Let's go through each problem one by one:

Problem 13:

In $\triangle MNP$, the angle measures are given as:

• $m\angle M = (4x - 3)^\circ$
• $m\angle N = (9x - 6)^\circ$
• $m\angle P = (6x - 1)^\circ$

Since the sum of the angles in a triangle is $180^\circ$, we can set up the equation: $(4x - 3) + (9x - 6) + (6x - 1) = 180$ $19x - 10 = 180$ $19x = 190$ $x = 10$

Now, let's find each angle:

• $m\angle M = 4x - 3 = 4(10) - 3 = 40 - 3 = 37^\circ$
• $m\angle N = 9x - 6 = 9(10) - 6 = 90 - 6 = 84^\circ$
• $m\angle P = 6x - 1 = 6(10) - 1 = 60 - 1 = 59^\circ$

Solution for Problem 13:

• $x = 10$
• $m\angle M = 37^\circ$
• $m\angle N = 84^\circ$
• $m\angle P = 59^\circ$

Problem 14:

In $\triangle RST$, the angle measures are:

• $m\angle R = 2x + 5$
• $m\angle S = x + 1$
• $m\angle T = 7x - 16$

The sum of the angles must be $180^\circ$: $(2x + 5) + (x + 1) + (7x - 16) = 180$ $10x - 10 = 180$ $10x = 190$ $x = 19$

Now, let's find each angle:

• $m\angle R = 2(19) + 5 = 38 + 5 = 43^\circ$
• $m\angle S = 19 + 1 = 20^\circ$
• $m\angle T = 7(19) - 16 = 133 - 16 = 117^\circ$

Solution for Problem 14:

• $x = 19$
• $m\angle R = 43^\circ$
• $m\angle S = 20^\circ$
• $m\angle T = 117^\circ$

Problem 15:

In $\triangle ABC$, the angle measures are:

• $m\angle A = m\angle C - 13$
• $m\angle B = 4m\angle C - 11$

Let $m\angle C = x$. Then: $m\angle A = x - 13, \quad m\angle B = 4x - 11$

The sum of the angles must be $180^\circ$: $(x - 13) + (4x - 11) + x = 180$ $6x - 24 = 180$ $6x = 204$ $x = 34$

Now, let's find each angle:

• $m\angle C = 34^\circ$
• $m\angle A = 34 - 13 = 21^\circ$
• $m\angle B = 4(34) - 11 = 136 - 11 = 125^\circ$

Solution for Problem 15:

• $m\angle A = 21^\circ$
• $m\angle B = 125^\circ$
• $m\angle C = 34^\circ$

Problem 16:

In $\triangle JKL$, the angle measures are:

• $m\angle K = m\angle J + 9$
• $m\angle L = 2m\angle J - 21$

Let $m\angle J = x$. Then: $m\angle K = x + 9, \quad m\angle L = 2x - 21$

The sum of the angles must be $180^\circ$: $x + (x + 9) + (2x - 21) = 180$ $4x - 12 = 180$ $4x = 192$ $x = 48$

Now, let's find each angle:

• $m\angle J = 48^\circ$
• $m\angle K = 48 + 9 = 57^\circ$
• $m\angle L = 2(48) - 21 = 96 - 21 = 75^\circ$

Solution for Problem 16:

• $m\angle J = 48^\circ$
• $m\angle K = 57^\circ$
• $m\angle L = 75^\circ$

Do you need further details or have any questions?

Here are five related questions:

1. How would the problem change if the sum of the angles was given in a different type of polygon?
2. Can you solve for unknown angles in a quadrilateral with similar angle expressions?
3. What happens if one of the angles is defined in terms of another unknown variable?
4. How do you verify the correctness of your calculated angles?
5. Can you determine the measure of an exterior angle given the interior angles?

Tip: Always double-check your calculations by ensuring the sum of the angles matches the known properties of the shape.