Answers:
- A = 116
- B = 67
- A = 169
- A = 50, B = 142 , C = 30
- A = 83 , B = 36 , C = 144
==============================================
Explanations:
Problem 1
The interior angles of any quadrilateral always add to 360.
So we solve 82+84+A+78 = 360 for A and we'll have these steps
82+84+A+78 = 360
A+244 = 360
A = 360-244
A = 116
----------------
Problem 2
Same idea as the previous problem. This time we need to solve the equation 110+130+B+53 = 360
So,
110+130+B+53 = 360
B+293 = 360
B = 360-293
B = 67
----------------
Problem 3
Same idea as the other previous problems.
The equation to solve is 40+A+71+80 = 360
I'll skip the steps for it unless you need me to show them.
You should get a final answer of A = 169
----------------
Problem 4
Focus on the triangle with interior angles of: A, 68, 62
For any triangle, the three angles always add to 180
A+68+62 = 180
A+130 = 180
A = 180-130
A = 50
Now focus on the quadrilateral with interior angles of: B, 68, 62, 88
Like with problems 1 through 3, the quadrilateral angles add to 360
The equation to solve is B+68+62+88 = 360 and you should isolate B to get B = 142
Lastly, focus on the triangle with interior angles of: 88, C, 62
The equation to solve is 88+C+62 = 180. You should get C = 30
Let me know if you need to see the steps. They are similar to the other set of steps shown earlier.
----------------
Problem 5
Focus on the triangle with interior angles of: A, 62, 35
We have these steps that will find angle A
A+62+35 = 180
A+97 = 180
A = 180-97
A = 83
Move your attention to the quadrilateral with interior angles of: C, 62, 35, 119
Those four angles add to 360
C+62+35+119 = 360
C+216 = 360
C = 360-216
C = 144
Notice how this angle C is adjacent and supplementary to angle B
I.e. the linear pair of angles add to 180
B+C = 180
B+144 = 180
B = 180-144
B = 36