Question Video: Finding the Measure of the Exterior Angle to a Triangle Using the Chords’ Theorem | Nagwa Question Video: Finding the Measure of the Exterior Angle to a Triangle Using the Chords’ Theorem | Nagwa

# Question Video: Finding the Measure of the Exterior Angle to a Triangle Using the Chordsβ Theorem Mathematics • Third Year of Preparatory School

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Given that πβ πΆπ΄π΅ = 39Β°, and πΈ is the midpoint of line segment π΄πΆ, find πβ π΅πΉπΈ.

03:26

### Video Transcript

Given that the measure of angle πΆπ΄π΅ equals 39 degrees and πΈ is the midpoint of line segment π΄πΆ, find the measure of angle π΅πΉπΈ.

So letβs have a look at this figure that weβre given, and we can fill in the measure of angle πΆπ΄π΅, which is 39 degrees. The angle measure that we wish to find out is the measure of angle π΅πΉπΈ, which is here. The center of the circle is here at point π, and we have two chords in the circle. The line segment π΄π΅ is a chord and so is the line segment π΄πΆ. They are chords because both of these lines is a line segment joining two distinct points on the circumference.

Taking a closer look at the chord π΄πΆ, we can see that itβs divided into two equal or congruent pieces. In fact, we could also say that this chord π΄πΆ has been bisected.

Letβs recall that there is an important property which involves a line from the center of a circle to a chord that has been bisected. This property tells us that if we have a circle with center π΄ containing a chord, line segment π΅πΆ, then the straight line that passes through π΄ and bisects line segment π΅πΆ is perpendicular to line segment π΅πΆ. We can even say by changing the letters in the general statement that this is exactly what we have here. The chord π΄πΆ has been bisected by this line from the center π. And so that means that this line is perpendicular to the chord.

Letβs consider how this piece of information is useful. Well, we can take this triangle π΄πΉπΈ and consider that we have two angles that we know and one which we donβt know. In fact, if we did know this angle measure of π΄πΉπΈ, then we could work out the measure of the angle π΅πΉπΈ that weβre asked to calculate. We can use the property that the interior angle measures in a triangle add to 180 degrees to help us work out the angle measure of π΄πΉπΈ.

So we can say that 90 degrees plus 39 degrees plus the measure of angle π΄πΉπΈ must be equal to 180 degrees. We can simplify 90 degrees plus 39 degrees as 129 degrees and then subtract this value from both sides of the equation, giving us that the measure of angle π΄πΉπΈ is 51 degrees. Next, we can work out the required angle measure of angle π΅πΉπΈ. And we can do that by remembering that the angles on a straight line add up to 180 degrees.

That means that 51 degrees plus the measure of angle π΅πΉπΈ will be equal to 180 degrees. This means, of course, that the measure of angle π΅πΉπΈ can be found by subtracting 51 degrees from both sides, giving us an answer of 129 degrees. And so we find the answer using this very important property, which allowed us to work out that the line which bisected the chord and passes through the center is also perpendicular to the chord.

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