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Lines of symmetry Rotational symmetry Types of angles Angles on a straight line Angles in a triangle How to work out perimeterThis topic is relevant for:
Here we will learn about triangles, including what a triangle is and how to solve problems involving their sides and their angles.
There are also triangles worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if youβre still stuck.
Triangles are 2D shapes with three straight sides and three vertices. These are the only two key properties of triangles.
To solve problems involving triangles, we need to be able to gather information about its side lengths and/or its interior angles and apply these to the problem.
The sum of interior angles of a triangle is
For example,
If a triangleβs angles are all acute angles (less than ), it is sometimes known as an acute triangle.
If a given triangle has one obtuse angle it is sometimes known as an obtuse triangle.
Step-by-step guide: Angles in a triangle
There are different types of triangles that have different properties.
Being able to recognise different types of triangles is known as classifying triangles.
Step-by-step guide: Types of triangles
An equilateral triangle has the following properties,
An equilateral triangle has lines of symmetry. The lines go from one vertex to the middle of the opposite side.
The lines of symmetry are angle bisectors of the vertices and are also perpendicular bisectors of the sides of the equilateral triangle.
Step-by-step guide: Equilateral triangles
An isosceles triangle has the following properties,
The two equal angles are called base angles and they are opposite the sides of equal length. The unequal side is known as the base and is opposite the unequal angle.
Step-by-step guide: Isosceles triangles
A scalene triangle has the following properties,
Step-by-step guide: Scalene triangles
A right-angled triangle has the following property,
The longest side of a right-angled triangle is opposite the right-angle. It is called the hypotenuse.
The Pythagorean theorem is used to determine missing side lengths of a right-angled triangle. If a question involves both the sides of the triangle and one of the angles, we could use trigonometry.
A special case of a right-angled triangle is a right-angled triangle which also has two equal sides. It can be known as an isosceles right triangle or an isosceles right-angled triangle. The two equal angles will both be
Step-by-step guide: Right angle triangle
The area of a triangle can be found by using the formula,
,
where is the base length and is the perpendicular height of the triangle.
Sometimes these values need to be calculated.
The area of a triangle is given in square units.
Step-by-step guide: Area of a triangle
It is impossible for a triangle to contain more than one obtuse angle as the sum of angles in a triangle is
For example,
In order to solve problems involving triangles:
Get your free triangles worksheet of 20+ questions and answers. Includes reasoning and applied questions.
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This problem involves the internal angles of a triangle. The sum of interior angles of a triangle is One angle is given.
As two sides have a dash through them indicating they are the same length, the triangle is isosceles and so the triangle contains the two base angles and
2Locate known sides and calculate any necessary unknown side lengths.
As we need to determine the size of angle and we have information about the other two angles, we can use these to find the missing angle. We do not need to calculate any length of the triangle.
3Solve the problem using any necessary values.
Using the two known angles and the angle fact that the sum of angles in a triangle is
Therefore angle
Determine the size of angle
Locate known angles and calculate any necessary unknown angles.
This problem involves the interior and exterior angles of a triangle. The sum of interior angles of a triangle is One of the angles in the triangle is a right-angle which is
As the sum of angles on a straight line total we can find the third angle in the triangle,
Labelling this onto the diagram, we now have,
The three interior angles of the triangle are and
Locate known sides and calculate any necessary unknown side lengths.
This step is not needed for this question as we know two of the three angles within the triangle and we are finding the value of the missing angle
Solve the problem using any necessary values.
We use the angle fact that interior angles of a triangle sum to to work out the third angle.
Therefore angle
Calculate the missing angle
Locate known angles and calculate any necessary unknown angles.
One of the angles in the triangle is a right-angle which is As there are two sides of the triangle with a dash through them, these lines are equal in length and so we have a right-angled isosceles triangle.
This means that the remaining two angles are the same, and the three angles must total The two remaining angles are therefore
Locate known sides and calculate any necessary unknown side lengths.
We know that two sides are of equal length but we do not need to determine the length of any side of the triangle and so we can move on to step 3.
Solve the problem using any necessary values.
As adjacent angles on a straight line sum to we can use this to find the angle as it is on a straight line with the angle
Therefore angle
The sides of a triangle are and What is the perimeter of the triangle?
Locate known angles and calculate any necessary unknown angles.
The angles are not involved in this question as we want to find the perimeter which is a length, and we do not know any angles in the triangle.
Locate known sides and calculate any necessary unknown side lengths.
The three sides of the triangle are given and so we have the information needed to answer the question.
Note: The sides are different lengths and so it is a scalene triangle.
Solve the problem using any necessary values.
The perimeter can be found by finding the total of the sides. We therefore add together the side lengths to get the perimeter.
