# Question Video: The Triangle Inequality Theorem Mathematics

Use <, =, or > to fill in the blank: π΅πΆ οΌΏ π΄πΆ.

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### Video Transcript

Use less than, equal to, or greater than to fill in the blank to compare side length π΅πΆ to side length π΄πΆ.

First, we can identify which side lengths weβre trying to compare. We want to compare side length π΅πΆ to side length π΄πΆ. One strategy we can try to use is the angle-side inequality in a triangle. To do that, weβd need to compare the angles opposite the two side lengths weβre interested in. Now you might be thinking we donβt have any information about the angles. However, we can use some properties of triangles to find out some information about the angle measures. In a triangle, if two sides have the same length, then their opposite angles are the same, which means here angle π΄π΅πΉ will be equal in measure to angle πΉπ΄π΅ and angle π΄π·πΉ will be equal to angle π·π΄πΉ.

Both triangle π΄π΅πΉ and triangle π΄π·πΉ are isosceles triangles, and they have two equal sides. This means we can say that segment π΄π· is going to be equal in length to segment π΄π΅ as these two triangles are congruent. It also means angle π΄πΉπ΅ will be equal to angle π΄πΉπ·. This means we found that side length π΄π΅ must be smaller than side length π΄πΆ. But how can we say something about side length π΅πΆ? For this, weβre going to think about the line segment π΄πΉ. Weβve seen that the line segment π΄πΉ bisects the segment π΅π·. The line segment π΄πΉ is a bisector of the isosceles triangle π΄π΅π·. And when that happens, it is a perpendicular bisector, which means both of these angles must measure 90 degrees. And if thatβs the case, in the smaller isosceles triangles, the smaller blue angles must be equal to each other and must therefore be equal 45 degrees each.

If all of these smaller angles measure 45 degrees, the big angle we were looking for, angle π΅π΄πΆ, is a right angle. And since line segment π΅πΆ is opposite that right angle, itβs the hypotenuse of the larger triangle π΄π΅πΆ, and it is therefore the longest side length of this triangle. The order of the side lengths for the larger triangle π΄π΅πΆ must be π΄π΅ is smaller than π΄πΆ, which is smaller than π΅πΆ. Since π΅πΆ is the hypotenuse and is the longest side length in this triangle, we can say π΅πΆ is greater than π΄πΆ.