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: Define given right angle

$\angle FCD = 90^\circ$

Step2: Use congruent angles

$\angle FCE \cong \angle ECD$, so $\angle FCE = \angle ECD = \frac{90^\circ}{2} = 45^\circ$

Step3: Use complementary angles

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

Step4: Calculate $\angle ACF$

$\angle ACF = 45^\circ$ (given in diagram)

Step5: Compare the two angles

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

Answer:

They are congruent and complementary.