Question

what is the reason for statement 5 in this proof?
a. asa
b. sss
c. aas
d. sas

statement | reason
--- | ---

1. let $\triangle abc$ be an isosceles triangle with $ab = bc$. | given
2. create point $d$ on side $\overline{ac}$ so $\overline{bd}$ bisects $\angle abc$. | constructing an angle bisector
3. $m\angle abd = m\angle cbd$ | definition of angle bisector
4. $bd = bd$ | reflexive property of equality
5. $\triangle abd \cong \triangle cbd$ |
6. $m\angle bac = m\angle bca$ | corresponding angles of congruent triangles have equal measures.

Explanation:

To determine the reason for \(\triangle ABD \cong \triangle CBD\) (statement 5), we analyze the given information:

• \(AB = BC\) (given, isosceles triangle).
• \(BD\) is the angle bisector, so \(\angle ABD=\angle CBD\) (definition of angle bisector).
• \(BD = BD\) (reflexive property, common side).

This matches the SAS (Side - Angle - Side) congruence criterion: two sides (\(AB = BC\), \(BD = BD\)) and the included angle (\(\angle ABD=\angle CBD\)) of one triangle are congruent to the corresponding parts of the other triangle.

Answer:

D. SAS