Question

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

Explanation:

Step1: Define angle - bisector property

An angle - bisector divides an angle into two congruent angles. Since $\overline{AC}$ bisects $\angle BAD$, we have $\angle BAC\cong\angle DAC$ (by the definition of an angle - bisector).

Step2: Use angle - bisector property again

Since $\overline{AC}$ bisects $\angle BCD$, we have $\angle BCA\cong\angle DCA$ (by the definition of an angle - bisector).

Step3: Identify common side

$\overline{AC}\cong\overline{AC}$ (by the reflexive property of congruence).

Step4: Apply 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 the Angle - Side - Angle (ASA) congruence criterion.

Step5: Use CPCTC

Corresponding parts of congruent triangles are congruent. Since $\triangle ABC\cong\triangle ADC$, then $\overline{AB}\cong\overline{AD}$ (by CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

Answer:

Reasons:

• $\angle BAC\cong\angle DAC$: Definition of angle - bisector
• $\angle BCA\cong\angle DCA$: Definition of angle - bisector
• $\triangle ABC\cong\triangle ADC$: ASA (Angle - Side - Angle)
• $\overline{AB}\cong\overline{AD}$: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)