Unit 1, Lesson 4
Big Angles, Long Sides. Small Angles, Short Sides.
Integrated Math 2
4.1
Warm-up
Which three go together? Why do they go together?
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Which three go together? Why do they go together?
Is this triangle a right triangle? Explain or show your reasoning.
In a triangle, larger angles are always opposite, or across from, longer sides. In triangle , the largest angle is and it is opposite the longest side . Similarly, the smallest angle, , is opposite the shortest side, . The converse is also true—longer sides are opposite larger angles. Since angle is the largest angle, we know side is the longest.
In triangle , which side is the shortest? Using the Triangle Angle Sum Theorem, we find angle is 32 degrees, so side must be the shortest side in the triangle.
In right triangles, we have additional methods to help us figure out information about the angles and sides. You may have learned in an earlier grade that the hypotenuse of a right triangle is always the longest side. Now we can say that the hypotenuse is the longest side in a right triangle because it is opposite the right angle, which is always the biggest angle in a right triangle.
In triangle , which side is shorter: or ?
Method 1
Using Pythagorean Theorem, we can solve for .
This method is helpful if you need to know the length of and you have a right triangle.
Method 2
Using the Triangle Angle Sum Theorem, we know the measure of angle is 56 degrees.
Since angle is the smallest angle, the shortest side must be .
This method doesn’t give us the length of , but it does answer the question of which side in the triangle is shortest.