Question

5 multiple choice 14 points 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) m∠abd = m∠dbc 2) (1/2)(m∠abc) = m∠abd 3) 2(m∠dbc) = m∠abc 4) 2(m∠abc) = m∠cbd

Explanation:

Step1: Analyze the construction

The construction with straightedge and compass here is an angle bisector construction. So, ray \( BD \) bisects \( \angle ABC \), meaning \( \angle ABD=\angle DBC \), and each is half of \( \angle ABC \).

Step2: Evaluate each statement

1. \( m\angle ABD = m\angle DBC \): True, since \( BD \) bisects \( \angle ABC \).
2. \( \frac{1}{2}(m\angle ABC)=m\angle ABD \): True, because \( BD \) bisects \( \angle ABC \), so \( \angle ABD=\frac{1}{2}\angle ABC \).
3. \( 2(m\angle DBC)=m\angle ABC \): True, since \( \angle DBC=\frac{1}{2}\angle ABC \), multiplying both sides by 2 gives \( 2\angle DBC = \angle ABC \).
4. \( 2(m\angle ABC)=m\angle CBD \): False, because \( \angle CBD \) is half of \( \angle ABC \), not twice. So this statement is false.

Answer:

4