The problem in the image states:

"The sum of the angles of a triangle is 180 degrees. Angle B is twice as large as angle A, and angle C is 12 degrees less than the sum of the other two angles. Find the measure of each angle."

Let's define the angles:

• Let angle A be $A$.
• Angle B is twice as large as angle A, so $B = 2A$.
• Angle C is 12 degrees less than the sum of angles A and B, so $C = (A + B) - 12$.

The sum of the angles in a triangle is always 180 degrees, so we can set up the equation:

$A + B + C = 180$

Substitute $B = 2A$ and $C = (A + B) - 12$ into the equation:

$A + 2A + [(A + 2A) - 12] = 180$

Simplifying:

$A + 2A + (3A - 12) = 180$

$6A - 12 = 180$

Add 12 to both sides:

$6A = 192$

Now, solve for $A$:

$A = \frac{192}{6} = 32$

So, $A = 32^\circ$.

Now, find $B$:

$B = 2A = 2(32) = 64^\circ$

And find $C$:

$C = (A + B) - 12 = (32 + 64) - 12 = 84^\circ$

Thus, the measures of the angles are:

• $A = 32^\circ$
• $B = 64^\circ$
• $C = 84^\circ$

Would you like further explanation on any of the steps?

Here are 5 related questions to explore further:

1. What is the general rule for the sum of angles in any triangle?
2. How would the problem change if angle B was three times angle A instead of twice?
3. Can you solve a triangle problem using only one angle and the difference between the others?
4. How can this problem be applied to real-life geometry problems?
5. How does the formula for exterior angles differ from interior angles in a triangle?

Tip: Always start by identifying relationships between angles or sides when solving for unknowns in a triangle!