Question

find the measure of ∠tdc.

Explanation:

Step1: Note angle - sum property

The sum of angles $\angle TDC$ and $\angle CDF$ is $90^{\circ}$ (assuming $\angle TDF = 90^{\circ}$ as it is a right - angle situation not explicitly stated but inferred from the context).

Step2: Set up equation

We know $\angle CDF=54^{\circ}$ and $\angle TDC = 4y$. So, $4y+54^{\circ}=90^{\circ}$.

Step3: Solve for $y$

Subtract $54^{\circ}$ from both sides of the equation: $4y=90^{\circ}- 54^{\circ}$, so $4y = 36^{\circ}$, then $y = 9^{\circ}$.

Step4: Find measure of $\angle TDC$

Since $\angle TDC=4y$, substituting $y = 9^{\circ}$, we get $\angle TDC=4\times9^{\circ}=36^{\circ}$.

Answer:

$36^{\circ}$