# Question Video: The Triangle Inequality Theorem Mathematics

Use <, =, or > to complete the statement: If πβ π΄πΆπ΅ = 62Β° and πβ π΄ = 57Β°, then πβ π΄π΅π· οΌΏ πβ π΄πΆπ·.

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### Video Transcript

Use the less than, is equal to, or greater than symbol to complete the statement. If the measure of angle π΄πΆπ΅ equals 62 degrees and the measure of angle π΄ equals 57 degrees, then the measure of angle π΄π΅π· what the measure of angle π΄πΆπ·.

We notice first that πΆπ· is equal to π΅π·, so the triangle π΅πΆπ· 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 π΅πΆπ· and πΆπ΅π· are equal.

Now, weβre told that the measure of angle π΄πΆπ΅ is 62 degrees and that the measure of the angle at π΄ 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 π΄ and those of angles π΄π΅πΆ and π΄πΆπ΅ sum to 180 degrees.

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

Now, if we call the measure of our two congruent angles π₯. And making some space, we have π₯ plus the measure of angle π΄πΆπ· equals 62 degrees. And π₯ plus the measure of angle π΄π΅π· equals 61 degrees. Now, if we add one to both sides of the second equation, we have that π₯ plus the measure of angle π΄π΅π· 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 π₯ plus the measure of angle π΄πΆπ· equals π₯ plus the measure of angle π΄π΅π· plus one. Subtracting π₯ from both sides, we then have the measure of angle π΄πΆπ· equals the measure of angle π΄π΅π· plus one. This means that the measure of angle π΄π΅π· is one degree smaller than that of angle π΄πΆπ·. Hence, the measure of angle π΄π΅π· is less than the measure of angle π΄πΆπ·. And the less than symbol completes the given statement.