Question

a straightedge and compass were used to create the construction below. arc ef was drawn from point b, and arcs with equal radii were drawn from e and f. which 3 statements are true? statement 1: $\frac{1}{2}(mangle abc)=mangle abd$ statement 2: $2(mangle abc)=mangle cbd$ statement 3: $2(mangle dbc)=mangle abc$ statement 4: $mangle abd = mangle dbc$

Explanation:

Step1: Recall angle - bisector construction

The construction shown is of an angle - bisector. When we construct an angle - bisector of an angle $\angle ABC$ using a straightedge and compass as described, the ray $BD$ divides $\angle ABC$ into two equal angles.

Step2: Analyze Statement 1

Since $BD$ is the angle - bisector of $\angle ABC$, by the definition of an angle - bisector, $\angle ABD=\frac{1}{2}\angle ABC$. So, $\frac{1}{2}(m\angle ABC)=m\angle ABD$ is True.

Step3: Analyze Statement 2

Since $BD$ bisects $\angle ABC$, $m\angle CBD = m\angle ABD=\frac{1}{2}m\angle ABC$. So, $2(m\angle ABC)
eq m\angle CBD$. This statement is False.

Step4: Analyze Statement 3

Since $BD$ bisects $\angle ABC$, $m\angle DBC=\frac{1}{2}m\angle ABC$, so $2(m\angle DBC)=m\angle ABC$ is True.

Step5: Analyze Statement 4

Since $BD$ bisects $\angle ABC$, $m\angle ABD = m\angle DBC$ is True.

Answer:

Statement 1: $\frac{1}{2}(m\angle ABC)=m\angle ABD$
Statement 3: $2(m\angle DBC)=m\angle ABC$
Statement 4: $m\angle ABD = m\angle DBC$