Question

select all the congruence statements that are true.

$squareoverline{hx}congoverline{hz}$
$squareoverline{wx}congoverline{zh}$
$squareangle xwhcongangle zhk$
$squareangle hxwcongangle hzk$
$squareoverline{hw}congoverline{wr}$

Explanation:

Step1: Recall the properties of congruent triangles

If \(\triangle XHW\cong\triangle ZHK\), then corresponding parts (sides and angles) are congruent.
For angles: \(\angle HXW\) corresponds to \(\angle HZK\), \(\angle XWH\) corresponds to \(\angle ZHK\), \(\angle HWX\) corresponds to \(\angle HKZ\).
For sides: \(\overline{HX}\) corresponds to \(\overline{HZ}\), \(\overline{HW}\) corresponds to \(\overline{HK}\), \(\overline{XW}\) corresponds to \(\overline{ZK}\).

Step2: Check each option

• Option \(\angle HXW\cong\angle HZK\):

Since \(\triangle XHW\cong\triangle ZHK\), by the definition of congruent triangles (corresponding angles of congruent triangles are congruent), \(\angle HXW\) and \(\angle HZK\) are corresponding angles. So \(\angle HXW\cong\angle HZK\) is True.

• Option \(\angle XWH\cong\angle ZHK\):

Since \(\triangle XHW\cong\triangle ZHK\), \(\angle XWH\) and \(\angle ZHK\) are corresponding angles. So \(\angle XWH\cong\angle ZHK\) is True.

• Option \(\angle HWX\cong\angle HKZ\):

Since \(\triangle XHW\cong\triangle ZHK\), \(\angle HWX\) and \(\angle HKZ\) are corresponding angles. So \(\angle HWX\cong\angle HKZ\) is True.

• Option \(\overline{HX}\cong\overline{HZ}\):

Since \(\triangle XHW\cong\triangle ZHK\), \(\overline{HX}\) and \(\overline{HZ}\) are corresponding sides. So \(\overline{HX}\cong\overline{HZ}\) is True.

• Option \(\overline{WX}\cong\overline{ZH}\):

Since \(\triangle XHW\cong\triangle ZHK\), \(\overline{WX}\) corresponds to \(\overline{ZK}\), not \(\overline{ZH}\). So \(\overline{WX}\cong\overline{ZH}\) is False.

• Option \(\overline{HW}\cong\overline{WR}\):

There is no information from \(\triangle XHW\cong\triangle ZHK\) that would imply \(\overline{HW}\cong\overline{WR}\). In fact, \(\overline{HW}\) corresponds to \(\overline{HK}\) in the congruent triangles. So \(\overline{HW}\cong\overline{WR}\) is False.

Answer:

\(\angle HXW\cong\angle HZK\), \(\angle XWH\cong\angle ZHK\), \(\angle HWX\cong\angle HKZ\), \(\overline{HX}\cong\overline{HZ}\)