Question

proofs

1. given: $overline{ac}$ bisects $angle dab$

$overline{ac}$ bisects $angle bcd$
prove: $\triangle abc cong \triangle adc$

possible statements:
(a) segment $ac$ bisects $angle bcd$, (b) $angle b cong angle d$
(c) $angle bac cong angle dac$, (d)$angle bca cong angle dca$

possible reasons:
(e) given, (f) sss, (g) sas, (h) asa, (i) reflexive property, (j) cpctc, (k) prove

reason #1: ____

statement #2: ____

statement #3: ____

statement #4: ____

reason #5: ____

reason #6 ____

statements | reasons
① $overline{ac}$ bisects $angle dab$ | ① ____
② ____ | ② def of $angle$ bisector
③ ____ | ③ given
④ ____ | ④ def of $angle$ bisector
⑤ $overline{ac} cong overline{ac}$ | ⑤ ____
⑥ $\triangle abc cong \triangle adc$ | ⑥ ____

Explanation:

Step1: Reason #1

The first statement is "AC bisects ∠DAB", and the reason for this is that it is given in the problem. So we use the "Given" reason, which is option (E).

Step2: Statement #2

By the definition of an angle bisector, if AC bisects ∠DAB, then ∠BAC ≅ ∠DAC. So the statement here is ∠BAC ≅ ∠DAC, which is option (C).

Step3: Statement #3

The third reason is "Given", and the given information is that AC bisects ∠BCD. So the statement is "AC bisects ∠BCD", which is option (A).

Step4: Statement #4

By the definition of an angle bisector, if AC bisects ∠BCD, then ∠BCA ≅ ∠DCA. So the statement here is ∠BCA ≅ ∠DCA, which is option (D).

Step5: Reason #5

The statement is "AC ≅ AC", and the reason for a segment being congruent to itself is the Reflexive Property, which is option (I).

Step6: Reason #6

We have two angles and the included side congruent (∠BAC ≅ ∠DAC, AC ≅ AC, ∠BCA ≅ ∠DCA), so the congruence criterion is ASA (Angle - Side - Angle), which is option (H).

Answer:

Reason #1: E
Statement #2: C
Statement #3: A
Statement #4: D
Reason #5: I
Reason #6: H