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 angie bcd is congruent to angle ace

Explanation:

Step1: Analyze supplementary angles

Two angles are supplementary if their sum is \(180^{\circ}\).
Since \(ACE + BCE=180^{\circ}\) (linear - pair), so angle \(BCE\) is supplementary to angle \(ACE\).
Also, \(BCD + BCE = 180^{\circ}\) (linear - pair), so angle \(BCD\) is supplementary to angle \(BCE\).

Step2: Analyze congruent angles

Given that \(\angle ACD\) is supplementary to \(\angle ACE\) and \(\angle BCD\) (\(\angle ACD+\angle ACE = 180^{\circ}\) and \(\angle ACD+\angle BCD=180^{\circ}\)).
By the congruent - supplements theorem (if two angles are supplementary to the same angle, then they are congruent), \(\angle BCD\cong\angle ACE\)

Step3: Check other options

• For \(\angle ACE\) and \(\angle BCD\):

Let \(\angle ACD\) be \(x\). Since \(\angle ACD+\angle ACE = 180^{\circ}\), \(\angle ACE=180 - x\). And \(\angle BCD\) is congruent to \(\angle ACE\) (from above), not supplementary.

• For \(\angle ACE\) and \(\angle BCE\):

We know \(\angle ACE+\angle BCE = 180^{\circ}\) (supplementary), not congruent.

Answer:

• Angle BCE is supplementary to angle ACE
• Angle BCD is supplementary to angle BCE
• Angle BCD is congruent to angle ACE