Triangle Angle Calculation: Solving for ∢C Where ∢C=2∢B and ∢B=5∢A

To solve this problem, we will follow these steps:

• Step 1: Express all angles in terms of a single variable.
• Step 2: Apply the angle sum property of triangles.
• Step 3: Calculate the measures of the individual angles.

Now, let's proceed with the detailed solution:

Step 1: We know that:

• $B = 5A$
• $C = 2B = 2(5A) = 10A$

Thus, all angles are expressed in terms of $A$.

Step 2: Use the angle sum property:

$A + B + C = 180^\circ$

Substituting for $B$ and $C$:

$A + 5A + 10A = 180^\circ$

$16A = 180^\circ$

Solve for $A$:

$A = \frac{180^\circ}{16} = 11.25^\circ$

Step 3: Calculate $B$ and $C$:

$B = 5A = 5 \times 11.25^\circ = 56.25^\circ$

$C = 2B = 2 \times 56.25^\circ = 112.5^\circ$

Therefore, the measure of angle $C$ is $56\frac{1}{4}°$, which matches the provided correct answer.