Answers
- angle 1 = 111 degrees
- angle 2 = 69 degrees
- angle 3 = 60 degrees
- angle 4 = 60 degrees
- angle 5 = 51 degrees
Be sure to use the actual degree symbol instead of typing in "degrees" after each number.
========================================================
Explanation:
To find angle 1, we can use the remote interior angle theorem. The yellow angles you've highlighted (46 and 65) add up to the measure of angle 1, which is an exterior angle. So angle 1 is 46+65 = 111 degrees
The slightly longer method is to make x be the missing angle of the left-most triangle. Solve x+46+65 = 180 to get x = 69 degrees. Then note how angle x and angle 1 are supplementary, meaning x+(angle1) = 180 leads to angle 1 = 111 degrees (because 180-69 = 111)
------------------------
Angle 2 is 69 degrees since angle x = 69, which is a vertical angle to angle 2. Or you could note that angle 1 and angle 2 are supplementary
(angle1)+(angle2) = 180
(111)+(angle2) = 180
angle2 = 180-111
angle 2 = 69 degrees
This method is used to prove the vertical angles theorem is always true.
-------------------------
Angle 3 can be found using the remote interior angle theorem, but we'll be going in reverse this time. Let y be the measure of angle 3
y+82 = 142
y = 142-82
y = 60
angle 3 = 60 degrees
Like with angle 1, there's also a slightly longer method that follows the same idea as before. If you follow this method, you'll need to find the missing piece of the green angle you highlighted (which his 180-142 = 38 degrees), then use the idea that A+B+C = 180 where A,B,C are the three interior angles of any triangle.
---------------------------
Angle 3 and angle 4 are vertical angles, so they are always congruent and angle 4 is also 60 degrees
---------------------------
Let z = measure of angle 5
Focusing on the smaller triangle in the middle, we can say,
(angle2)+(angle4)+(angle5) = 180
(69) + (60) + (z) = 180
z+129 = 180
z = 180-129
z = 51
angle 5 = 51 degrees
