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4.8 H) Alternate Segment Theorem
4.8 H) Alternate Segment Theorem
The final circle theorem involves a tangent and a chord. A tangent is a straight line that touches a curve. A chord is a straight line that goes from one point on the circumference of a circle to another point on the circumference of a circle.
The last circle theorem is that the angle between a tangent and a chord is equal to the angle in the alternate segment. The line ABC is a straight line that is a tangent to the circle at point B. The two green angles are equal to one another and the two orange angles are equal to one another.
The last circle theorem is that the angle between a tangent and a chord is equal to the angle in the alternate segment. The line ABC is a straight line that is a tangent to the circle at point B. The two green angles are equal to one another and the two orange angles are equal to one another.
Example 1
The straight line DEF is a tangent to the circle and it touches the circle at point E. What is the size of angle x and y?
The straight line DEF is a tangent to the circle and it touches the circle at point E. What is the size of angle x and y?
This question is fairly straight forward because we know the alternate angle circle theorem, which states that the angle between a tangent and a chord is equal to the angle in the alternate segment.
This means that angle x is 68° and angle y is 56°.
This means that angle x is 68° and angle y is 56°.
Example 2
The straight line DCE is a tangent to the circle and it touches the circle at point C. The line AOC is the dimeter of the circle and the centre of the circle is the point labelled O.
The straight line DCE is a tangent to the circle and it touches the circle at point C. The line AOC is the dimeter of the circle and the centre of the circle is the point labelled O.
Find the size of the following angles:
Part 1 – angle BCD
Angle BCD and angle BAC (the one that is 63°) will be the same size due to the alternate segment theorem. Therefore, angle BCD is 63°.
Part 2 – angle ABC
We were told at the start of the question that the line AOC is the diameter of the circle. This means that we have a semicircle. We know that the angle at the circumference of a semicircle is a right angle (90°). This means that angle ABC is 90°.
Part 3 – angle OCE
There are two different method that we can use to find the size of angle OCE. One of the methods is to use the alternate segment theorem, which tells us that angle ABC and angle OCE are the same. We found out in part B that angle ABC is 90° and this means that angle OCE is also 90°.
The second method that we can use to find the size of angle OCE is the circle theorem that states that the angle between a tangent and a radius is 90°. We were told the line DCE is a tangent to the circle and it touches the circle at point C. We were also told that O is the centre of the circle and this means that OC is the radius of the circle. This means that angle OCE is 90°. Sometimes with circle theorems it will be the case that there are multiple different reasons/ theorems that you can use to find a particular angle.
Part 4 – angle BCA
The last part of this question asks you to find the size of angle BCA. Like the previous question, there are two different ways that we can find the size of this angle. Before I go through the two different methods, I am going to insert a labelled diagram with all of the angles that we have found in the previous parts. Also, I am going to let the angle that we are looking for equal x.
- BCD
- ABC
- OCE
- BCA
Part 1 – angle BCD
Angle BCD and angle BAC (the one that is 63°) will be the same size due to the alternate segment theorem. Therefore, angle BCD is 63°.
Part 2 – angle ABC
We were told at the start of the question that the line AOC is the diameter of the circle. This means that we have a semicircle. We know that the angle at the circumference of a semicircle is a right angle (90°). This means that angle ABC is 90°.
Part 3 – angle OCE
There are two different method that we can use to find the size of angle OCE. One of the methods is to use the alternate segment theorem, which tells us that angle ABC and angle OCE are the same. We found out in part B that angle ABC is 90° and this means that angle OCE is also 90°.
The second method that we can use to find the size of angle OCE is the circle theorem that states that the angle between a tangent and a radius is 90°. We were told the line DCE is a tangent to the circle and it touches the circle at point C. We were also told that O is the centre of the circle and this means that OC is the radius of the circle. This means that angle OCE is 90°. Sometimes with circle theorems it will be the case that there are multiple different reasons/ theorems that you can use to find a particular angle.
Part 4 – angle BCA
The last part of this question asks you to find the size of angle BCA. Like the previous question, there are two different ways that we can find the size of this angle. Before I go through the two different methods, I am going to insert a labelled diagram with all of the angles that we have found in the previous parts. Also, I am going to let the angle that we are looking for equal x.
The first method that we can use to find the value of x is to use the triangle ABC and the fact that all angles inside a triangle add up to 180°. We are able to create the following equation from this information:
We now solve in the usual way to find the value of x. The first step is to collect the two numbers that are on the left side of the equation.
We now need to move the 153 that is currently on the left to the right side of the equation. We are able to do this by doing the opposite; we take 153 from both sides of the equation.
Therefore, angle BCA is 27°.
The other method that we can use to find the size of angle BCA (or angle x) is to use the rule that angles on a straight line add up to 180°. This means that angle DCB (63°), angle BCA (x) and angle OCE (90°) will all add to give 180°. This gives us the equation below:
The other method that we can use to find the size of angle BCA (or angle x) is to use the rule that angles on a straight line add up to 180°. This means that angle DCB (63°), angle BCA (x) and angle OCE (90°) will all add to give 180°. This gives us the equation below:
We would then solve in the usual way. I am not going to do this because the equation above is exactly the same equation as the equation that we obtained when using the other method.