Problem Analysis

We are solving for the angles in triangle $\triangle IJK$, which is isosceles with base $\overline{JI}$. Since the triangle is isosceles, the angles opposite the congruent sides are equal. This means: $\angle J = \angle K.$

Given: $m\angle J = (3x + 8)^\circ \quad \text{and} \quad m\angle K = (2x + 52)^\circ.$

The sum of all angles in a triangle is $180^\circ$, so: $m\angle J + m\angle K + m\angle I = 180^\circ.$

We'll solve for $x$, find the measures of $\angle J$, $\angle K$, and $\angle I$.

Step 1: Equate $\angle J$ and $\angle K$

Since $\angle J = \angle K$: $3x + 8 = 2x + 52.$ Simplify: $3x - 2x = 52 - 8,$ $x = 44.$

Step 2: Calculate $m\angle J$ and $m\angle K$

Substitute $x = 44$ into $m\angle J$ and $m\angle K$: $m\angle J = 3x + 8 = 3(44) + 8 = 132 + 8 = 140^\circ,$ $m\angle K = 2x + 52 = 2(44) + 52 = 88 + 52 = 140^\circ.$

Step 3: Calculate $m\angle I$

Since the sum of the angles is $180^\circ$: $m\angle J + m\angle K + m\angle I = 180^\circ,$ $140 + 140 + m\angle I = 180,$ $m\angle I = 180 - 280 = 40^\circ.$

Final Answer:

The degree measures of the angles in the triangle are: $m\angle J = 140^\circ, \quad m\angle K = 140^\circ, \quad m\angle I = 40^\circ.$

Would you like me to go into further details?

Related Questions:

1. What would change if $\triangle IJK$ were not isosceles?
2. How can we verify our results geometrically?
3. Can this approach be extended to scalene triangles?
4. What happens if $x$ gives a negative angle? Is it a valid triangle?
5. How do we compute the exterior angles of $\triangle IJK$?

Tip:

When solving equations involving triangles, always double-check that the angle measures add up to $180^\circ$.