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4.8 B) Angles in the Same Segment are Equal
4.8 B) Angles in the Same Segment are Equal
Angles at the circumference that are subtended by the same arc are equal to one another. This theorem is known as the “angles in the same segment theorem”.
Both of the angles that are a in the above circle are subtended by the same arc. The arc that they are subtended by has been colour in green.
In addition to the two top angles at the circumference on the circle above being the same as each other, the two bottom angles at the circumference are also the same as each other. Both of these angles have been labelled as b on the diagram below.
In addition to the two top angles at the circumference on the circle above being the same as each other, the two bottom angles at the circumference are also the same as each other. Both of these angles have been labelled as b on the diagram below.
The two angles labelled b are the same because they are both subtended by the same arc. The arc that they are both subtended by is coloured in orange.
Example 1
What is the size of angle c and d in the circle below?
What is the size of angle c and d in the circle below?
The two top angles at the circumference are subtended by the same arc and this means that angle c will be 60° because the other angle is 60°.
The two bottom angles at the circumference are subtended by the same arc. This means that angle d is 30°.
The two bottom angles at the circumference are subtended by the same arc. This means that angle d is 30°.
Example 2
What is the size of angle e in the diagram below?
What is the size of angle e in the diagram below?
There are quite a few steps in finding the size of angle e. We are able to find the top left angle at the circumference because we know that this angle is going to be the same size at the top right angle at the circumference because they are subtended by the same arc. This means that the top angle left is 65°.
We also know that the bottom two angles at the circumference are going to be the same because they are both subtended by the same arc. Therefore, I am going to label the bottom left angle as the circumference as e.
We can find the bottom left angle at the circumference because we know that all of the angles in a triangle add up to 180° and we have two angles in this triangle. Therefore, we can create the following equation.
We can collect the two numbers that are on the left side of the equation.
We find the value of e by moving the 125 from the left side of the equation to the right. We move this term from the left to the right by doing the opposite; we take 125 from both sides of the equation.
Therefore, the size of the angle at the circumference at the bottom left and the bottom right is 55° (e is 55°).