Question

triangles kzn, byh, lwf, and xrc are shown.
determine if the given information is sufficient to determine whether the triangles are congruent. if so, state the congruence theorem or postulate.

1. ( overline{by} cong overline{xr}, overline{yh} cong overline{cx} )
2. ( angle nzk cong angle wlf, overline{nk} cong overline{fw}, angle zkn cong angle fwl )
3. ( overline{hb} cong overline{rc}, overline{by} cong overline{xr}, overline{hy} cong overline{cx} )
4. ( angle wfl cong angle znk, overline{wf} cong overline{nk}, angle nzk cong angle wlf )
5. ( angle zkn cong angle fwl, overline{nz} cong overline{fl}, overline{kz} cong overline{lw} )

. ( angle ybh cong angle crx, angle bhy cong angle xcr, overline{hy} cong overline{cx} )

Explanation:

5.

Step1: Analyze the given sides

We are given \( \overline{BY}\cong\overline{XR}\) and \( \overline{YH}\cong\overline{CX}\). But we need at least three pairs of congruent parts (either SSS, SAS, ASA, AAS, or HL for right - angled triangles). Here, we have only two pairs of sides.

6.

Step1: Check the congruence criteria

We have \( \angle NZK\cong\angle WLF\), \( \overline{NK}\cong\overline{FW}\), \( \angle ZKN\cong\angle FWL\). By the ASA (Angle - Side - Angle) congruence postulate (two angles and the included side of one triangle are congruent to two angles and the included side of another triangle).

7.

Step1: Check the congruence criteria

We have \( \overline{HB}\cong\overline{RC}\), \( \overline{BY}\cong\overline{XR}\), \( \overline{HY}\cong\overline{CX}\). By the SSS (Side - Side - Side) congruence postulate (all three sides of one triangle are congruent to all three sides of another triangle).

8.

Step1: Check the congruence criteria

We have \( \angle WFL\cong\angle ZNK\), \( \overline{WF}\cong\overline{NK}\), \( \angle NZK\cong\angle WLF\). By the ASA (Angle - Side - Angle) congruence postulate (two angles and the included side of one triangle are congruent to two angles and the included side of another triangle).

9.

Step1: Check the congruence criteria

We have \( \angle ZKN\cong\angle FWL\), \( \overline{NZ}\cong\overline{FL}\), \( \overline{KZ}\cong\overline{LW}\). By the SAS (Side - Angle - Side) congruence postulate (two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle).

10.

Step1: Check the congruence criteria

We have \( \angle YBH\cong\angle CRX\), \( \angle BHY\cong\angle XCR\), \( \overline{HY}\cong\overline{CX}\). By the AAS (Angle - Angle - Side) congruence theorem (two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle).

Answer:

1. No
2. Yes, ASA
3. Yes, SSS
4. Yes, ASA
5. Yes, SAS
6. Yes, AAS