Triangle angle sums | Year 7

01 / LEARN

Understand the idea.

Three corners make a straight angle

The three interior angles of a triangle on a flat plane total $180^{\circ}$ . To see why, draw a line through a vertex parallel to the opposite side. Alternate angles place the other two corner sizes beside the vertex angle, filling a straight angle.

Use the given values rather than the apparent size of the sketch. Add the two known interior angles, then subtract their total from $180^{\circ}$ . Each of the three angles is part of that single total.

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

Use extra properties before calculating

An isosceles triangle has equal angles opposite its equal sides. After subtracting its vertex angle from $180^{\circ}$ , divide the remainder by $2$ : that remainder belongs to both base angles together. In a right triangle, the two non-right angles total $90^{\circ}$ .

Extending a side forms an exterior angle supplementary to the adjacent interior angle. Comparing their $180^{\circ}$ total with the triangle total shows that the exterior angle equals the sum of the two non-adjacent interior angles.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Find the third corner

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

Worked solution

1
Write the triangle-angle total.
$52^{\circ} + 71^{\circ} + C = 180^{\circ}$
The three interior angles of a triangle on a flat plane sum to 180 degrees.
2
Add the known angles and subtract.
$\begin{aligned} 52^{\circ} + 71^{\circ} & = 123^{\circ} \\ C & = 180^{\circ} - 123^{\circ} = 57^{\circ} \end{aligned}$
The third angle fills the part of the total not used by the first two.
3
Check all three corners.
$52^{\circ} + 71^{\circ} + 57^{\circ} = 180^{\circ}$
The proposed third angle gives the required triangle sum.

Answer

$C = 57^{\circ}$ .

EXAMPLE 02

Share the remaining turn

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

Worked solution

1
Remove the vertex angle from the total.
$180^{\circ} - 36^{\circ} = 144^{\circ}$
The two base angles together use the rest of the triangle angle sum.
2
Share the remaining total equally.
$\frac{144^{\circ}}{2} = 72^{\circ}$
Base angles opposite the two equal sides are equal.
3
Check the whole triangle.
$72^{\circ} + 72^{\circ} + 36^{\circ} = 180^{\circ}$
Both equal base angles and the vertex angle give the required total.

Answer

Each base angle is $72^{\circ}$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$63^{\circ}$ .

The right angle uses $90^{\circ}$ of the triangle total, leaving $90^{\circ}$ for the other two. Subtracting $27^{\circ}$ gives the third angle $63^{\circ}$ .

Keep practising

6 worksheet questions · 15 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Three corners make a straight angle

Use extra properties before calculating

See the method in action.

Find the third corner

Write the triangle-angle total.

Add the known angles and subtract.

Check all three corners.

Share the remaining turn

Remove the vertex angle from the total.

Share the remaining total equally.

Check the whole triangle.

Try it yourself.

Check your understanding

Keep practising