Let's go through the problem set:

1. Classify Each Triangle by Its Angles and Sides
   I'll provide a brief classification based on angles (acute, right, obtuse) and sides (scalene, isosceles, equilateral).

2. Problems 10 and 11 - Finding Side Lengths
   Use the distance formula to determine the side lengths of triangles $\Delta DEF$ and classify each by its side lengths.

Let's begin:

Classification of Triangles

1. Triangle 1:

   • Angles: 73°, 73°, 34° (One obtuse angle)
   • Sides: Since two angles are the same, it’s isosceles.
   • Classification: Isosceles obtuse triangle.

2. Triangle 2:

   • Angles: Includes a right angle.
   • Sides: Different side lengths.
   • Classification: Scalene right triangle.

3. Triangle 3:

   • Angles: 127° (obtuse)
   • Sides: All sides are different lengths.
   • Classification: Scalene obtuse triangle.

4. Triangle 4:

   • Angles: All angles are the same (60° each).
   • Sides: All sides are the same length (3 mm).
   • Classification: Equilateral acute triangle.

5. Triangle 5:

   • Angles: Not explicitly given but can assume based on sides.
   • Sides: All sides different.
   • Classification: Scalene triangle (angle type unknown from image alone).

6. Triangle 6:

   • Angles: Given as 41°, 65°, and 74°.
   • Sides: All sides are different.
   • Classification: Scalene acute triangle.

7. Triangle 7:

   • Angles: All 60°.
   • Sides: All sides are the same.
   • Classification: Equilateral acute triangle.

8. Triangle 8:

   • Angles: One angle is 118° (obtuse).
   • Sides: Two sides are equal (15 in).
   • Classification: Isosceles obtuse triangle.

Problem 9: Classify Triangles with Bisector Information

Using the provided triangle properties and midpoint/bisector information for $\overline{QN}$ and $\overline{PR}$:

• a. $\Delta PQR$: Based on given diagram.
• b. $\Delta PRT$: Based on given diagram.
• c. $\Delta TQS$: Based on given diagram.
• d. $\Delta QNP$: Based on given diagram.

Problems 10 and 11: Finding Side Lengths and Classifying $\Delta DEF$

We can apply the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ for each pair of points to find the side lengths of triangles in Problems 10 and 11.

Would you like detailed calculations for Problems 10 and 11, or additional explanations for each classification?

Additional Questions to Explore

1. How is the triangle classification affected if we change one angle?
2. What formulas can help verify if a triangle is isosceles?
3. How can we determine the type of angle in a triangle given only the side lengths?
4. What steps are involved in using the midpoint to classify triangles?
5. How does the distance formula help in classifying triangles based on coordinates?

Tip: Remember that the sum of angles in any triangle is always 180°. This can help check your classifications!