Triangle Angle Relationships: Given Conditions

We are given a triangle, $\Delta ABC$, with the following angle relationships:

• Condition 1: $\angle C = \frac{9}{4} \angle B$
• Condition 2: $\angle C = \frac{3}{2} (\angle A+\angle B)$
• Fundamental Property: The sum of angles in any triangle is $180^{\circ}$, so $\angle A + \angle B + \angle C = 180^{\circ}$.

Solving Angle Equations for ΔABC

First, let's express $\angle B$ in terms of $\angle C$ using Condition 1:

$\angle C = \frac{9}{4} \angle B \implies \angle B = \frac{4}{9} \angle C$

Next, rearrange Condition 2:

$\frac{2}{3} \angle C = \angle A + \angle B$

Substitute the expression for $\angle B$ (from Condition 1) into this rearranged equation:

$\frac{2}{3} \angle C = \angle A + \frac{4}{9} \angle C$

Now, isolate $\angle A$ by moving the $\angle B$ term to the other side:

$\angle A = \frac{2}{3} \angle C - \frac{4}{9} \angle C$

To subtract, find a common denominator:

$\angle A = \frac{6}{9} \angle C - \frac{4}{9} \angle C = \frac{2}{9} \angle C$

Calculating Angle C Value

Utilize the Triangle Sum Property: $\angle A + \angle B + \angle C = 180^{\circ}$. Substitute the expressions we found for $\angle A$ and $\angle B$ in terms of $\angle C$:

$\frac{2}{9} \angle C + \frac{4}{9} \angle C + \angle C = 180^{\circ}$

Combine the terms with $\angle C$:

$(\frac{2}{9} + \frac{4}{9} + \frac{9}{9}) \angle C = 180^{\circ}$

$\frac{15}{9} \angle C = 180^{\circ}$

Simplify the fraction $\frac{15}{9}$ to $\frac{5}{3}$:

$\frac{5}{3} \angle C = 180^{\circ}$

Solve for $\angle C$:

$\angle C = 180^{\circ} \times \frac{3}{5} = 36^{\circ} \times 3 = 108^{\circ}$

Finding Angles A and B

With the value of $\angle C$, we can now find $\angle B$ using its relationship with $\angle C$:

$\angle B = \frac{4}{9} \angle C = \frac{4}{9} \times 108^{\circ} = 4 \times 12^{\circ} = 48^{\circ}$

Similarly, find $\angle A$ using its relationship with $\angle C$:

$\angle A = \frac{2}{9} \angle C = \frac{2}{9} \times 108^{\circ} = 2 \times 12^{\circ} = 24^{\circ}$

Final Angle Measures for ΔABC

The measures of the angles in $\Delta ABC$ are:

$\angle A = 24^{\circ}$

$\angle B = 48^{\circ}$

$\angle C = 108^{\circ}$

Verification: $24^{\circ} + 48^{\circ} + 108^{\circ} = 180^{\circ}$. The sum is correct.