Question

given: point m is the midpoint of \\( \overline{lo} \\) and \\( \overline{np} \\) and \\( \angle lnm \\) and \\( \angle opm \\) are right angles. prove: \\( \triangle lmn \cong \triangle opm \\) the students are asked to construct an argument to prove \\( \triangle lmn \cong \triangle opm \\). some of the students responses are given below. which of the arguments prove the congruence of the two triangles? select all that apply. a • \\( \angle lnm \\) and \\( \angle opm \\) are right angles: given • \\( \overline{lm} \\) is the hypotenuse of \\( \triangle lnm \\) and \\( \overline{om} \\) is the hypotenuse of \\( \triangle opm \\): definition of a hypotenuse • point m is the midpoint of \\( \overline{lo} \\) and \\( \overline{np} \\): given • \\( \overline{lm} \cong \overline{om} \\) and \\( \overline{nm} \cong \overline{pm} \\): definition of midpoint and definition of congruent segments. • \\( \triangle lnm \cong \triangle opm \\): hypotenuse - leg congruence theorem b • \\( \angle lnm \\) and \\( \angle opm \\) are right angles: given • \\( \angle lnm \cong \angle opm \\): right angles are congruent • point m is the midpoint of \\( \overline{lo} \\) and \\( \overline{np} \\): given • \\( \overline{lm} \cong \overline{om} \\) and \\( \overline{nm} \cong \overline{pm} \\): definition of midpoint and definition of congruent • \\( \triangle lnm \cong \triangle opm \\): sas theorem c • point m is the midpoint of \\( \overline{lo} \\) and \\( \overline{np} \\): given • \\( \overline{lm} \cong \overline{om} \\) and \\( \overline{nm} \cong \overline{pm} \\): definition of midpoint and definition of congruent segments. • \\( \angle lmn \cong \angle omp \\): vertical angles are congruent • \\( \triangle lnm \cong \triangle opm \\): sas theorem d • point m is the midpoint of \\( \overline{lo} \\) and \\( \overline{np} \\): given • \\( \overline{lm} \cong \overline{om} \\) and \\( \overline{nm} \cong \overline{pm} \\): definition of midpoint and definition of congruent segments. • \\( \overline{ln} \cong \overline{op} \\): congruent parts of congruent triangles are congruent • \\( \triangle lnm \cong \triangle opm \\): sss theorem e • \\( \angle lnm \\) and \\( \angle opm \\) are right angles: given • \\( \angle lnm \cong \angle opm \\): right angles are congruent • \\( \angle lmn \cong \angle omp \\): vertical angles are congruent • point m is the midpoint of \\( \overline{lo} \\): given • \\( \overline{lm} \cong \overline{om} \\): definition of midpoint and definition of congruent segments • \\( \triangle lmn \cong \triangle omp \\): aas theorem

Explanation:

Step1: Analyze Option A

• Hypotenuse - Leg (HL) Congruence Theorem requires two right - angled triangles, with the hypotenuse and one leg congruent. Here, \(\angle LNM\) and \(\angle OPM\) are right angles. \(LM\) and \(OM\) are hypotenuses (\(M\) is the mid - point of \(LO\), so \(LM = OM\)), and \(NM\) and \(PM\) are legs (\(M\) is the mid - point of \(NP\), so \(NM=PM\)). But the HL theorem is used for right - angled triangles, and the step “\(\overline{LM}\) is the hypotenuse of \(\triangle LNM\) and \(\overline{OM}\) is the hypotenuse of \(\triangle OPM\)” is incorrect. In \(\triangle LNM\), \(LM\) is a side, not the hypotenuse (the hypotenuse of a right - angled triangle is the side opposite the right angle. In \(\triangle LNM\), \(LN\) and \(NM\) are legs and \(LM\) is the hypotenuse is wrong. The hypotenuse of \(\triangle LNM\) should be \(LM\) if we consider \(\angle LNM = 90^{\circ}\), but the reasoning about hypotenuse - leg is misapplied here.

Step2: Analyze Option B

• We know that \(\angle LNM\cong\angle OPM\) (right angles are congruent), \(LM\cong OM\) (since \(M\) is the mid - point of \(LO\)), and \(NM\cong PM\) (since \(M\) is the mid - point of \(NP\)). The Side - Angle - Side (SAS) Theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. Here, the included angles of the pairs of sides (\(\angle LNM\) and \(\angle OPM\)) are congruent.

Step3: Analyze Option C

• We have \(LM\cong OM\) (mid - point of \(LO\)), \(\angle LMN\cong\angle OMP\) (vertical angles), and \(NM\cong PM\) (mid - point of \(NP\)). By the Side - Angle - Side (SAS) Theorem, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Step4: Analyze Option D

• The statement “\(\overline{LN}\cong\overline{OP}\): Congruent Parts of Congruent Triangles are Congruent” is circular. We are trying to prove \(\triangle LMN\cong\triangle OPM\), so we cannot use the fact that \(LN\cong OP\) (which would be a result of the congruence of the triangles) to prove the congruence of the triangles using SSS (Side - Side - Side).

Step5: Analyze Option E

• We have \(\angle LNM\cong\angle OPM\) (right angles), \(\angle LMN\cong\angle OMP\) (vertical angles), and \(LM\cong OM\) (mid - point of \(LO\)). By the Angle - Angle - Side (AAS) Theorem, if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.

Answer:

B. Option B: \(\angle LNM\) and \(\angle OPM\) are right angles (Given); \(\angle LNM\cong\angle OPM\) (Right angles are congruent); Point \(M\) is the midpoint of \(\overline{LO}\) and \(\overline{NP}\) (Given); \(\overline{LM}\cong\overline{OM}\) and \(\overline{NM}\cong\overline{PM}\) (Definition of midpoint and definition of congruent); \(\triangle LNM\cong\triangle OPM\) (SAS Theorem)
C. Option C: Point \(M\) is the midpoint of \(\overline{LO}\) and \(\overline{NP}\) (Given); \(\overline{LM}\cong\overline{OM}\) and \(\overline{NM}\cong\overline{PM}\) (Definition of midpoint and definition of congruent segments); \(\angle LMN\cong\angle OMP\) (Vertical angles are congruent); \(\triangle LNM\cong\triangle OPM\) (SAS Theorem)
E. Option E: \(\angle LNM\) and \(\angle OPM\) are right angles (Given); \(\angle LNM\cong\angle OPM\) (Right angles are congruent); \(\angle LMN\cong\angle OMP\) (Vertical angles are congruent); Point \(M\) is the midpoint of \(\overline{LO}\) (Given); \(\overline{LM}\cong\overline{OM}\) (Definition of midpoint and definition of congruent segments); \(\triangle LMN\cong\triangle OMP\) (AAS Theorem)