ray ce is the angle bisector of ∠acd. which statement about the figure must be true?\n○ (mangle…

ray ce is the angle bisector of ∠acd. which statement about the figure must be true?\n○ (mangle ecd=\frac{1}{2}mangle ecb)\n○ (mangle ace = \frac{1}{2}mangle acd)\n○ (angle acecongangle dcb)\n○ (angle ecdcongangle acd)

ray ce is the angle bisector of ∠acd. which statement about the figure must be true?\n○ (mangle ecd=\frac{1}{2}mangle ecb)\n○ (mangle ace = \frac{1}{2}mangle acd)\n○ (angle acecongangle dcb)\n○ (angle ecdcongangle acd)

Answer

Explanation:

Step1: Recall angle - bisector definition

An angle bisector divides an angle into two equal - measure angles. Since ray $CE$ is the angle bisector of $\angle ACD$, we know that $\angle ACE$ and $\angle ECD$ are equal in measure and each is half of $\angle ACD$.

Step2: Analyze each option

  • Option 1: $m\angle ECD=\frac{1}{2}m\angle ECB$ is not correct because $CE$ is the bisector of $\angle ACD$, not related to $\angle ECB$ in this way.
  • Option 2: Since $CE$ bisects $\angle ACD$, by the definition of an angle bisector, $m\angle ACE = \frac{1}{2}m\angle ACD$. This is correct.
  • Option 3: $\angle ACE\cong\angle DCB$ is not correct. There is no information from the angle - bisector property to suggest this equality.
  • Option 4: $\angle ECD\cong\angle ACD$ is not correct. $\angle ECD$ is half of $\angle ACD$.

Answer:

$m\angle ACE=\frac{1}{2}m\angle ACD$ (the second option)