Question Video: Finding the Measure of an Angle in an Isosceles Triangle Using Its Properties Mathematics

Video Transcript

Find the measure of angle 𝐷𝐴𝐵.

When we’re given questions like this, it’s always good to start with what we’re given. We have triangle 𝐴𝐵𝐶. In this triangle, line segment 𝐴𝐶 is equal in length to line segment 𝐴𝐵. We know the measure of angle 𝐴𝐷𝐵 equals 90 degrees, and we know that the measure of angle 𝐶𝐴𝐷 equals 25 degrees. We want to know the measure of angle 𝐷𝐴𝐵. That’s this angle. To do that, we take the information we were given and draw some conclusions. Because line segment 𝐴𝐶 is equal to line segment 𝐴𝐵 and because the measure of angle 𝐴𝐷𝐵 is 90 degrees, we can say that line segment 𝐴𝐷 is a perpendicular bisector.

We based that on the converse of the perpendicular bisector theorem, which tells us that if a point is equidistant from the ends of two line segments — for us, that would be the line segments 𝐴𝐶 and 𝐴𝐵 that are equal — then the point 𝐴 must fall along the perpendicular bisector. Because line segment 𝐴𝐷 is a perpendicular bisector, we can say that line segment 𝐶𝐷 is equal in length to line segment 𝐵𝐷. We know that this perpendicular bisector creates two smaller triangles. And we can say that the smaller triangle 𝐴𝐷𝐶 must be congruent to the smaller triangle 𝐴𝐷𝐵.

We say this based on side-side-side congruence. Three sides of triangle 𝐴𝐷𝐶 are equal to the corresponding three sides of triangle 𝐴𝐷𝐵. And since these two triangles are congruent, we can say that the measure of angle 𝐶𝐴𝐷 will be equal to the measure of angle 𝐷𝐴𝐵. We’re saying these two angles are congruent, which makes the measure of angle 𝐷𝐴𝐵 25 degrees.