Question

write a two - column proof. given (overrightarrow{bd}) bisects (angle abc). prove (mangle dba = \frac{1}{2}mangle abc)

Explanation:

Step1: Recall Angle Bisector Definition

An angle bisector divides an angle into two congruent angles, each equal to half the measure of the original angle. Given \(\overrightarrow{BD}\) bisects \(\angle ABC\), by the definition of an angle bisector, \(\angle DBA \cong \angle DBC\) and \(m\angle DBA = m\angle DBC\).

Step2: Use Angle Addition Postulate

The Angle Addition Postulate states that \(m\angle ABC = m\angle DBA + m\angle DBC\). Since \(m\angle DBA = m\angle DBC\) (from the angle bisector definition), we can substitute \(m\angle DBC\) with \(m\angle DBA\) in the equation: \(m\angle ABC = m\angle DBA + m\angle DBA\).

Step3: Simplify the Equation

Simplifying \(m\angle ABC = m\angle DBA + m\angle DBA\) gives \(m\angle ABC = 2m\angle DBA\).

Step4: Solve for \(m\angle DBA\)

Divide both sides of the equation \(m\angle ABC = 2m\angle DBA\) by 2. We get \(m\angle DBA=\frac{1}{2}m\angle ABC\).

Two - Column Proof:

Statements	Reasons
2. \(m\angle DBA = m\angle DBC\)	Definition of Angle Bisector
3. \(m\angle ABC=m\angle DBA + m\angle DBC\)	Angle Addition Postulate
4. \(m\angle ABC=m\angle DBA + m\angle DBA\)	Substitution (from Step 2)
5. \(m\angle ABC = 2m\angle DBA\)	Simplification (Combining like terms)
6. \(m\angle DBA=\frac{1}{2}m\angle ABC\)	Division Property of Equality

Answer:

The two - column proof as shown above proves that \(m\angle DBA=\frac{1}{2}m\angle ABC\) given that \(\overrightarrow{BD}\) bisects \(\angle ABC\).