Question Video: Applying Inequalities to the angles in triangles Mathematics

Video Transcript

Look at the figure. Use the symbols for less than, equals, or greater than to fill in the blanks in the following. The measure of angle 𝐴𝐶𝐷 what the measure of angle 𝐴𝐷𝐸. The measure of angle 𝐴𝐶𝐷 what the measure of angle 𝐴𝐵𝐶. The measure of angle 𝐴𝐷𝐶 what the measure of angle 𝐴𝐶𝐵. And the measure of angle 𝐴𝐷𝐸 what the measure of angle 𝐶𝐴𝐷.

To compare the given measures, we begin by recalling that the measures of the internal angles of a triangle sum to 180 degrees and that the measures of the angles that make up a straight line also sum to 180 degrees. We can use the first of these facts to find the measure of angle 𝐴𝐶𝐵 in triangle 𝐴𝐵𝐶 by noting that its sum with 17 and 24 degrees must equal 180 degrees.

Now, subtracting 17 and 24 degrees from both sides, we have the measure of angle 𝐴𝐶𝐵 is equal to 139 degrees, and making a note of this. Now, the measure of angle 𝐴𝐶𝐷 combines with the measure of angle 𝐴𝐶𝐵 to make a straight angle. And using our second fact, their sum must be 180 degrees. Substituting 139 degrees for the measure of angle 𝐴𝐶𝐵 and then subtracting this from both sides, we have the measure of angle 𝐴𝐶𝐷 equals 180 degrees minus 139 degrees, which is 41 degrees.

Now that we have two of the internal angles of triangle 𝐴𝐶𝐷, we can find the other again using our first fact. The measures of angles 𝐴𝐷𝐶, 𝐴𝐶𝐷, and 𝐶𝐴𝐷 must sum to 180 degrees. Hence, we have that the measure of angle 𝐴𝐷𝐶 equals 180 degrees minus 41 degrees minus 71 degrees. And that’s 68 degrees. Finally, we see that the measures of angles 𝐴𝐷𝐶 and 𝐴𝐷𝐸 combine to make a straight angle, so their measures sum to 180 degrees. With the measure of angle 𝐴𝐷𝐶 equal to 68 degrees, we find that the measure of angle 𝐴𝐷𝐸 equals 112 degrees.

Now we have all the information we need to fill in the blanks. So making some space, we have the measure of angle 𝐴𝐶𝐷 equals 41 degrees and the measure of angle 𝐴𝐷𝐸 equals 112 degrees. Hence, the measure of angle 𝐴𝐶𝐷 is less than the measure of angle 𝐴𝐷𝐸. And we can fill in our first blank with the symbol for less than.

Next, again, the measure of angle 𝐴𝐶𝐷 equals 41 degrees, and that of angle 𝐴𝐵𝐶 equals 17 degrees. Hence, the measure of angle 𝐴𝐶𝐷 is greater than the measure of angle 𝐴𝐵𝐶. Third, we have the measures of angles 𝐴𝐷𝐶 and 𝐴𝐶𝐵 are 68 and 139 degrees, respectively. So our third symbol is less than, since 68 is less than 139. And finally, the measure of angle 𝐴𝐷𝐸 at 112 degrees is greater than the measure of angle 𝐶𝐴𝐷, which is 71 degrees.