Question

right angle fcd intersects ab and ce at point c. ∠ fce is congruent to ∠ecd. ∠ecd is complementary to ∠dcb. which statement is true about ∠dcb and ∠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: Analyze given angles

Given \(\angle FCD\) is a right angle (\(90^\circ\)), so \(\angle FCE+\angle ECD = 90^\circ\). Also, \(\angle FCE\cong\angle ECD\), so each is \(45^\circ\) (since \(90^\circ\div2 = 45^\circ\)). \(\angle ECD\) is complementary to \(\angle DCB\), so \(\angle ECD+\angle DCB=90^\circ\), which means \(\angle DCB = 90^\circ-\angle ECD = 45^\circ\). \(\angle ACF\) is given as \(45^\circ\) (from the diagram).

Step2: Check congruence and complementarity

\(\angle DCB = 45^\circ\) and \(\angle ACF = 45^\circ\), so they are congruent. Their sum is \(45^\circ + 45^\circ=90^\circ\), so they are complementary.

Answer:

They are congruent and complementary.