Question

angle acd is supplementary to angles ace and bcd and congruent to angle bce. which statements are true about the angles in the diagram? select three options. angle ace is supplementary to angle bcd. angle bce is supplementary to angle ace. angle bcd is supplementary to angle bce. angle ace is congruent to angle bce. angle bcd is congruent to angle ace.

Explanation:

Step1: Recall supplementary - angle definition

Two angles are supplementary if their sum is 180 degrees. Since \(\angle ACD\) is supplementary to \(\angle ACE\) and \(\angle BCD\), and \(\angle ACD+\angle ACE = 180^{\circ}\), \(\angle ACD+\angle BCD=180^{\circ}\), we know that \(\angle ACE\) and \(\angle BCD\) are congruent (because \(\angle ACD + \angle ACE=\angle ACD+\angle BCD\), so \(\angle ACE=\angle BCD\)). Also, \(\angle BCE\) and \(\angle ACD\) are congruent. And \(\angle BCE+\angle BCD = 180^{\circ}\) as they form a linear - pair.

Step2: Analyze each option

• Option 1: \(\angle ACE\) is supplementary to \(\angle BCD\). False, they are congruent.
• Option 2: \(\angle BCE\) is supplementary to \(\angle ACE\). True, because \(\angle BCE+\angle ACD = 180^{\circ}\) and \(\angle ACE\) and \(\angle BCD\) are congruent, and \(\angle BCE+\angle BCD = 180^{\circ}\).
• Option 3: \(\angle BCD\) is supplementary to \(\angle BCE\). True, they form a linear - pair.
• Option 4: \(\angle ACE\) is congruent to \(\angle BCE\). False, \(\angle ACD\) is congruent to \(\angle BCE\).
• Option 5: \(\angle BCD\) is congruent to \(\angle ACE\). True, as shown above.

Answer:

B. Angle BCE is supplementary to angle ACE.
C. Angle BCD is supplementary to angle BCE.
E. Angle BCD is congruent to angle ACE.