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Point B is the midpoint of Line segment A C .
Horizontal line with points A, B, and C. A vertical line extends from point B through point D to create two right angles.
Which statements about the figure must be true? Select three options.
AngleDBC is bisected by ray BD.
AngleABC is bisected by ray BD.
BC = One-halfAC
Line segment D B is-congruent-to Line segment B C
2mAngleDBC = mAngleABC
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Correct choices:
- Angle ABC is bisected by ray BD.
- BC = one-half AC.
- 2·m∠DBC = m∠ABC.
Reasoning: B is the midpoint of AC so AB = BC, hence BC = (1/2)AC. BD is perpendicular to AC, so ∠ABD = ∠DBC = 90°. Thus BD bisects the straight angle ∠ABC (180°) into two 90° angles, and 2·m∠DBC = 2·90° = 180° = m∠ABC. The other statements are false.
- Angle ABC is bisected by ray BD.
- BC = one-half AC.
- 2·m∠DBC = m∠ABC.
Reasoning: B is the midpoint of AC so AB = BC, hence BC = (1/2)AC. BD is perpendicular to AC, so ∠ABD = ∠DBC = 90°. Thus BD bisects the straight angle ∠ABC (180°) into two 90° angles, and 2·m∠DBC = 2·90° = 180° = m∠ABC. The other statements are false.
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