Question Video: Finding the Measure of an Angle given the Measure of an Arc by Using the Properties of Tangents to the Circle | Nagwa Question Video: Finding the Measure of an Angle given the Measure of an Arc by Using the Properties of Tangents to the Circle | Nagwa

Question Video: Finding the Measure of an Angle given the Measure of an Arc by Using the Properties of Tangents to the Circle Mathematics • Third Year of Preparatory School

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Find π‘₯.

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Video Transcript

Find π‘₯.

In this question, we’re asked to find the value of π‘₯. And we can see that π‘₯ is the angle between two tangent lines to our circle. That’s the line from 𝐴 to 𝐢 and the line from 𝐴 to 𝐡. They just touch the circle at a single point, so these are tangent lines. And we can find the value of π‘₯ by recalling the following property for the angle between two tangent lines which intersect at a point outside of our circle.

We recall the angle between two tangent lines which intersect at a point is 180 degrees minus the measure of the arc between the two points of tangency. In our diagram, the points of tangency are the points 𝐡 and 𝐢. And the arc between 𝐡 and 𝐢 will be the minor arc shown. And we know the measure of this arc; its measure is 151 degrees. Then, our property tells us that the value of π‘₯ is equal to 180 degrees minus the measure of arc 𝐡𝐢. So we can substitute the measure of arc 𝐡𝐢, being 151 degrees, to get π‘₯ is equal to 180 degrees minus 151 degrees, which we can calculate is 29 degrees.

Therefore, by using the fact that the angle between two tangent lines which intersect at a point outside of a circle is 180 degrees minus the measure of the arc between the two points of tangency, we were able to show that π‘₯ is equal to 29 degrees.

Finally, let’s try and find the angle between a tangent line and a secant line which intersect outside of a circle. In this diagram, the tangent line is 𝐴𝐷 and the secant line is 𝐢𝐡. And we want to find the measure of the angle 𝐴𝐷𝐡. We’ll do this by using a very similar method to the last three proofs. We’ll start by connecting 𝐴 and 𝐡 to construct a triangle 𝐴𝐡𝐷. We see that angle 𝐢𝐡𝐴 and angle 𝐴𝐡𝐷 are on a straight line, so their measures add to 180 degrees. So we have the measure of angle 𝐢𝐡𝐴 plus the measure of angle 𝐴𝐡𝐷 is equal to 180 degrees.

We then also have that the sum of the measures of the internal angles in a triangle add to 180 degrees. So we have the measure of angle 𝐡𝐷𝐴 plus the measure of angle 𝐡𝐴𝐷 plus the measure of angle 𝐴𝐡𝐷 is equal to 180 degrees. And now we have two different expressions which when added to the measure of angle 𝐴𝐡𝐷 is equal to 180 degrees. So these two expressions must be equal. The measure of angle 𝐢𝐡𝐴 is equal to the measure of angle 𝐡𝐷𝐴 added to the measure of angle 𝐡𝐴𝐷.

We can subtract the measure of angle 𝐡𝐴𝐷 from both sides to find an expression for the measure of angle 𝐡𝐷𝐴. We have the measure of angle 𝐡𝐷𝐴 is equal to the measure of angle 𝐡𝐴𝐷 minus the measure of angle 𝐢𝐡𝐴. We can find an expression for the measure of angle 𝐡𝐴𝐷 by first adding the following two radii to our diagram. And then we’ll use the fact that the measure of the internal angles of quadrilateral 𝑀𝐴𝐷𝐡 add to 360 degrees. Since 𝐴 is a point of tangency for our tangent line, angle 𝑀𝐴𝐷 is a right angle. So the sum of the internal angles of this quadrilateral β€” the measure of angle 𝐴𝑀𝐡 plus 90 degrees plus the measure of angle 𝐡𝐷𝐴 plus the measure of angle 𝐷𝐡𝑀 β€” is equal to 360 degrees.

We know the measure of the central angle 𝐴𝑀𝐡 will be equal to the measure of the arc 𝐴𝐡. So we can substitute this into our expression to give us the following. And by considering the internal angles of triangle 𝐴𝐡𝐷, the internal angles sum to 180 degrees. So the measure of angle 𝐴𝐡𝐷 is 180 degrees minus the sum of the other two angles, the measure of angle 𝐡𝐴𝐷 and the measure of angle 𝐡𝐷𝐴. Finally, since 𝑀𝐴 and 𝑀𝐡 are radii, this means 𝑀𝐴𝐡 is an isosceles triangle. Therefore, the measure of angle 𝑀𝐴𝐡 and the measure of angle 𝑀𝐡𝐴 are equal. In particular, since angle 𝑀𝐴𝐷 is a right angle, we have the measure of angle 𝑀𝐴𝐡 is equal to 90 minus the measure of angle 𝐡𝐴𝐷.

Now, all we need to use is the fact that the measure of angle 𝐷𝐡𝑀 is the sum of the measure of angle 𝐴𝐡𝐷 and the measure of angle 𝑀𝐴𝐡. We would substitute these into our expression and then simplify. And we would be able to find the following result. The measure of angle 𝐡𝐴𝐷 is one-half the measure of the arc from 𝐴 to 𝐡. To do this, let’s clear some space and go back to the following equation.

We can find an expression for the measure of angle 𝐢𝐡𝐴 from our diagram. Angle 𝐢𝐡𝐴 is subtended by the major arc from 𝐴 to 𝐢. And the measure of an inscribed angle is one-half the measure of the arc it’s subtended by. So this is one-half the measure of the arc from 𝐴 to 𝐢. We can substitute our expression for the measure of angle 𝐡𝐴𝐷, giving us the following equation, which we can rearrange for the measure of angle 𝐡𝐷𝐴, which gives us the following. The measure of angle 𝐡𝐷𝐴 is one-half the measure of the major arc from 𝐴 to 𝐢 minus the measure of the arc from 𝐴 to 𝐡.

An easy way to remember this is the measure of the angle is one-half the difference of the measures of the two arcs intercepted by the sides of the angle. And of course we take the positive value for this difference.

Before we finish, there’s one more property we can show. We’ve already shown the measure of the angle between two tangents of a circle which meet at a point is 180 degrees minus the measure of the minor arc between the two points of tangency. We can relate this to our other results by considering the measure of the other arc; let’s call this 𝑦. These two arcs make up a full circle, so the sum of their measures is 360 degrees. Subtracting π‘₯ from both sides of the equation gives us 𝑦 is 360 degrees minus π‘₯. And we want to use this to consider one-half the difference between these two arcs. That’s one-half 𝑦 minus π‘₯.

We’ll substitute this expression for 𝑦 into one-half the difference. This gives us one-half 360 minus π‘₯ minus π‘₯, which if we simplify is 180 degrees minus π‘₯, which by using our first result is the measure of angle 𝐴𝐢𝐡. In other words, we can also think of the measure of the angle between two tangents which meet outside of a circle as one-half the difference between the two arcs between the points of tangency.

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