How To Show The Angles In A Triangle Add Up To 180 Degrees - BBC

Home » What Do Triangles Add Up To » How To Show The Angles In A Triangle Add Up To 180 Degrees - BBC

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  1. Key points
  2. Video
  3. Angles in a complete turn and angles in triangles
    1. Example
    2. Question
  4. Interior angles of equilateral and isosceles triangles
    1. Examples
    2. Question
  5. Interior angles in quadrilaterals
    1. Examples
    2. Question
  6. Practise finding angles in triangles and quadrilaterals
    1. Quiz

Key points

An image of a triangle. Each of the three interior angles are labelled, A, B and C. Written below: A plus B plus C equals one hundred and eighty degrees.
Image caption, Interior angles in a triangle add up to 180˚.

  • All polygons have interior angles. The number of angles is equal to the number of sides it has.

  • Triangles have three sides, therefore they have three angles. Interior angles in a triangle sum to a half turn (180˚).

  • Interior angles in an equilateral triangle are equal. Base interior angles in an isosceles triangle are equal.

  • Quadrilaterals have four sides and four angles. Interior angles in a quadrilateral sum to a full turn (360˚).

An image of a triangle. Each of the three interior angles are labelled, A, B and C. Written below: A plus B plus C equals one hundred and eighty degrees.
Image caption, Interior angles in a triangle add up to 180˚.
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Video

Watch the video to learn how it is possible to prove mathematically that the angle sum of every triangle is 180°.

Video Transcript

It's possible to demonstrate and prove mathematically that the angle sum of every triangle is 180 degrees. The techniques used are useful for solving all kinds of other geometry problems too.

There are two rules to remember. First, when a diagonal line is drawn crossing two parallel lines, it forms alternate angles that are equal. That's these two.

Second, angles on a straight line add up to 180 degrees. Using these two rules it's possible to prove that angles in a triangle add up to 180

Back to the bridge. To start, draw two parallel lines. One that follows an edge of the triangle and one through the vertex opposite that edge. Now both rules are in play - the alternate angles rule and the rule the angles in a straight line add up to 180

If angle 𝑎 equals angle 𝑥 and angle 𝑦 equals angle 𝑏, our first rule. And if angles 𝑥, 𝑦 and 𝑐 equal 180, our second rule. Then angles 𝑎, 𝑏 and 𝑐 must equal 180 as well.

This proves that the angles of a triangle must add up to 180 degrees.

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Angles in a complete turn and angles in triangles

A complete turn is a 360˚ rotation.

  • To find unknown angles in a complete turn:

    1. Add the known angles.

    2. Form an equation equal to 360˚.

    3. Subtract the sum of the known angles from the sum of a complete turn (360˚).

Triangles have three interior angles. They are formed at the vertices (corners).

  • The sum of these angles is 180˚.

  • In scalene triangles, each angle is a different size because each side is a different length.

  • To find the missing angles in a triangle:

    1. Add the known angles.

    2. Subtract the sum of the known angles from the sum of interior angles in a triangle (180˚).

Example

Image gallerySkip image gallery
  1. An image of a horizontal line. The left end of the line is marked with a point. An arc with a clockwise arrow shows a complete turn. It is labelled as three hundred and sixty degrees. Written above: A complete turn. The arc is coloured orange.
    Image caption,

    A complete turn is 360˚.

1 of 8

Previous imageNext imageSlide 1 of 8, An image of a horizontal line. The left end of the line is marked with a point. An arc with a clockwise arrow shows a complete turn. It is labelled as three hundred and sixty degrees. Written above: A complete turn. The arc is coloured orange., A complete turn is 360˚.End of image gallery

Question

Find the size of the missing angle, \(a\).

An image of a triangle. Each of the three interior angles are labelled, sixty degrees, forty degrees and a.

Show answer

The same image as the previous. The angle marked a has been replaced with eighty degrees. Written right: a plus forty degrees plus sixty degrees equals one hundred and eighty degrees. Written below: a plus one hundred degrees equals one hundred and eighty degrees. Written beneath: a equals eighty degrees. Written between both equations: subtract one hundred degrees, on both the left and right sides of the equation. A blue box is drawn around the forty degrees plus sixty degrees. The eighty degrees is coloured orange and the subtract one hundred degrees is coloured blue.

Angles in a triangle add up to 180˚.

First form an equation that is equal to 180˚. The equation that can be formed is \(a\) + 40 + 60 = 180

Then, we can solve the equation:

\(a\) + 100 = 180

\(a\) = 180 - 100

\(a\) = 80

The size of the missing angle, \(a\), is 80˚.

To check that the value of \(a\) is correct, add the three angles together.

40 + 60 + 80 = 180˚.

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Interior angles of equilateral and isosceles triangles

The sum of interior angles in a triangle is 180˚.

  • In an equilateral triangle, all three angles are equal to 60˚.

  • To find the missing angles in an equilateral triangle, divide the sum of interior angles in a triangle by 3

  • In an isosceles triangle, two of the sides are equal.

  • This means two of the angles are equal. These are the base angles.

  • The base angles are opposite the equal sides.

