Let's solve each problem one by one.

Problem 8

1. Given Information:

   • Triangle with angles: $40^\circ$, $20^\circ$, and $2x$.

2. Solution:

   • The sum of angles in a triangle is $180^\circ$.
   • Therefore, we set up the equation: $40^\circ + 20^\circ + 2x = 180^\circ$
   • Simplify this equation: $60^\circ + 2x = 180^\circ$
   • Subtract $60^\circ$ from both sides: $2x = 120^\circ$
   • Divide by 2: $x = 60^\circ$

3. Classification by Angles:

   • Since all angles are less than $90^\circ$, it is an acute triangle.

4. Theorem Used:

   • Sum of Interior Angles Theorem.

Problem 9

1. Given Information:

   • Triangle with angles: $40^\circ$, $60^\circ$, and an exterior angle $x$.

2. Solution:

   • By the Exterior Angle Theorem, the exterior angle is equal to the sum of the two non-adjacent interior angles.
   • Therefore: $x = 40^\circ + 60^\circ$
   • Calculate $x$: $x = 100^\circ$

3. Classification by Angles:

   • The interior angles sum up to $180^\circ$ with one of them being $100^\circ$, so this is an obtuse triangle.

4. Theorem Used:

   • Exterior Angle Theorem.

Problem 10

1. Given Information:

   • Triangle with one angle marked as $110^\circ$, an exterior angle labeled $3x - 20$, and a missing angle $x$.

2. Solution:

   • By the Exterior Angle Theorem, the exterior angle $3x - 20$ is equal to the sum of the two opposite interior angles, one of which is $x$ and the other is $110^\circ$.
   • Set up the equation: $3x - 20 = x + 110^\circ$
   • Solve for $x$:
     • First, bring all $x$-terms to one side: $3x - x = 110^\circ + 20$ $2x = 130^\circ$
     • Now, divide by 2: $x = 65^\circ$

3. Classification by Angles:

   • The interior angles include one angle of $110^\circ$, making it an obtuse triangle.

4. Theorem Used:

   • Exterior Angle Theorem.

Would you like further clarification on any step, or details on any theorem?

Related Questions:

1. How do you classify a triangle with one angle greater than $90^\circ$?
2. What is the Exterior Angle Theorem and when is it useful in solving triangles?
3. Why do the angles of a triangle always sum up to $180^\circ$?
4. Can a triangle have two obtuse angles? Why or why not?
5. How can you determine the type of triangle based solely on its side lengths?

Tip:

When working with exterior angles, remember they are always equal to the sum of the two opposite interior angles. This is very helpful for finding unknown angles in triangles.