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:

• For supplementary angles: Two angles are supplementary if their sum is \(180^{\circ}\).
• \(\angle BCD+\angle BCE = 180^{\circ}\) (linear - pair of angles, since \(DCE\) is a straight line).
• \(\angle ACE+\angle ACD = 180^{\circ}\) and \(\angle BCD+\angle ACD=180^{\circ}\), so \(\angle ACE\cong\angle BCD\) (congruent angles that are supplementary to the same angle).
• For congruent angles:
• \(\angle ACE\) and \(\angle BCE\) are not supplementary (\(\angle ACE+\angle BCE

eq180^{\circ}\)), \(\angle ACE\) and \(\angle BCD\) are congruent (as shown above), \(\angle ACE\) and \(\angle BCD\) are not supplementary (\(\angle ACE+\angle BCD
eq180^{\circ}\) unless they are right - angles, which is not indicated), \(\angle BCE\) and \(\angle ACE\) are not supplementary (no linear - pair relationship), \(\angle BCD\) and \(\angle BCE\) form a linear pair (\(\angle BCD+\angle BCE = 180^{\circ}\)), and \(\angle ACE\) and \(\angle BCD\) are congruent (because \(\angle ACE+\angle ACD=180^{\circ}\) and \(\angle BCD+\angle ACD = 180^{\circ}\), so by the congruent supplements theorem \(\angle ACE\cong\angle BCD\)).

Answer:

• Angle \(BCD\) is supplementary to angle \(BCE\).
• Angle \(BCD\) is congruent to angle \(ACE\).