Examples

Image gallerySkip image gallery
  1. An image of an equilateral triangle. Hatch marks are drawn in the midpoint of each side. Arcs are drawn in each angle. Written above: Equilateral triangle.
    Image caption,

    An equilateral triangle has three equal sides. This is shown by the hatch marks on each side. This means that the three angles will also be equal.

1 of 9

Previous imageNext imageSlide 1 of 9, An image of an equilateral triangle. Hatch marks are drawn in the midpoint of each side. Arcs are drawn in each angle. Written above: Equilateral triangle., An equilateral triangle has three equal sides. This is shown by the hatch marks on each side. This means that the three angles will also be equal.End of image gallery

Question

Find the size of the missing angle, \(x\).

An image of an isosceles triangle. Two of the interior angles are labelled, fifty six degrees and x. There are hatches on the two sides that meet at the angle marked fifty six degrees.

Show answer

The same image as the previous. The angle marked x and the unlabelled angle have both been replaced by sixty two degrees. Written right: x plus x plus fifty six degrees equals one hundred and eighty degrees. Written below: two x plus fifty six degrees equals one hundred and eighty degrees. Written beneath: two x equals one hundred and twenty four degrees. Written between both equations: subtract fifty six degrees, on both the left and right sides of the equation. Written below: x equals sixty two degrees. Written between both equations: divide by two, on both the left and right sides of the equation. A blue box is drawn around the x plus x. The sixty two degrees is coloured orange and the subtract fifty six degrees and divide by two are coloured blue.

As two of the sides are equal, the base angles are equal sizes.

The equation that can be formed to find the value of the base angles is:

\(x\) + \(x\) + 56 = 180

The equation can then be solved:

2\(x\) + 56 = 180

2\(x\) = 180 – 56

2\(x\) = 124

\(x\) = 124 ÷ 2

\(x\) = 62

The size of the missing angle, \(x\), is 62˚.

To check that the value of \(x\) is correct, add the three angles together.

56 + 62 + 62 = 180˚.

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Interior angles in quadrilaterals

  • The sum of interior angles in a quadrilateral is 360˚.

  • In a square or rectangle, each interior angle is 90˚.

  • In irregular quadrilaterals, each angle is a different size.

  • To find a missing angle in an irregular quadrilateral:

    1. Add the known angles.

    2. Subtract the sum of the known angles from the sum of interior angles in a quadrilateral (360˚).

Examples

Image gallerySkip image gallery
  1. An image of a square and an irregular quadrilateral. In the square each angle has been marked as a right-angle, ninety degrees. Written below: four multiplied by ninety degrees equals three hundred and sixty degrees. In the irregular quadrilateral each of the four interior angles are labelled, sixty eight degrees, one hundred and eighteen degrees, ninety four degrees and eighty degrees. Written above: sixty eight degrees plus one hundred and eighteen degrees plus ninety four degrees plus eighty degrees equals three hundred and sixty degrees.
    Image caption,

    A quadrilateral has 4 angles. If the quadrilateral is irregular, all the angles are different.

1 of 8

Previous imageNext imageSlide 1 of 8, An image of a square and an irregular quadrilateral. In the square each angle has been marked as a right-angle, ninety degrees. Written below: four multiplied by ninety degrees equals three hundred and sixty degrees. In the irregular quadrilateral each of the four interior angles are labelled, sixty eight degrees, one hundred and eighteen degrees, ninety four degrees and eighty degrees. Written above: sixty eight degrees plus one hundred and eighteen degrees plus ninety four degrees plus eighty degrees equals three hundred and sixty degrees., A quadrilateral has 4 angles. If the quadrilateral is irregular, all the angles are different.End of image gallery

Question

Find the size of the missing angle, \(x\).

An image of an irregular quadrilateral. Each of the four interior angles are labelled, one hundred degrees, ninety five degrees, sixty degrees and x.

Show answer

The same image as the previous. The angle marked x has been replaced by one hundred and five degrees. Written above: x plus sixty degrees plus ninety five degrees plus one hundred degrees equals three hundred and sixty degrees. Written below: x plus two hundred and fifty five degrees equals three hundred and sixty degrees. Written beneath: x equals one hundred and five degrees. Written between both equations: subtract two hundred and fifty five degrees, on both the left and right sides of the equation. A blue box is drawn around the sixty degrees plus ninety five degrees plus one hundred degrees. The one hundred and five degrees is coloured orange and the subtract two hundred and fifty five degrees is coloured blue.

The equation that can be formed to find the value of \(x\) is:\(x\) + 60 + 95 + 100 = 360˚.

The equation can then be solved:

\(x\) + 255 = 360

\(x\) = 360 – 255

\(x\) = 105

The size of the missing angle, \(x\), is 105˚.

To check that the value of \(x\) is correct, add the four angles together.

105 + 60 + 95 + 100 = 360

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Practise finding angles in triangles and quadrilaterals

Quiz

Practise finding angles in triangles and quadrilaterals with this quiz. You may need a pen and paper to help you with your answers.

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More on Angles

Find out more by working through a topic

  • Intersecting and parallel lines

    • count3 of 7

  • Constructing triangles

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  • Angles in polygons

    • count5 of 7

  • Polygons - sum of interior angles

    • count6 of 7

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