Question

right angle fcd intersects $overleftrightarrow{ab}$ and $overrightarrow{ce}$ at point c. $angle$ fce is congruent to $angle ecd$. $angle ecd$ is complementary to $angle dcb$.
which statement is true about $angle dcb$ and $angle acf$?
○ they are congruent and complementary.
○ they are congruent and supplementary.
○ they are complementary but not necessarily congruent.
○ they are supplementary but not necessarily congruent.

Explanation:

Step1: Find ∠ACF value

Given $\angle ACF = 45^\circ$.

Step2: Calculate ∠FCD and ∠FCE

$\angle FCD$ is a right angle, so $\angle FCD = 90^\circ$. Since $\angle FCE \cong \angle ECD$, $\angle FCE = \angle ECD = \frac{90^\circ}{2} = 45^\circ$.

Step3: Find ∠DCB using complement rule

$\angle ECD$ is complementary to $\angle DCB$, so $\angle ECD + \angle DCB = 90^\circ$. Substitute $\angle ECD=45^\circ$:
$45^\circ + \angle DCB = 90^\circ \implies \angle DCB = 45^\circ$.

Step4: Compare ∠DCB and ∠ACF

$\angle DCB = 45^\circ$, $\angle ACF = 45^\circ$, so they are congruent. Check complementarity: $45^\circ + 45^\circ = 90^\circ$, so they are complementary.

Answer:

They are congruent and complementary.