Question

given $kn \cong ko$ and $np \cong op$, select the two true statements that are used to prove that $\overline{kp}$ bisects $\angle jkl$.

(image of a geometric figure with points k, n, o, p, j, l, m; segments and arcs as part of the diagram)

show your work here

$np \perp op$ because $n$ and $o$ are equidistant from $kp$
$\triangle nkp \cong \triangle okp$ by the side - side - side criterion
$\triangle kno \cong \triangle npo$ by the side - side - side criterion
$jk \cong kl$ because corresponding parts of congruent triangles are congruent
$\angle nkp \cong \angle okp$ because corresponding parts of congruent triangles are congruent

Explanation:

Step1: Identify congruent triangles

We know \( KN = KO \), \( NP = OP \), and \( KP \) is common to both \( \triangle NKP \) and \( \triangle OKP \). By the SSS (Side - Side - Side) criterion, \( \triangle NKP\cong\triangle OKP \).

Step2: Corresponding angles of congruent triangles

Since \( \triangle NKP\cong\triangle OKP \), their corresponding angles \( \angle NKP \) and \( \angle OKP \) are congruent (by CPCTC - Corresponding Parts of Congruent Triangles are Congruent). This means that \( KP \) bisects \( \angle JKL \) because it divides the angle into two congruent angles.

So the two true statements are:

• \( \triangle NKP\cong\triangle OKP \) by the side - side - side criterion
• \( \angle NKP\cong\angle OKP \) because corresponding parts of congruent triangles are congruent

Answer:

\( \triangle NKP\cong\triangle OKP \) by the side - side - side criterion; \( \angle NKP\cong\angle OKP \) because corresponding parts of congruent triangles are congruent