Question

question 1: 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 statement is false? (1) $\frac{1}{2}(mangle abc)=mangle abd$ (2) $mangle abd = mangle dbc$ (3) $2(mangle abc)=mangle cbd$ (4) $2(mangle dbc)=mangle abc$

Explanation:

Step1: Recall angle - bisector construction

The construction with a straightedge and compass shown is the construction of an angle - bisector. Here, ray $BD$ is the angle - bisector of $\angle ABC$.

Step2: Analyze angle - bisector properties

By the definition of an angle - bisector, if $BD$ bisects $\angle ABC$, then $\angle ABD=\angle DBC$ and $\angle ABC = 2\angle ABD=2\angle DBC$.

Step3: Evaluate each statement

• Statement (1): $\frac{1}{2}(m\angle ABC)=m\angle ABD$. Since $\angle ABC = 2\angle ABD$, this statement is True.
• Statement (2): $m\angle ABD = m\angle DBC$. This is True by the definition of an angle - bisector.
• Statement (3): $2(m\angle ABC)=m\angle CBD$. Since $\angle ABC = 2\angle CBD$, this statement is False.
• Statement (4): $2(m\angle DBC)=m\angle ABC$. This is True by the definition of an angle - bisector.

Answer:

(3)