠ of △)

Practise this →

2. Constructing triangles and quadrilaterals

Given	Tools	Fixes one triangle?
three sides (SSS)	ruler, compasses	yes, if the two shorter sides add up to more than the longest
two sides and the angle between (SAS)	ruler, protractor, compasses	yes
two angles and the side between (ASA)	ruler, protractor	yes, if the two angles add up to less than $180^\circ$
three angles		no: any size

📖 Worked example

Jun Jie is constructing triangle $ABC$ with $AB = 7$ cm, $AC = 5$ cm and $BC = 6$ cm. The diagram shows the construction so far. Which of the following is the next step?

Step 1 of 2

How many steps are done?

Done: draw AB, arc about A.

📖 Worked example

Construct triangle $ABC$ with $AB = 6$ cm, $\angle ABC = 115^\circ$ and $BC = 5$ cm. Measure the length of $AC$ , to the nearest $0.1$ cm.

Step 1 of 4

Draw the base AB with a ruler

Draw

AB = 6

cm.

📖 Worked example

Construct quadrilateral $ABCD$ with $AB = 4$ cm, $BC = 6$ cm, $CD = 5$ cm, $DA = 6$ cm and the diagonal $AC = 7$ cm. Measure the length of the diagonal $BD$ , to the nearest $0.1$ cm.

Step 1 of 4

Start with the diagonal: it splits ABCD into two triangles

Draw

AC = 7

cm.

Practise this →