Question

complete the proof that in rhombus abcd the diagonal bd bisects angle abc. choose your answers from the drop-down menus.
a. a rhombus has ▼, so segments ab, cb, ad, and cd are congruent.
b. segment ▼ is congruent to segment ▼ because segments are congruent to themselves.
c. triangle dab is congruent to triangle dcb by ▼ triangle congruence theorem.
d. angle abd is congruent to angle cbd because ▼.
e. therefore, diagonal bd bisects angle abc because ▼.

Explanation:

Step1: Recall rhombus properties

A rhombus has all sides congruent, so segments \(AB\), \(CB\), \(AD\), and \(CD\) are congruent. So for part a, the answer is "all sides congruent".

Step2: Identify congruent segments (Reflexive property)

Segment \(BD\) is congruent to segment \(BD\) (reflexive property of congruence, a segment is congruent to itself). So for part b, the first blank is \(BD\) and the second blank is \(BD\).

Step3: Triangle congruence theorem

Triangle \(DAB\) and triangle \(DCB\): \(AB = CB\), \(AD = CD\), \(BD = BD\), so by SSS (Side - Side - Side) Triangle Congruence Theorem, triangle \(DAB\) is congruent to triangle \(DCB\). So for part c, the answer is "SSS".

Step4: Corresponding parts of congruent triangles

Since \(\triangle DAB\cong\triangle DCB\) (by SSS), then \(\angle ABD\cong\angle CBD\) because corresponding parts of congruent triangles are congruent (CPCTC). So for part d, the answer is "CPCTC".

Step5: Definition of angle bisector

If \(\angle ABD\cong\angle CBD\), then diagonal \(BD\) bisects \(\angle ABC\) because a bisector of an angle divides the angle into two congruent angles. So for part e, the reasoning is that \(\angle ABD\) and \(\angle CBD\) are congruent (from part d), so \(BD\) bisects \(\angle ABC\).

Answer:

s:
a. all sides congruent
b. \(BD\); \(BD\)
c. SSS
d. CPCTC
e. \(\angle ABD\cong\angle CBD\) (from part d), so \(BD\) bisects \(\angle ABC\)