Question Video: The Triangle Inequality Theorem Mathematics

Video Transcript

Use the less than, is equal to, or greater than symbol to complete the statement. If the measure of angle 𝐴𝐶𝐵 equals 62 degrees and the measure of angle 𝐴 equals 57 degrees, then the measure of angle 𝐴𝐵𝐷 what the measure of angle 𝐴𝐶𝐷.

We notice first that 𝐶𝐷 is equal to 𝐵𝐷, so the triangle 𝐵𝐶𝐷 is an isosceles triangle. Then recalling that the angles opposite the congruent sides of an isosceles triangle are congruent, we see that the measures of angles 𝐵𝐶𝐷 and 𝐶𝐵𝐷 are equal.

Now, we’re told that the measure of angle 𝐴𝐶𝐵 is 62 degrees and that the measure of the angle at 𝐴 is 57 degrees. And recalling that the measures of the interior angles of a triangle sum to 180 degrees, we have that the measure of the angle at 𝐴 and those of angles 𝐴𝐵𝐶 and 𝐴𝐶𝐵 sum to 180 degrees.

Substituting the given values into our equation, we have 57 degrees plus the measure of angle 𝐴𝐵𝐶 plus 62 degrees equals 180 degrees. Now, subtracting 57 and 62 degrees from both sides, we find that the measure of angle 𝐴𝐵𝐶 equals 61 degrees. We can mark this on our diagram as shown.

Now, if we call the measure of our two congruent angles 𝑥. And making some space, we have 𝑥 plus the measure of angle 𝐴𝐶𝐷 equals 62 degrees. And 𝑥 plus the measure of angle 𝐴𝐵𝐷 equals 61 degrees. Now, if we add one to both sides of the second equation, we have that 𝑥 plus the measure of angle 𝐴𝐵𝐷 plus one is equal to 62 degrees.

And now since both of our right-hand sides equal 62, we can equate our left-hand sides to give 𝑥 plus the measure of angle 𝐴𝐶𝐷 equals 𝑥 plus the measure of angle 𝐴𝐵𝐷 plus one. Subtracting 𝑥 from both sides, we then have the measure of angle 𝐴𝐶𝐷 equals the measure of angle 𝐴𝐵𝐷 plus one. This means that the measure of angle 𝐴𝐵𝐷 is one degree smaller than that of angle 𝐴𝐶𝐷. Hence, the measure of angle 𝐴𝐵𝐷 is less than the measure of angle 𝐴𝐶𝐷. And the less than symbol completes the given statement.