We are given a triangle with three angles:

• $m \angle I = (5x - 9)^\circ$,
• $m \angle J = (6x - 3)^\circ$,
• $m \angle K = x^\circ$.

We need to find the degree measures of each angle. Since the sum of the angles in any triangle is always 180 degrees, we can set up the following equation:

$(5x - 9) + (6x - 3) + x = 180$

Let's solve this equation step-by-step.

1. Combine like terms: $5x + 6x + x - 9 - 3 = 180$ $12x - 12 = 180$

2. Add 12 to both sides: $12x = 192$

3. Divide both sides by 12: $x = 16$

Now that we have the value of $x$, we can find the measure of each angle:

• $m \angle I = 5x - 9 = 5(16) - 9 = 80 - 9 = 71^\circ$,
• $m \angle J = 6x - 3 = 6(16) - 3 = 96 - 3 = 93^\circ$,
• $m \angle K = x = 16^\circ$.

So, the measures of the angles are:

• $m \angle I = 71^\circ$,
• $m \angle J = 93^\circ$,
• $m \angle K = 16^\circ$.

Would you like more details on any part of this solution or clarification on any step?

Follow-up questions:

1. How do we know that the sum of the angles in a triangle is always 180 degrees?
2. Can you solve for $x$ in other triangles where angle expressions involve different variables?
3. What are some properties of triangles that help in solving such problems?
4. How can you verify that these angles indeed form a valid triangle?
5. How would the solution change if one of the angles were expressed as a quadratic?

Tip:

Always check if the sum of calculated angles equals 180° to verify the correctness of your solution when dealing with triangle angle problems.