Question

△ abc ≅△ def
df = 12, de = 10, m∠e = 117°, and m∠d = 27°
what are the values of ac and m∠c?
enter your answers in the boxes.
ac =

m∠c = °

Explanation:

Step1: Use congruent triangle properties

Since $\triangle ABC \cong \triangle DEF$, corresponding sides and angles are equal. So, $AC$ corresponds to $DF$, and $\angle C$ corresponds to $\angle F$.

Step2: Find length of $AC$

Given $DF = 12$, and $AC \cong DF$ (corresponding sides of congruent triangles), so $AC = DF = 12$.

Step3: Find measure of $\angle C$

First, find $m\angle F$ in $\triangle DEF$. The sum of angles in a triangle is $180^\circ$. So, $m\angle F = 180^\circ - m\angle D - m\angle E$. Substitute $m\angle D = 27^\circ$ and $m\angle E = 117^\circ$: $m\angle F = 180 - 27 - 117 = 36^\circ$. Since $\angle C \cong \angle F$ (corresponding angles of congruent triangles), $m\angle C = 36^\circ$.

Answer:

$AC = 12$
$m\angle C = 36^\circ$