Let's solve for the values of the variables $x$ in each figure by using the fact that the sum of the angles in a triangle is $180^\circ$.

Problem 16

The angles given are $71^\circ$, $46^\circ$, and $x$.

Step 1: Set up the equation: $71^\circ + 46^\circ + x = 180^\circ$

Step 2: Simplify and solve for $x$: $117^\circ + x = 180^\circ$ $x = 180^\circ - 117^\circ = 63^\circ$

Answer for Problem 16: $x = 63^\circ$

Problem 17

The angles given are $56^\circ$, $76^\circ$, and $x$.

Step 1: Set up the equation: $56^\circ + 76^\circ + x = 180^\circ$

Step 2: Simplify and solve for $x$: $132^\circ + x = 180^\circ$ $x = 180^\circ - 132^\circ = 48^\circ$

Answer for Problem 17: $x = 48^\circ$

Problem 18

The triangle is split into two parts. We need to find $x$ in the smaller triangle where the given angles are $46^\circ$, $36^\circ$, and $x$.

Step 1: Set up the equation: $46^\circ + 36^\circ + x = 180^\circ$

Step 2: Simplify and solve for $x$: $82^\circ + x = 180^\circ$ $x = 180^\circ - 82^\circ = 98^\circ$

Answer for Problem 18: $x = 98^\circ$

Problem 19

The given angles are $39^\circ$, $71^\circ$, and $x$. The angle $91^\circ$ is the sum of the angles in the right triangle on the left.

Step 1: Set up the equation: $71^\circ + 39^\circ + x = 180^\circ$

Step 2: Simplify and solve for $x$: $110^\circ + x = 180^\circ$ $x = 180^\circ - 110^\circ = 70^\circ$

Answer for Problem 19: $x = 70^\circ$

Would you like more details or have any questions?

Related Questions

1. How do the properties of different types of triangles affect the calculation of their angles?
2. Can the value of an exterior angle in a triangle be used to find interior angles? How?
3. What is the difference between solving for angles in acute, obtuse, and right triangles?
4. How can you use supplementary and complementary angles to solve for unknown angles in geometric figures?
5. How does the Triangle Inequality Theorem relate to the angles within a triangle?

Tip

When dealing with multiple triangles or complex figures, consider solving smaller sections or using known angle relationships to simplify the process.