Question

given $nq = nr$ and $qs = rs$, select the two true statements that are used to prove that $overline{ns}$ bisects $\angle mno$.
show your work here
$qs \perp rs$ because $q$ and $r$ are equidistant from $ns$
$\triangle nqr \cong \triangle qsr$ by the side - side - side criterion
$\triangle qns \cong \triangle rns$ by the side - side - side criterion
$\angle qns \cong \angle rns$ because corresponding parts of congruent triangles are congruent
$mn = no$ because corresponding parts of congruent triangles are congruent

Explanation:

Step1: Analyze triangle congruence

We have \( NQ = NR \), \( QS = RS \), and \( NS \) is a common side. So, for triangles \( \triangle QNS \) and \( \triangle RNS \), we can apply the SSS (Side - Side - Side) congruence criterion. That is, \( \triangle QNS\cong\triangle RNS \) by SSS because \( NQ = NR \), \( QS = RS \), and \( NS=NS \).

Step2: Corresponding parts of congruent triangles

Since \( \triangle QNS\cong\triangle RNS \), by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, \( \angle QNS=\angle RNS \). This means that \( \overline{NS} \) bisects \( \angle MNO \) because it divides the angle into two equal angles \( \angle QNS \) and \( \angle RNS \).

Now let's analyze the given statements:

• " \( QS\perp RS \) because \( Q \) and \( R \) are equidistant from \( NS \)" is incorrect. There is no basis for this perpendicularity from the given information.
• " \( \triangle NQR\cong\triangle QSR \) by the side - side - side criterion" is incorrect. The sides of \( \triangle NQR \) and \( \triangle QSR \) do not match the SSS criteria with the given information.
• " \( \triangle QNS\cong\triangle RNS \) by the side - side - side criterion" is correct as we proved above.
• " \( \angle QNS\cong\angle RNS \) because corresponding parts of congruent triangles are congruent" is correct as it follows from the congruence of \( \triangle QNS \) and \( \triangle RNS \).
• " \( MN = NO \) because corresponding parts of congruent triangles are congruent" is incorrect. There is no information to suggest that \( MN \) and \( NO \) are corresponding parts of congruent triangles.

Answer:

The two true statements are:

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