A triangle has two corners of angles #pi /12# and #pi/6 #. What are the complement and supplement of the third corner?

Question

A triangle has two corners of angles #pi /12# and #pi/6  #.  What are the complement and supplement of the third corner?

Violet Aldrich · Accepted Answer

Complementary of third angle #= -pi/4#

Supplementary of third angle # = pi/4# #hat A = pi/12, hat B = pi/6, hat C = pi - pi/12 - pi/6 = (3pi)/4# Complementary angle of #hat C = pi/2 - (3pi)/4 = -(pi/4)# Supplementary angle of #hat C = pi - (3pi)/4 = pi/4#

Savannah Aguilar · Answer

undefined The sum of the angles in any triangle is always $ \pi $ radians (or 180 degrees).

Given:
Angle 1: $ \frac{\pi}{12} $
Angle 2: $ \frac{\pi}{6} $

To find the measure of the third angle, $ \theta $, use the fact that the sum of the angles in a triangle is $ \pi $:

$$ \frac{\pi}{12} + \frac{\pi}{6} + \theta = \pi $$

First, find a common denominator for $ \frac{\pi}{12} $ and $ \frac{\pi}{6} $, which is $ \frac{\pi}{12} $.

$$ \frac{\pi}{12} + \frac{2\pi}{12} + \theta = \pi $$
$$ \frac{3\pi}{12} + \theta = \pi $$
$$ \frac{\pi}{4} + \theta = \pi $$

Now, solve for $ \theta $:

$$ \theta = \pi - \frac{\pi}{4} $$
$$ \theta = \frac{3\pi}{4} $$

The complement of $ \frac{3\pi}{4} $ is the angle that, when added to $ \frac{3\pi}{4} $, equals $ \pi $. Therefore, the complement is $ \frac{\pi}{4} $.

The supplement of $ \frac{3\pi}{4} $ is the angle that, when added to $ \frac{3\pi}{4} $, equals $ 2\pi $. Therefore, the supplement is $ \frac{5\pi}{4} $.