Question

given $overline{ac}$ is the angle bisector of $angle bad$ and $angle bcd$, complete the flowchart proof below.
$overline{ac}$ bisects $angle bad$ given
$overline{ac}$ bisects $angle bcd$ given
$angle baccongangle dac$
$angle bcacongangle dca$
$overline{ab}congoverline{ad}$
$\triangle abccong\triangle adc$

Explanation:

Step1: Definition of angle - bisector

If a ray bisects an angle, it divides the angle into two congruent angles. Since $\overline{AC}$ bisects $\angle BAD$, then $\angle BAC\cong\angle DAC$.

Step2: Definition of angle - bisector

Since $\overline{AC}$ bisects $\angle BCD$, then $\angle BCA\cong\angle DCA$.

Step3: Reflexive property

$\overline{AC}\cong\overline{AC}$ (any segment is congruent to itself).

Step4: Angle - Side - Angle (ASA) congruence criterion

In $\triangle ABC$ and $\triangle ADC$, we have $\angle BAC\cong\angle DAC$, $\overline{AC}\cong\overline{AC}$, and $\angle BCA\cong\angle DCA$. So, $\triangle ABC\cong\triangle ADC$ by ASA.

Step5: Corresponding parts of congruent triangles are congruent (CPCTC)

Since $\triangle ABC\cong\triangle ADC$, then $\overline{AB}\cong\overline{AD}$.

Answer:

• Reason for $\angle BAC\cong\angle DAC$: Definition of angle - bisector
• Reason for $\angle BCA\cong\angle DCA$: Definition of angle - bisector
• Reason for $\triangle ABC\cong\triangle ADC$: ASA (Angle - Side - Angle)
• Reason for $\overline{AB}\cong\overline{AD}$: CPCTC (Corresponding parts of congruent triangles are congruent)