Question

in triangle abc, \\(\overline{cd}\\) is an altitude and \\(\overline{ad}\cong\overline{bd}\\). move options into the table to complete the proof that triangle abc is an isosceles triangle.

statement	reason
\\(\angle adc\cong\angle bdc\\)	right angles are congruent.
\\(\triangle adc\cong\triangle bdc\\)	side - angle - side congruence
\\(\triangle abc\\) is an isosceles triangle.

Explanation:

Step1: Identify reflexive side

Since $\overline{CD}$ is common to both $\triangle ADC$ and $\triangle BDC$, $\overline{CD}\cong\overline{CD}$ by the reflexive property of congruence.

Step2: Use CPCTC

After proving $\triangle ADC\cong\triangle BDC$ by SAS (Side - Angle - Side) congruence criterion, the corresponding sides $\overline{AC}$ and $\overline{BC}$ are congruent because corresponding parts of congruent triangles are congruent.

Step3: Apply isosceles triangle definition

A triangle is isosceles if it has two congruent sides. Since $\overline{AC}\cong\overline{BC}$, $\triangle ABC$ is an isosceles triangle.

Answer:

Statement	Reason
$\angle ADC\cong\angle BDC$	Right - angles are congruent
$\overline{CD}\cong\overline{CD}$	Reflexive property of congruence
$\triangle ADC\cong\triangle BDC$	Side - Angle - Side congruence
$\overline{AC}\cong\overline{BC}$	Corresponding parts of congruent triangles are congruent (CPCTC)
$\triangle ABC$ is an isosceles triangle	Definition of isosceles triangle (a triangle with two congruent sides)