Question Video: Identifying the Pair of Segments in a Triangle with Equal Length Using Vertically Opposite Angles Mathematics • 11th Grade

Which pair of segments have the same length?

02:54

Video Transcript

Which pair of segments have the same length?

Here, we are given a triangle which has been constructed from the lines 𝐵𝐶 and 𝐴𝐶 and the line segment 𝐴𝐵. We need to find a pair of segments, or line segments, which are the same length. We aren’t given any lengths on the diagram, so we will need to use the angle properties of triangles to help us. We are given that the measure of angle 𝐶𝐴𝐵 is 84 degrees. If, for example, this angle was 60 degrees, we could check if the other angles might also have a measure of 60 degrees and then we would have an equilateral triangle. However, we know that this cannot be the case here.

We might then wonder if triangle 𝐴𝐵𝐶 is an isosceles triangle. We can recall that isosceles triangles have two congruent sides. And by the isosceles triangle theorem, we know that the angles opposite the congruent sides are congruent. And the converse of this is true. That is, if two angles in a triangle are congruent, then the sides opposite those angles are congruent. So let’s see what angle measures we have and that we can calculate.

We are given this angle measure of 48 degrees. So the angle opposite it, that’s the angle 𝐴𝐶𝐵, will have a measure of 48 degrees, as opposite angles are equal in measure. Next, we can recall that the internal angle measures in a triangle sum to 180 degrees. So the three angle measures in this triangle of 48 degrees, 84 degrees, and the measure of angle 𝐴𝐵𝐶 will sum to 180 degrees. We can simplify this to 132 degrees plus the measure of angle 𝐴𝐵𝐶 equals 180 degrees. And subtracting 132 degrees from both sides, we have that the measure of angle 𝐴𝐵𝐶 is 48 degrees.

Now we can observe that triangle 𝐴𝐵𝐶 has two congruent angles, which means that triangle 𝐴𝐵𝐶 is an isosceles triangle. And importantly, we can identify the line segments that are congruent. They are the sides opposite the congruent angles. So that’s line segment 𝐴𝐵 and line segment 𝐴𝐶. This is the answer for the pair of line segments that have the same length.

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