Side and Angle Conditions for a Triangle

Pharaoh Ahmose enjoys a luxurious lifestyle, but there's something missing. He has heard stories of the kites which thrill the emperors in China. He wants a kite to call his own. So, Ahmose summons his engineers, and challenges them to build a triangular kite that will soar like a falcon. The engineers aren't sure they can actually form a triangle from the materials they have. They'll need to experiment to understand Side and Angle Conditions for a Triangle. The engineers are given three rods to form the supporting elements for the kite. The rods are 150 cm 70 cm and 30 cm long. We'll label the rods 'a', 'b,' and 'c,' respectively. Let's leave 'a,' the largest segment, here. Now let's try and assemble a triangle by placing the remaining segments on its ends. Can we complete a triangle? No! We can change the angles of 'b' and 'c' as much as we like, but no matter how much we try, they won't reach each other to complete a triangle. Together, they are just too short. Notice that 'b' and 'c' sum to be 100 cm which is less than 150 cm, the length of 'a'. What can we do to make this a complete triangle? Let's replace the 30 cm rod with one of 100 cm. Look! That makes a complete triangle! The two shorter sides of 70 cm and 100 cm sum to be 170 cm. That's greater than the longest side of 150 cm. In general, the two other sides must sum to be more than the longest side. Given three sides this is one of the conditions we must meet in order to form a triangle. The engineers want to try out different side lengths given two sides. Let's remove the 100 cm side and experiment. Given any two sides, what are the lower and upper limits for the third side? Let's start by finding the smallest measurement that this side, 'c,' can be. Let's rotate 'a' so that its angle is really small. Can 'c' be 80 cm? No! Because that would just give us a straight line and not a triangle. But could 'c' be 81 cm? Or even 85 cm? Yes! In fact, 'c' can be anything more than 80 cm. Because 80 is the difference of the two given sides, 150 and 70. Anything greater than that difference should be fine, right? Oh, but wait! What's the biggest that 'c' can be? If we swing side 'a' so that this angle is really big, then the side 'c' gets longer. Can 'c' be 220 cm? No! That just makes another straight line and not a triangle. So 'c' has to be less than 220 which is the SUM of the two given sides, 150 and 70. Anything less than the sum of the other two sides works. Now we can put these limits together. Given any two sides we can find the range of values for the third side. Any third side must be greater than the difference and less than the sum of the two given sides. In this case, 'c' must be greater than 80 and less than 220. But a triangle consists of more than just three sides. It also has three angles! How can we determine a triangle by just looking at two angles? Let's experiment again. First, let's construct a horizontal line segment 'AB'. Now, let's place our protractor at point A and mark a measure of 50 degrees. We'll use this mark to construct a ray, that is a line with a start point but no end point. Now from point 'B' let's measure a 150 degree angle. We'll construct a ray here as well. Do these two angles give us a triangle? No, clearly the rays will never intersect! Notice that the sum of the two angles is 200 degrees! But what if we made angle 'B' smaller? Notice that as the sum of the angles gets smaller we get closer to a triangle. In fact, any pair of angles in a triangle must sum to less than 180 degrees in order to form a triangle. Now the engineers understand the limits on the sides and angles they can use to build the triangular kite! Let's review side and angle conditions for forming a triangle as they complete their construction. Given three sides, the longest side must be less than the sum of the other two sides. Given any two sides, the third side must be greater than their difference, and less than their sum. Finally, two angle measurements determine many triangles provided that the sum of any two angles is less than 180 degrees. With all this knowledge the engineers construct a beautiful kite that soars like a falcon! But Ahmose, you can't just watch a kite, you've got to fly it.