Question Video: Applying Inequalities to the angles in triangles Mathematics

Look at the figure. Use <, =, or > to fill in the blanks in the following. πβ π΄πΆπ· οΌΏ πβ π΄π·πΈ. πβ π΄πΆπ· οΌΏ πβ π΄π΅πΆ. πβ π΄π·πΆ οΌΏ πβ π΄πΆπ΅. πβ π΄π·πΈ οΌΏ πβ πΆπ΄π·.

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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 π΄πΆπ· what the measure of angle π΄π·πΈ. The measure of angle π΄πΆπ· what the measure of angle π΄π΅πΆ. The measure of angle π΄π·πΆ what the measure of angle π΄πΆπ΅. And the measure of angle π΄π·πΈ what the measure of angle πΆπ΄π·.

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 π΄πΆπ΅ in triangle π΄π΅πΆ 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 π΄πΆπ΅ is equal to 139 degrees, and making a note of this. Now, the measure of angle π΄πΆπ· combines with the measure of angle π΄πΆπ΅ to make a straight angle. And using our second fact, their sum must be 180 degrees. Substituting 139 degrees for the measure of angle π΄πΆπ΅ and then subtracting this from both sides, we have the measure of angle π΄πΆπ· equals 180 degrees minus 139 degrees, which is 41 degrees.

Now that we have two of the internal angles of triangle π΄πΆπ·, we can find the other again using our first fact. The measures of angles π΄π·πΆ, π΄πΆπ·, and πΆπ΄π· must sum to 180 degrees. Hence, we have that the measure of angle π΄π·πΆ equals 180 degrees minus 41 degrees minus 71 degrees. And thatβs 68 degrees. Finally, we see that the measures of angles π΄π·πΆ and π΄π·πΈ combine to make a straight angle, so their measures sum to 180 degrees. With the measure of angle π΄π·πΆ equal to 68 degrees, we find that the measure of angle π΄π·πΈ 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 π΄πΆπ· equals 41 degrees and the measure of angle π΄π·πΈ equals 112 degrees. Hence, the measure of angle π΄πΆπ· is less than the measure of angle π΄π·πΈ. And we can fill in our first blank with the symbol for less than.

Next, again, the measure of angle π΄πΆπ· equals 41 degrees, and that of angle π΄π΅πΆ equals 17 degrees. Hence, the measure of angle π΄πΆπ· is greater than the measure of angle π΄π΅πΆ. Third, we have the measures of angles π΄π·πΆ and π΄πΆπ΅ 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 π΄π·πΈ at 112 degrees is greater than the measure of angle πΆπ΄π·, which is 71 degrees.