Question

what are the values of the variables in each figure? see examples 1 - 3 16. 17. 18. 19.

Explanation:

Step1: Use the triangle angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\).
For problem 16:
\(x = 180-(71 + 46)\)
\(x=180 - 117\)
\(x = 63\)

Step2: Use the triangle angle - sum theorem

For problem 17:
\(x=180-(56 + 76)\)
\(x = 180-132\)
\(x = 48\)

Step3: Use the triangle angle - sum theorem and vertical angles

For problem 18:
First, find \(x\) in the left - hand triangle. \(x=180-(46 + 36)\)
\(x = 180 - 82\)
\(x = 98\)
Since \(x\) and the non - labeled angle in the right - hand triangle are vertical angles, they are equal. Then find \(y\) in the right - hand triangle. \(y=180-(98 + 44)\)
\(y=180 - 142\)
\(y = 38\)

Step4: Use the triangle angle - sum theorem

For problem 19:
First, find the non - labeled angle in the left - hand triangle. Let it be \(z\), \(z = 180-(39+71)\)
\(z=180 - 110\)
\(z = 70\)
Then \(x = 180 - 70\) (linear pair)
\(x = 110\)
Next, find \(y\) in the right - hand triangle. \(y=180-(91+(180 - 110))\)
\(y=180-(91 + 70)\)
\(y=180 - 161\)
\(y = 19\)

Answer:

1. \(x = 63\)
2. \(x = 48\)
3. \(x = 98,y = 38\)
4. \(x = 110,y = 19\)