Question

use the following picture to classify the following statements as true or false.
drag & drop the answer
△abc≅△cda by sas
△abc≅△cda by hl
ad≅cb
∠acb≅∠cda
∠bac≅∠cad
true
false

Explanation:

Step1: Check $\angle BAC\cong\angle CAD$

In the given figure, $\angle BAC$ and $\angle CAD$ are not congruent. $\angle BAC$ is part of $\triangle ABC$ and $\angle CAD$ is part of $\triangle ACD$. There is no indication of equality of these angles from the given information (like no markings for equal angles). So, $\angle BAC\cong\angle CAD$ is False.

Step2: Check $\angle ACB\cong\angle CDA$

$\angle ACB$ is an angle in $\triangle ABC$ and $\angle CDA$ is a right - angle ($\angle D = 90^{\circ}$). $\angle ACB$ is not a right - angle (since $\angle B = 90^{\circ}$ in $\triangle ABC$ and the sum of angles in a triangle is $180^{\circ}$). So, $\angle ACB\cong\angle CDA$ is False.

Step3: Check $\overline{AD}\cong\overline{CB}$

In the figure, $\overline{AD}$ and $\overline{CB}$ are the non - hypotenuse sides of the right - triangles $\triangle ABC$ and $\triangle CDA$. There is no information (like no markings for equal length) to suggest that $\overline{AD}\cong\overline{CB}$. So, $\overline{AD}\cong\overline{CB}$ is False.

Step4: Check $\triangle ABC\cong\triangle CDA$ by $HL$

For the Hypotenuse - Leg ($HL$) congruence criterion in right - triangles, we need the hypotenuse and one leg of one right - triangle to be congruent to the hypotenuse and one leg of another right - triangle.
Let's assume $\triangle ABC$ and $\triangle CDA$ are right - triangles ($\angle B=\angle D = 90^{\circ}$). The hypotenuse of $\triangle ABC$ is $\overline{AC}$ and the hypotenuse of $\triangle CDA$ is also $\overline{AC}$ (common side). But we don't know if $\overline{AB}\cong\overline{CD}$ or $\overline{BC}\cong\overline{DA}$. So, $\triangle ABC\cong\triangle CDA$ by $HL$ is False.

Step5: Check $\triangle ABC\cong\triangle CDA$ by $SAS$

For the Side - Angle - Side ($SAS$) congruence criterion, we need two sides and the included angle of one triangle to be congruent to two sides and the included angle of another triangle.
We know $\angle B=\angle D = 90^{\circ}$. But we don't know if $\overline{AB}\cong\overline{CD}$ and $\overline{BC}\cong\overline{DA}$ (no markings for equal length). So, $\triangle ABC\cong\triangle CDA$ by $SAS$ is False.

Answer:

$\angle BAC\cong\angle CAD$: False, $\angle ACB\cong\angle CDA$: False, $\overline{AD}\cong\overline{CB}$: False, $\triangle ABC\cong\triangle CDA$ by $HL$: False, $\triangle ABC\cong\triangle CDA$ by $SAS$: False