A horizontal line has points D, B, C. A line extends vertically from point B to point A.
Given that Ray B A bisects ∠DBC, which statement must be true?
m∠ABD = m∠ABC
AB ≅ BC
B is the midpoint of DC.
m∠DBC = 90°
The correct statement is: m∠ABD = m∠ABC.
The statement that must be true is: B is the midpoint of DC.
Here's how we can determine this:
Since Ray BA bisects angle DBC, it means that angle ABD is equal to angle CBD. This is a property of angle bisectors.
From the given information, we know that line DB is horizontal, and a vertical line extends from point B to point A. This vertical line intersects line DB at point B, forming a right angle.
So, angle DBC is a right angle (90°). This is because there is a horizontal line and a vertical line intersecting at point B, forming a right angle.
Since angles ABD and CBD are equal and angle DBC is a right angle, it implies that angle ABD is also a right angle. This means that line BA is perpendicular to line DB.
Now, if B is the midpoint of DC, it means that line BA is also perpendicular to line BC. This is because the midpoint of a segment divides it into two equal parts.
Therefore, we can conclude that B is the midpoint of DC.
To summarize:
- m∠ABD = m∠ABC is not necessarily true, since we don't have enough information about the angles.
- AB ≅ BC is not necessarily true, since we don't have enough information about the lengths of the sides.
- B is the midpoint of DC is the correct statement, as explained above.
- m∠DBC = 90° is true, based on the information given.