Question

given $overline{ac}$ bisects $angle bad$, complete the flowchart proof below.
$overline{ac}$ bisects $angle bad$, reason: given
$angle baccongangle dac$, reason: select reason
$angle bcacongangle dca$, reason: select reason
$overline{bc}congoverline{dc}$, reason: select reason
$\triangle abccong\triangle adc$, reason: select reason

Explanation:

Step1: Definition of angle - bisector

By the definition of an angle - bisector, if a line bisects an angle, it divides the angle into two congruent angles. Since $\overline{AC}$ bisects $\angle BAD$, we have $\angle BAC\cong\angle DAC$.

Step2: Given congruent angles

The reason for $\angle BCA\cong\angle DCA$ is likely given in the problem context (if it is a given condition). If not, we may need more information about the figure. But assuming it is given.

Step3: Given congruent sides

The reason for $\overline{BC}\cong\overline{DC}$ is likely given in the problem context (if it is a given condition).

Step4: ASA congruence criterion

In $\triangle ABC$ and $\triangle ADC$, we have $\angle BAC\cong\angle DAC$ (from angle - bisector definition), $\overline{AC}=\overline{AC}$ (common side), and $\angle BCA\cong\angle DCA$ (given). By the Angle - Side - Angle (ASA) congruence criterion, $\triangle ABC\cong\triangle ADC$.

Answer:

• $\angle BAC\cong\angle DAC$ Reason: Definition of angle - bisector
• $\angle BCA\cong\angle DCA$ Reason: Given
• $\overline{BC}\cong\overline{DC}$ Reason: Given
• $\triangle ABC\cong\triangle ADC$ Reason: ASA congruence criterion