Question

1. complete the proof below given: \\( \overleftrightarrow { r c } \parallel \overleftrightarrow { t n }, \overline { r x } \cong \overline { n x } \\) \\( \angle r h x \cong \angle n k x \\) prove: \\( \delta r h x \cong \delta n k x \\) \\( \overleftrightarrow { r c } \parallel \overleftrightarrow { t n } \\) given \\( \angle r h x \cong \angle n k x \\) vertical \\( \angle s \\) are \\( \cong \\) alt int \\( \angle s \\) are \\( \cong \\) when lines are \\( \parallel \\) alt ext \\( \angle s \\) are \\( \cong \\) when lines are \\( \parallel \\) \\( \overline { r x } \cong \overline { n x } \\) given \\( \angle r x h \cong \angle n x k \\) vertical \\( \angle s \\) are \\( \cong \\) alt int \\( \angle s \\) are \\( \cong \\) when lines are \\( \parallel \\) alt ext \\( \angle s \\) are \\( \cong \\) when lines are \\( \parallel \\) \\( \delta r h x \cong \delta n k x \\) sss, sas, asa, aas, hl

Explanation:

Step1: Identify the reason for \(\angle RHX\cong\angle NKX\)

Since \(\overleftrightarrow{RC}\parallel\overleftrightarrow{TN}\), by the Alternate Interior Angles Theorem (Alt Int \(\angle s\) are \(\cong\) when lines are \(\parallel\)), \(\angle RHX\cong\angle NKX\)

Step2: Identify the reason for \(\angle RXH\cong\angle NXK\)

Vertical angles are congruent. So, \(\angle RXH\cong\angle NXK\) because they are vertical angles (Vertical \(\angle s\) are \(\cong\))

Step3: Use the ASA (Angle - Side - Angle) congruence criterion

We are given \(\overline{RX}\cong\overline{NX}\), \(\angle RHX\cong\angle NKX\) (from Step 1) and \(\angle RXH\cong\angle NXK\) (from Step 2). In \(\triangle RHX\) and \(\triangle NKX\), we have two angles and the included side congruent. So, by the ASA (Angle - Side - Angle) congruence criterion, \(\triangle RHX\cong\triangle NKX\)

Answer:

\(\triangle RHX\cong\triangle NKX\) by the ASA (Angle - Side - Angle) congruence criterion.