Question

which statements are true? select all that apply
a △fgk≅△fjk
b ∠gkh≅∠jkh
c. (overline{fg}congoverline{kg})
d ∠gfh≅∠jfh

Explanation:

Step1: Analyze congruent - triangle criteria

In the figure, we have \(FG = FJ\) (marked with one - hash), \(GK=JK\) (marked with two - hashes), and \(FK = FK\) (common side). By the SSS (Side - Side - Side) congruence criterion, \(\triangle FGK\cong\triangle FJK\). So, statement A is true.

Step2: Use congruent - triangle properties

Since \(\triangle FGK\cong\triangle FJK\), corresponding angles are congruent. \(\angle GFK=\angle JFK\) and \(\angle FKG=\angle FKJ\). Also, \(\angle GKH\) and \(\angle JKH\) are linear pairs and since \(\angle FKG=\angle FKJ\), \(\angle GKH\cong\angle JKH\) (because \(\angle GKH = 180^{\circ}-\angle FKG\) and \(\angle JKH=180^{\circ}-\angle FKJ\)). So, statement B is true.

Step3: Analyze side - length relationships

There is no information to suggest that \(FG\cong KG\). The markings only show \(FG = FJ\) and \(GK = JK\). So, statement C is false.

Step4: Analyze angle relationships

Since \(\triangle FGK\cong\triangle FJK\), \(\angle GFH\cong\angle JFH\) (corresponding angles of congruent triangles). So, statement D is true.

Answer:

A. \(\triangle FGK\cong\triangle FJK\)
B. \(\angle GKH\cong\angle JKH\)
D. \(\angle GFH\cong\angle JFH\)