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4.1 F) Angle Rules – Vertically Opposite
4.1 F) Angle Rules – Vertically Opposite
There are few different angle rules that we are going to look at in this section and the next few sections.
Vertically Opposite Angles are Equal
The first rule that we are going to look at is that vertically opposite angles are equal. On the diagram below you can see that there are two sets of vertically opposite angles; two of the angles are x and two of the angles are y.
The first rule that we are going to look at is that vertically opposite angles are equal. On the diagram below you can see that there are two sets of vertically opposite angles; two of the angles are x and two of the angles are y.
Example 1 & Some Proof
What are the values of angle a, b and c?
What are the values of angle a, b and c?
We are able to work out the value of a because we can use the rule that angles on a straight line add up to 180°. Therefore, the 50° angle and angle a must add up to 180°. We can create the following equation from this information:
We want to find the value of a and we find the value of a by moving the 50 from the left side of the equation to the right. We are able to do this by doing the opposite; we take 50 from both sides of the equation.
Therefore, angle a is 130°.
We are now able to find the value of angle b by using the same rule that angles on a straight line add up to 180°. This time the straight line will be made up of angle a (which is 130°) and angle b. We can create the following equation for this information:
We are now able to find the value of angle b by using the same rule that angles on a straight line add up to 180°. This time the straight line will be made up of angle a (which is 130°) and angle b. We can create the following equation for this information:
We can sub a in as 130°.
We find the value of b by taking 130 from both sides of the equation.
Therefore, b is equal to 50°, which is exactly the same value as the angle that is vertically opposite it.
We now need to find the value of angle c. According to the vertically opposite rule, angle c will be the same as angle a, thus meaning that c is 130°. However, let’s prove that this is the case. I am going to prove that angle c is 130° by using the rule that all of the angles in a full turn add up to 360°.
We now need to find the value of angle c. According to the vertically opposite rule, angle c will be the same as angle a, thus meaning that c is 130°. However, let’s prove that this is the case. I am going to prove that angle c is 130° by using the rule that all of the angles in a full turn add up to 360°.
We can sub the values for a and b in.
We can collect all of the numbers on the left side of the equation.
We find the value of c by moving the 230 from the left to the right side of the equation and we do this by taking 230 from both sides.
Angle c is 130°, which proves that the vertically opposite angle rule is true. The diagram with all of the angles on is shown below.
Example 2
What is the value of x, y and z?
What is the value of x, y and z?
Angles that are vertically opposite one another are equal. Therefore, angle y is 110°.
Also, angles along a straight line add up to 180°. I am going to use the straight line that involves angle x and the angle that we were given. We can create the following equation from this information:
Also, angles along a straight line add up to 180°. I am going to use the straight line that involves angle x and the angle that we were given. We can create the following equation from this information:
We find the value of x by moving the 110 from the left side of the equation to the right. We do this by doing the opposite; we take 110 from both sides.
Angle x is 70°. This means that angle z will also be 70° because angles that are vertically opposite are the same. Therefore, x is 70°, y is 110° and z is 70°.