Triangle angle sums | Year 7

01 / LEARN

The idea, step by step

Three corners make a straight angle

The interior angles of every ordinary triangle on a flat plane total $180^{\circ}$ . One explanation is to draw a line through one vertex parallel to the opposite side. Alternate angles move the other two corner sizes beside the angle at that vertex; together they fill a straight angle.

To find one missing angle, add the known two first, then subtract from 180. A sketch can suggest an answer but is not the reason. Use given angle values and marked equal sides, not the apparent size of a printed shape.

A + B + C = 180^{\circ}

Use extra properties before calculating

In an isosceles triangle, angles opposite the two equal sides are equal. Subtract the vertex angle from 180, then divide the remaining total equally between the base angles. In a right-angled triangle, the two remaining angles total $90^{\circ}$ .

An exterior angle is formed when a side is extended. It and the adjacent interior angle total $180^{\circ}$ . Combining this with the triangle angle sum shows that the exterior angle equals the sum of the other two interior angles.

A common trap

Subtract the vertex angle from 180 and use that result as each base angle.

Instead: The remainder belongs to both base angles together. Divide it by two when the triangle is isosceles.

02 / WORKED EXAMPLES

See the thinking

Example 1

Find the third corner

Triangle ABC has $A = 52^{\circ}$ and $B = 71^{\circ}$ . Find C.

The interior angles total $180^{\circ}$ .
$52 + 71 = 123$ .
$C = 180^{\circ} - 123^{\circ} = 57^{\circ}$ .

C = 57^{\circ}

Example 2

Share the remaining turn

An isosceles triangle has vertex angle $36^{\circ}$ . Find each equal base angle.

The base angles total $180^{\circ} - 36^{\circ} = 144^{\circ}$ .
They are equal because they are opposite equal sides.
$144 \div 2 = 72$ .

Each base angle is

72^{\circ}

Check your understanding

A right-angled triangle has another angle of $27^{\circ}$ . Find the third angle.

Show answer

$63^{\circ}$ .

The two non-right angles total 90 degrees, so $90 - 27 = 63$ .

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Two interior angles of a triangle are $66^{\circ}$ and $47^{\circ}$ . Find the third.
2 marks
An isosceles triangle has vertex angle $42^{\circ}$ . Find each base angle.
2 marks
A triangle contains a right angle and an angle of $58^{\circ}$ . Find the third angle.
2 marks
A triangle has an exterior angle of $118^{\circ}$ . One of the two non-adjacent interior angles is $43^{\circ}$ .
4 marks
1. (a)
  Find the interior angle next to the exterior angle.
  2 marks
2. (b)
  Find the other non-adjacent interior angle.
  2 marks
Can a triangle have angles $90^{\circ}$ , $60^{\circ}$ and $40^{\circ}$ ? Give a numerical reason.
2 marks
An isosceles triangle has vertex angle $70^{\circ}$ . Each base angle is ${(2 x + 5)}^{\circ}$ . Find x.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Two interior angles of a triangle are $66^{\circ}$ and $47^{\circ}$ . Find the third.
2 marks
$66 + 47 = 113$ .
$180^{\circ} - 113^{\circ} = 67^{\circ}$ .
$67^{\circ}$ .
An isosceles triangle has vertex angle $42^{\circ}$ . Find each base angle.
2 marks
$180 - 42 = 138$ .
$138 \div 2 = 69$ .
$69^{\circ}$ .
A triangle contains a right angle and an angle of $58^{\circ}$ . Find the third angle.
2 marks
$180^{\circ} - 90^{\circ} - 58^{\circ} = 32^{\circ}$ .
$32^{\circ}$ .
A triangle has an exterior angle of $118^{\circ}$ . One of the two non-adjacent interior angles is $43^{\circ}$ .
4 marks
Use a straight line for the adjacent angle and the exterior-angle relationship for the remote angle.
1. (a)
  Find the interior angle next to the exterior angle.
  2 marks
  $180 - 118 = 62$ .
  $62^{\circ}$ .
2. (b)
  Find the other non-adjacent interior angle.
  2 marks
  The two remote interior angles total the exterior angle.
  $118 - 43 = 75$ .
  $75^{\circ}$ .
Adjacent interior angle = 62 degrees; remaining remote angle = 75 degrees.
Can a triangle have angles $90^{\circ}$ , $60^{\circ}$ and $40^{\circ}$ ? Give a numerical reason.
2 marks
$90 + 60 + 40 = 190$ , not 180 degrees.
These three angles cannot be the interior angles of one triangle.
No.
An isosceles triangle has vertex angle $70^{\circ}$ . Each base angle is ${(2 x + 5)}^{\circ}$ . Find x.
3 marks
$70 + 2 (2 x + 5) = 180$ .
$4 x + 80 = 180$ , so $4 x = 100$ and $x = 25$ .
Each base angle is 55 degrees; $55 + 55 + 70 = 180$ .
$x = 25$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Mathematics

Year 7 angle relationships, polygon properties and classification algorithms. Chapter grouping follows the approved Ascent outline.

Official curriculum source ↗

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