Question

n exercises 1 and 2, decide whether the congruence statement is true. explain your reasoning.

1. $\triangle wxycong\triangle yzw$ 2. $\triangle abccong\triangle cde$
2. write a proof.

given $overline{ab}congoverline{cd},angle bca$ and $angle dac$ are right angles.
prove $\triangle abccong\triangle cda$

Explanation:

Step1: Analyze the first congruence statement

For \(\triangle WXY\) and \(\triangle YZW\), by SSS (Side - Side - Side) congruence criterion.
In \(\triangle WXY\) and \(\triangle YZW\), \(WX = YZ\), \(XY=ZW\), \(WY = WY\) (common side).

Step2: Analyze the second congruence statement

For \(\triangle ABC\) and \(\triangle CDE\), \(\angle ACB\) is a right angle and there is no information about \(\angle DCE\) being a right angle. Also, the side lengths do not match in a way that would satisfy a congruence criterion (like SSS, SAS, ASA, AAS).

Step3: Prove \(\triangle ABC\cong\triangle CDA\)

• Given \(\overline{AB}\cong\overline{CD}\) (side).
• \(\angle BCA = \angle DAC = 90^{\circ}\) (angle).
• \(\overline{AC}\cong\overline{AC}\) (common side).

By SAS (Side - Angle - Side) congruence criterion, \(\triangle ABC\cong\triangle CDA\).

Answer:

1. The congruence statement \(\triangle WXY\cong\triangle YZW\) is true by SSS congruence criterion.
2. The congruence statement \(\triangle ABC\cong\triangle CDE\) is false as there is no valid congruence criterion (like SSS, SAS, ASA, AAS) to support it.
3. Since \(\overline{AB}\cong\overline{CD}\), \(\angle BCA=\angle DAC = 90^{\circ}\), and \(\overline{AC}\cong\overline{AC}\), by SAS congruence criterion \(\triangle ABC\cong\triangle CDA\).