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 \\( \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 \\) 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 \\) sss, sas, asa, aas, hl

Explanation:

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

Since \(\overrightarrow{RC}\parallel\overrightarrow{TN}\), 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 \(\angle RXH\cong\angle NXK\)

\(\angle RXH\) and \(\angle NXK\) are vertical angles. By the Vertical Angles Theorem (Vertical \(\angle s\) are \(\cong\)), \(\angle RXH\cong\angle NXK\).

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).
By the ASA (Angle - Side - Angle) congruence criterion (\(ASA\): If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent), \(\triangle RHX\cong\triangle NKX\).

Answer:

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