Unit 1, Lesson 5
Unknown Angles
Integrated Math 2
5.1
Warm-up
Find the value of each variable mentally.
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Find the value of each variable mentally.
| triangle | longest side(s) | shortest side(s) |
Suppose you need to find the unknown angles in a complicated figure and figure out which side is the longest. Here is a figure with several unknown angles.
One way to start is to consider the equilateral triangle . Since all sides are congruent, all angles in that triangle are congruent. Since the angles must add up to 180 degrees, each angle in triangle is 60 degrees, because .
Next, consider the angles with a vertex at . Using rules for supplementary and vertical angles, we can find these unknown angles. Angle is supplementary to angle , so the measure of angle is 120 degrees because . Since angle forms a vertical pair with angle , these angles are congruent. Similarly, angle forms a vertical pair with angle , so the measures of angles and are equal.
Triangle is isosceles, so we can use the Isosceles Triangle Theorem to find the measure of the two base angles. The measures of angles and are both 30 degrees because .
Finally, we can use the Triangle Angle Sum Theorem to find the two unknown angles. The measure of angle is 45 degrees, because , and the measure of angle is 25 degrees, because .
With this knowledge, we can say that is the longest side in triangle and is the longest side in triangle . This is not enough information to know which side is the longest overall or help us find the exact lengths of any of the sides.