Question

right angle fcd intersects (overleftrightarrow{ab}) and (overleftrightarrow{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 the measure of \(\angle FCE\) and \(\angle ECD\)

Since \(\angle FCD = 90^{\circ}\) and \(\angle FCE\cong\angle ECD\), then \(\angle FCE=\angle ECD = 45^{\circ}\) (because \(90\div2 = 45\)).

Step2: Find the measure of \(\angle DCB\)

Given \(\angle ECD\) is complementary to \(\angle DCB\), and \(\angle ECD = 45^{\circ}\). By the definition of complementary angles (\(\angle1+\angle2 = 90^{\circ}\)), we have \(\angle DCB=90^{\circ}-\angle ECD=45^{\circ}\).

Step3: Check the relationship between \(\angle DCB\) and \(\angle ACF\)

\(\angle ACF = 45^{\circ}\) (given in the figure). The sum of \(\angle DCB\) and \(\angle ACF\) is \(\angle DCB+\angle ACF=45^{\circ}+ 135^{\circ}=180^{\circ}\) (since \(\angle ACB = 180^{\circ}\), \(\angle ACF+\angle FCE+\angle ECD+\angle DCB=\angle ACB\), substituting values \(45^{\circ}+45^{\circ}+45^{\circ}+\angle DCB = 180^{\circ}\), \(\angle DCB = 45^{\circ}\), \(\angle ACF+\angle DCB=180^{\circ}\)). Also, \(\angle DCB=\angle ACF = 45^{\circ}\)

Answer:

They are congruent and supplementary.