Question

determine the
easons\ and \statements\ for each statement made (you may use the same reason multiple times) in order to complete the following proof. given: $overline{bd}$ bisects $angle abc$ and $overline{bd}$ bisects $angle adc$. prove: $overline{ad}congoverline{cd}$. note: quadrilateral properties are not permitted in this proof. step statement reason 1 $overline{bd}$ bisects $angle abc$ $overline{bd}$ bisects $angle adc$ 2 $angle abdcongangle cbd$ 3 $angle adbcongangle cdb$ 4 $overline{bd}congoverline{bd}$ 5 $\triangle abdcong\triangle cbd$ 6 $overline{ad}congoverline{cd}$ answer attempt 1 out of 3 step 1 reason = step 2 reason = step 3 reason =

Explanation:

Step1: Given information

Given

Step2: Definition of angle - bisector

If a ray bisects an angle, it divides the angle into two congruent angles. Since $\overline{BD}$ bisects $\angle ABC$, $\angle ABD\cong\angle CBD$.

Step3: Definition of angle - bisector

If a ray bisects an angle, it divides the angle into two congruent angles. Since $\overline{BD}$ bisects $\angle ADC$, $\angle ADB\cong\angle CDB$.

Step4: Reflexive property of congruence

Any segment is congruent to itself, so $\overline{BD}\cong\overline{BD}$.

Step5: Angle - Side - Angle (ASA) congruence criterion

In $\triangle ABD$ and $\triangle CBD$, we have $\angle ABD\cong\angle CBD$, $\overline{BD}\cong\overline{BD}$, and $\angle ADB\cong\angle CDB$. So, $\triangle ABD\cong\triangle CBD$ by ASA.

Step6: Corresponding parts of congruent triangles are congruent (CPCTC)

Since $\triangle ABD\cong\triangle CBD$, the corresponding sides $\overline{AD}\cong\overline{CD}$.

Answer:

Step 1 Reason = Given
Step 2 Reason = Definition of angle - bisector
Step 3 Reason = Definition of angle - bisector
Step 4 Reason = Reflexive property of congruence
Step 5 Reason = ASA congruence criterion
Step 6 Reason = CPCTC