Question

point e is added to the given figure so that point e lies on segment bd exactly halfway between points b and d. which statement correctly compares the angle measures? a. ( 2 cdot mangle abe = mangle dbc ) b. ( mangle ebc = mangle abd ) c. ( mangle abd = mangle abe ) d. ( mangle ebc = \frac{1}{2} cdot mangle dbc )

Explanation:

To determine the correct statement, we analyze the angle bisector property. If \( E \) is on the angle bisector of \( \angle DBC \), then it divides \( \angle DBC \) into two equal angles. So \( m\angle EBC=\frac{1}{2}m\angle DBC \) (or equivalently \( 2\cdot m\angle EBC = m\angle DBC \), but looking at the options, option D states \( m\angle EBC=\frac{1}{2}\cdot m\angle DBC \), which is consistent with the angle bisector definition. Let's check other options:

• Option A: \( 2\cdot m\angle ABE = m\angle DBC \) – There's no info to support this relation.
• Option B: \( m\angle EBC = m\angle ABD \) – No relation given between these angles.
• Option C: \( m\angle ABD = m\angle ABE \) – No info to support this.

Answer:

D. \( m\angle EBC=\frac{1}{2}\cdot m\angle DBC \)