Question

lesson 6 homework date____ period__ a diagram is shown where ( overline{tw}paralleloverline{lk} ), ( overline{xw}congoverline{ld} ), and ( angle mwxcongangle nld ). use the given information to answer the questions about the incomplete proof that is shown. 1. what is the statement for line 3 of the proof? 2. what is the reason for line 3 of the proof? 3. what is the statement for line 4 of the proof? 4. complete the congruence statement for the statement of line 5 of the proof. ( \triangle xdlcong\triangle )__ 5. what is the reason for line 5 of the proof?

Explanation:

Step1: Find the statement for line 3

Since \( \overline{TW}\parallel\overline{LK}\), by the alternate - interior angles theorem, \( \angle TXD=\angle LDK\) (or \( \angle WXD=\angle LDD\)). But considering the triangles \( \triangle XDL\) and the other triangle, because of the parallel lines \( \overline{TW}\parallel\overline{LK}\), we have \( \angle WXD=\angle LDD\).

Step2: State the reason for line 3

The reason for line 3 is the Alternate - Interior Angles Theorem. When two parallel lines (\( \overline{TW}\) and \( \overline{LK}\)) are cut by a transversal (\( \overline{MLN}\)), the alternate - interior angles are congruent.

Step3: Find the statement for line 4

By the reflexive property of congruence, \( \overline{XD}\cong\overline{XD}\)

Step4: Complete the congruence statement for line 5

We know that \( \overline{XW}\cong\overline{LD}\) (given), \( \angle WXD=\angle LDD\) (from step 1), and \( \overline{XD}\cong\overline{XD}\) (reflexive property). So, \( \triangle XDL\cong\triangle WDX\)

Step5: State the reason for line 5

The reason for line 5 is the Angle - Side - Angle (ASA) congruence criterion. We have two angles (\( \angle WXD=\angle LDD\) and \( \angle M WX=\angle NLD\) (given) which can be related to the angles of the triangles) and the included side (\( \overline{XW}\cong\overline{LD}\) and \( \overline{XD}\cong\overline{XD}\)) for the congruence of \( \triangle XDL\) and \( \triangle WDX\)

Answer:

1. \( \angle WXD\cong\angle LDD\)
2. Alternate - Interior Angles Theorem
3. \( \overline{XD}\cong\overline{XD}\)
4. \( \triangle WDX\)
5. Angle - Side - Angle (ASA)