Which pair of segments have the
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.