The perimeter of the triangle is
Here is a triangle.
Two of these triangles are joined on their shortest sides to make a parallelogram. Calculate the perimeter of the parallelogram.
Locate known angles and calculate any necessary unknown angles.
There is a right angle in the triangle, denoted using a square and so it is a right-angled triangle. As we want to find the perimeter of the parallelogram it produces, which is a length, we do not need to find the size of any other angle in the triangle.
Locate known sides and calculate any necessary unknown side lengths.
The three side lengths are known and and so we can move on to step
Solve the problem using any necessary values.
The question states that βtwo of these triangles are joined on their shortest sides to make a parallelogramβ.
The shortest side of the right-angled triangle is the perpendicular height, and so we need to join two of these triangles together at this side to form a parallelogram. It is useful to label each side length here.
The perimeter can be found by finding the total of the sides. Remember that the length is the perpendicular height of the parallelogram now and so it does not form part of the outside of the shape.
The perimeter is therefore the sum of the four side lengths, and
The perimeter of the parallelogram is
The perimeter of a triangle is
The ratio of the sides of the triangle is
Determine the length of each side of the triangle.
Locate known angles and calculate any necessary unknown angles.
The angles of the triangle are not involved in this question as we are calculating the side lengths of the triangle and have no information about the angles within the triangle.
Locate known sides and calculate any necessary unknown side lengths.
The three sides of the triangle are not given. But we have been given the ratio of the sides. Drawing a sketch of what the triangle may look like (although not essential), we have
Solve the problem using any necessary values.
Currently we have a triangle with a perimeter of As we want a similar triangle with a perimeter of we need to find the scale factor of enlargement. We do this by dividing the desired perimeter, by the scaled perimeter.
We therefore know that each side length must be times greater than our scaled triangle. We therefore need to multiply each side length by
This gives us the three side lengths, and
Note: This is exactly the same approach as if we were to divide a quantity into a ratio, as explained below.
We need to divide the perimeter by the total of the parts of the perimeter.
We then multiply the ratio parts by
The lengths of the sides of the triangle are and
Make sure you know your angle properties as getting these confused causes quite a few misconceptions.
β Angles in a triangle total .
β Angles in a quadrilateral total .
Make sure you know your angle properties.
β Adjacent angles on a straight line add up to .
β Vertically opposite angles are equal.
β Alternate angles are equal.
β Corresponding angles are equal.
1. Which of these shapes show an equilateral triangle?
An equilateral triangle has all equal sides and all equal angles.
2. The sides of a triangle are and What kind of triangle is it?
Scalene
Equilateral
Isosceles
Right-angled
is the same as Since two sides are the same, it is an isosceles triangle.
3. Find angle
The triangle has two equal side lengths and so it is an isosceles triangle. This means that the base angles will be equal. The interior angles of a triangle sum to Angle is therefore equal to
Angle
4. Determine the size of the angle in the diagram below.
As the sum of adjacent angles is the angle next to is equal to
As the sum of angles in a triangle is the unknown value of is equal to and so
5. Here is a triangle. Two of these triangles are joined on the longest sides to make a quadrilateral. Find its perimeter.
The two congruent triangles will make a rectangle when they are joined on their longest side (the hypotenuse).
The perimeter of the rectangle will be
6. The perimeter of an isosceles triangle is The unequal side is half the size of one of the equal sides. Find the length of the unequal side of the isosceles triangle.
The ratio of the sides of the isosceles triangle would be
The unequal side is the shortest side and can be found by
The sides of the triangles will be and The shortest side will be
1. Here is a triangle.
Draw the line(s) of symmetry of the shape.
(2 marks)
For one line of symmetry.
(1)
For three lines of symmetry.
(1)
2. Below is a diagram showing the triangle . The point lies on the line such that
(a) Determine the size of angle
(b) Find angle
(4 marks)
(a)
(1)
(1)
(b)
(1)
(1)
3. (a) Write down the coordinates of point
(b) Plot the point and label it
(c) The points and make a triangle.
Find the area of the triangle.
(4 marks)
(a)
(1)
(b)
(1)
(c)
(1)
(1)
4. The perimeter of a triangle is
The ratio of the sides of the triangle are
(a) What type of triangle is ?
(b) Find the smallest side of the triangle
(c) Heronβs formula can be used to find the area of a triangle.
Heronβs formula is
where is half the perimeter and and are the three side lengths.
Use Heronβs formula along with your solution to part b) to find the area of the triangle
Give your answer to significant figures.
State the units for your answer.
(8 marks)
(a) Isosceles
(1)
(b)
(1)
(1)
(c)
(1)
The second and third sides are
(1)
(1)
(1)
(1)
You have now learned how to:
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