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.
