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₯.

06:00

### Video Transcript

Find π₯.

In this question, weβre asked to find the value of π₯. And we can see that π₯ is the angle between two tangent lines to our circle. Thatβs the line from π΄ to πΆ and the line from π΄ to π΅. 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 π΅ and πΆ. And the arc between π΅ and πΆ 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 π΅πΆ. So we can substitute the measure of arc π΅πΆ, 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, letβ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 π΄π· and the secant line is πΆπ΅. And we want to find the measure of the angle π΄π·π΅. Weβll do this by using a very similar method to the last three proofs. Weβll start by connecting π΄ and π΅ to construct a triangle π΄π΅π·. We see that angle πΆπ΅π΄ and angle π΄π΅π· are on a straight line, so their measures add to 180 degrees. So we have the measure of angle πΆπ΅π΄ plus the measure of angle π΄π΅π· 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 π΅π·π΄ plus the measure of angle π΅π΄π· plus the measure of angle π΄π΅π· is equal to 180 degrees. And now we have two different expressions which when added to the measure of angle π΄π΅π· is equal to 180 degrees. So these two expressions must be equal. The measure of angle πΆπ΅π΄ is equal to the measure of angle π΅π·π΄ added to the measure of angle π΅π΄π·.

We can subtract the measure of angle π΅π΄π· from both sides to find an expression for the measure of angle π΅π·π΄. We have the measure of angle π΅π·π΄ is equal to the measure of angle π΅π΄π· minus the measure of angle πΆπ΅π΄. We can find an expression for the measure of angle π΅π΄π· by first adding the following two radii to our diagram. And then weβll use the fact that the measure of the internal angles of quadrilateral ππ΄π·π΅ add to 360 degrees. Since π΄ is a point of tangency for our tangent line, angle ππ΄π· is a right angle. So the sum of the internal angles of this quadrilateral β the measure of angle π΄ππ΅ plus 90 degrees plus the measure of angle π΅π·π΄ plus the measure of angle π·π΅π β is equal to 360 degrees.

We know the measure of the central angle π΄ππ΅ will be equal to the measure of the arc π΄π΅. So we can substitute this into our expression to give us the following. And by considering the internal angles of triangle π΄π΅π·, the internal angles sum to 180 degrees. So the measure of angle π΄π΅π· is 180 degrees minus the sum of the other two angles, the measure of angle π΅π΄π· and the measure of angle π΅π·π΄. Finally, since ππ΄ and ππ΅ are radii, this means ππ΄π΅ is an isosceles triangle. Therefore, the measure of angle ππ΄π΅ and the measure of angle ππ΅π΄ are equal. In particular, since angle ππ΄π· is a right angle, we have the measure of angle ππ΄π΅ is equal to 90 minus the measure of angle π΅π΄π·.

Now, all we need to use is the fact that the measure of angle π·π΅π is the sum of the measure of angle π΄π΅π· and the measure of angle ππ΄π΅. We would substitute these into our expression and then simplify. And we would be able to find the following result. The measure of angle π΅π΄π· is one-half the measure of the arc from π΄ to π΅. To do this, letβs clear some space and go back to the following equation.

We can find an expression for the measure of angle πΆπ΅π΄ from our diagram. Angle πΆπ΅π΄ is subtended by the major arc from π΄ to πΆ. And the measure of an inscribed angle is one-half the measure of the arc itβs subtended by. So this is one-half the measure of the arc from π΄ to πΆ. We can substitute our expression for the measure of angle π΅π΄π·, giving us the following equation, which we can rearrange for the measure of angle π΅π·π΄, which gives us the following. The measure of angle π΅π·π΄ is one-half the measure of the major arc from π΄ to πΆ minus the measure of the arc from π΄ to π΅.

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, thereβs one more property we can show. Weβ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; letβs call this π¦. 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 π¦ is 360 degrees minus π₯. And we want to use this to consider one-half the difference between these two arcs. Thatβs one-half π¦ minus π₯.

Weβll substitute this expression for π¦ 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 π΄πΆπ΅. 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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