Let's solve each of the triangles for the value of $x$ in the given figures.

Problem 20:

This is a triangle where we know two angles and need to find the third. The angles given are $98^\circ$ and $41^\circ$.

• Solution: Use the triangle sum property, which states that the sum of the angles in a triangle is always $180^\circ$. $x = 180^\circ - 98^\circ - 41^\circ$ $x = 41^\circ$

Problem 21:

This is a straight line with angles given as $42^\circ$ and $79^\circ$. $x$ is the angle adjacent to $79^\circ$.

• Solution: The angles on a straight line sum to $180^\circ$. Hence, $x = 180^\circ - 79^\circ$ $x = 101^\circ$

Problem 22:

This is a triangle with given angles $119^\circ$ and $46^\circ$. We need to find the third angle $x$.

• Solution: Again, using the triangle sum property: $x = 180^\circ - 119^\circ - 46^\circ$ $x = 15^\circ$

Problem 23:

This triangle has angles $74^\circ$, $23^\circ$, and $x$.

• Solution: Use the triangle sum property: $x = 180^\circ - 74^\circ - 23^\circ$ $x = 83^\circ$

Final Values:

• Problem 20: $x = 41^\circ$
• Problem 21: $x = 101^\circ$
• Problem 22: $x = 15^\circ$
• Problem 23: $x = 83^\circ$

Would you like any additional details or explanations on these problems?

Further Questions:

1. How does the triangle sum property work in non-Euclidean geometry?
2. What is the relationship between exterior and interior angles of a triangle?
3. How can you determine if a triangle is acute, obtuse, or right using its angles?
4. What happens if the sum of angles in a polygon is more than $180^\circ$?
5. Can the angles of a triangle be used to determine the type of triangle (equilateral, isosceles, scalene)?

Tip:

When solving for angles in any polygon, always remember that the sum of the interior angles in a triangle is $180^\circ$, which can be a useful checkpoint in your calculations.