Question

proving triangles congruent
digital textbook
click each link below to learn the concepts. each component includes a video and you can select your desired language.
student geometry -> triangle congruence -> proving triangles congruent -> 5.6.2 guided instruction
student geometry -> triangle congruence -> proving triangles congruent -> 5.6.3 guided practice
then, answer \check your understanding\ assignment below. you have three attempts.
given: \\( \overline { t r } \cong \overline { t d } \\), \\( \overline { t k } \\) bisects \\( \angle r t d \\)
prove: \\( \triangle t r k \cong \triangle t d k \\)
complete the paragraph proof.
it is given that \\( \overline { t k } \\) bisects \\( \angle r t d \\), so \\( \square \\) by definition of angle bisector. therefore,
\\( \square \\) by definition of congruence. also, it is given that \\( \overline { t r } \cong \overline { t d } \\). \\( \overline { t k } \cong \overline { t k } \\) by the
\\( \square \\). therefore, \\( \triangle t r k \cong \triangle t d k \\) by
\\( \square \\)

Explanation:

Step1: Use angle - bisector definition

By the definition of an angle bisector, if \(\overline{TK}\) bisects \(\angle RTD\), then \(\angle RTK\cong\angle DTK\).

Step2: Use SAS (Side - Angle - Side) criterion

We know that \(\overline{TR}\cong\overline{TD}\) (given), \(\angle RTK\cong\angle DTK\) (from the angle - bisector property), and \(\overline{TK}\cong\overline{TK}\) (reflexive property of congruence).

Answer:

It is given that \(\overline{TK}\) bisects \(\angle RTD\), so \(\angle RTK\cong\angle DTK\) by definition of angle bisector. Therefore, in \(\triangle TRK\) and \(\triangle TDK\), we have two sides and the included angle congruent. Also, it is given that \(\overline{TR}\cong\overline{TD}\). \(\overline{TK}\cong\overline{TK}\) by the reflexive property of congruence. Therefore, \(\triangle TRK\cong\triangle TDK\) by the \(SAS\) (Side - Angle - Side) congruence criterion.