Question

right angle fcd intersects \\( \overleftrightarrow { a b } \\) and \\( \overleftrightarrow { c e } \\) 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: Analizar ángulo FCD

Dado que ∠FCD es un ángulo recto ($90^\circ$), y ∠FCE ≅ ∠ECD, entonces ∠FCE = ∠ECD = $45^\circ$ (porque $90^\circ \div 2 = 45^\circ$).

Step2: Analizar ∠ECD y ∠DCB

Sabemos que ∠ECD es complementario a ∠DCB, lo que significa que ∠ECD + ∠DCB = $90^\circ$. Como ∠ECD = $45^\circ$, entonces ∠DCB = $90^\circ - 45^\circ = 45^\circ$.

Step3: Analizar ∠ACF

De la figura, ∠ACF = $45^\circ$ (dado).

Step4: Comparar ∠DCB y ∠ACF

∠DCB = $45^\circ$ y ∠ACF = $45^\circ$, por lo que son congruentes. Además, ∠DCB + ∠ACF = $45^\circ + 45^\circ = 90^\circ$, lo que significa que son complementarios.

Answer:

They are congruent and complementary.