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: Define supplementary angles

Supplementary angles sum to $180^\circ$.
Given $\angle ACD + \angle ACE = 180^\circ$ and $\angle ACD + \angle BCD = 180^\circ$.

Step2: Prove $\angle ACE \cong \angle BCD$

Set the two equalities equal: $\angle ACD + \angle ACE = \angle ACD + \angle BCD$.
Subtract $\angle ACD$: $\angle ACE = \angle BCD$.

Step3: Use $\angle ACD \cong \angle BCE$

Since $\angle ACD + \angle ACE = 180^\circ$, substitute $\angle ACD$ with $\angle BCE$: $\angle BCE + \angle ACE = 180^\circ$, so they are supplementary.

Step4: Use $\angle ACD \cong \angle BCE$

Since $\angle ACD + \angle BCD = 180^\circ$, substitute $\angle ACD$ with $\angle BCE$: $\angle BCE + \angle BCD = 180^\circ$, so they are supplementary.

Step5: Verify other statements

$\angle ACE$ and $\angle BCD$ are congruent, not supplementary, so that statement is false. $\angle ACE$ and $\angle BCE$ are supplementary, not congruent, so that statement is false.

Answer:

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