Question

if $overline{cd}congoverline{cf}$, $mangle ced = 5a$, and $mangle cef=a + 68^{circ}$, what is $mangle def? mangle def=square^{circ}$

Explanation:

Step1: Use property of congruent segments

Since $\overline{CD}\cong\overline{CF}$ and $\angle CDE = \angle CFE=90^{\circ}$, and $CE = CE$ (common side), then $\triangle CDE\cong\triangle CFE$ (HL - Hypotenuse - Leg congruence criterion). So, $\angle CED=\angle CEF$.

Step2: Set up equation for angles

Set $5a=a + 68^{\circ}$.
Subtract $a$ from both sides: $5a - a=a + 68^{\circ}-a$.
We get $4a=68^{\circ}$.
Then divide both sides by 4: $a=\frac{68^{\circ}}{4}=17^{\circ}$.

Step3: Calculate $\angle DEF$

Since $\angle DEF=\angle CED+\angle CEF$, and $\angle CED=\angle CEF = 5a$ or $a + 68^{\circ}$.
Substitute $a = 17^{\circ}$, then $\angle CED=\angle CEF=5\times17^{\circ}=85^{\circ}$.
So, $\angle DEF=85^{\circ}+85^{\circ}=170^{\circ}$.

Answer:

$170$