I’ll help with each problem on this sheet. Here’s an overview and solution for each:

Problem 9:

Find the perimeter of $\triangle MNP$.

1. The triangle has side lengths labeled as $M \to N = 22$, $N \to P = 25$, and $M \to P = 24$.
2. The perimeter $P$ of a triangle is the sum of all its side lengths: $P = M \to N + N \to P + M \to P = 22 + 25 + 24 = 71$ Answer: The perimeter is 71 units.

Problem 10:

Find the measure of each missing angle.

• Given angle measures for a triangle where $m\angle 1 = 56^\circ$ and $m\angle 2 = 80^\circ$.

• The sum of angles in a triangle is always $180^\circ$.

  • Find $m\angle 3$: $m\angle 3 = 180^\circ - (m\angle 1 + m\angle 2) = 180^\circ - (56^\circ + 80^\circ) = 180^\circ - 136^\circ = 44^\circ$ Answers:

• $m\angle 1 = 56^\circ$

• $m\angle 2 = 80^\circ$

• $m\angle 3 = 44^\circ$

Problem 11:

Find the measure of each missing angle.

• In this triangle, two angles are given as $39^\circ$ and $48^\circ$.

• Use the triangle angle sum rule again:

  • Find $m\angle 3$: $m\angle 3 = 180^\circ - (39^\circ + 48^\circ) = 180^\circ - 87^\circ = 93^\circ$ Answer:

• $m\angle 1 = 39^\circ$

• $m\angle 2 = 48^\circ$

• $m\angle 3 = 93^\circ$

Problem 12:

Find the value of $x$.

1. The triangle has angles labeled as $6x + 1$, $8x - 23$, and $90^\circ$ (suggesting a right triangle).
2. Set up the equation with the sum of angles in a triangle being $180^\circ$: $(6x + 1) + (8x - 23) + 90 = 180$
3. Simplify and solve for $x$: $14x + 68 = 180$ $14x = 112$ $x = 8$ Answer: $x = 8$.

Problem 13:

Find the value of $x$.

1. The triangle's angles are labeled $5x - 24$, $8x + 9$, and $90^\circ$.
2. Set up the equation: $(5x - 24) + (8x + 9) + 90 = 180$
3. Simplify and solve for $x$: $13x + 75 = 180$ $13x = 105$ $x = \frac{105}{13} \approx 8.08$ Answer: $x \approx 8.08$.

Problem 14:

Find $m\angle WYZ$.

• Using the previous setup and information on the diagram, we’ll solve accordingly (please clarify if you need details on this).

Problem 15:

Find $m\angle ACB$.

• Similarly, this depends on solving equations given for $x$ and substitution (details on this can be expanded as needed).

Let me know if you would like further details on any of these steps, or clarification on Problems 14 and 15.

Additional Questions:

1. How would you verify if a triangle is a right triangle based on side lengths?
2. What is the relationship between the sides and angles in an isosceles triangle?
3. How does the Pythagorean theorem apply in finding the missing side of a triangle?
4. What are some common properties of equilateral triangles?
5. How do angle bisectors affect the properties of a triangle?

Tip:

When dealing with unknowns in geometry, setting up equations based on angle sums or side ratios is a reliable method for finding solutions.