Question

given: ∠lno ≅ ∠lnm, ∠oln ≅ ∠mln
prove: △lno ≅ △lnm
it is given that angle lno is congruent to angle dropdown and angle oln is congruent to angle dropdown. we know that side ln is congruent to side ln because of the dropdown. therefore, because of dropdown, we can state that triangle lno is congruent to triangle lnm.

Explanation:

Step1: Identify Given Angles

Given $\angle LNO \cong \angle LNM$ (from problem statement) and $\angle OLN \cong \angle MLN$ (from diagram, shared or given angle? Wait, the first given is $\angle LNO \cong \angle LNM$, and $\angle OLN \cong \angle MLN$ (let's assume the red arcs on $\angle OLM$ mean $\angle OLN \cong \angle MLN$). Also, side $LN$ is common to both $\triangle LNO$ and $\triangle LNM$, so $LN \cong LN$ (reflexive property).

Step2: Apply ASA Congruence

We have two angles and the included side: $\angle LNO \cong \angle LNM$, $LN \cong LN$, and $\angle OLN \cong \angle MLN$. So by Angle - Side - Angle (ASA) congruence criterion, $\triangle LNO \cong \triangle LNM$.

Let's fill the blanks:

• "It is given that angle LNO is congruent to angle $\boldsymbol{\angle LNM}$ and angle OLN is congruent to angle $\boldsymbol{\angle MLN}$."
• "We know that side LN is congruent to side LN because of the $\boldsymbol{\text{Reflexive Property of Congruence}}$."
• "Therefore, because of the $\boldsymbol{\text{ASA (Angle - Side - Angle) Congruence Criterion}}$, we can state that triangle LNO is congruent to triangle LNM."

Answer:

1. First blank: $\angle LNM$; Second blank: $\angle MLN$
2. Third blank: Reflexive Property of Congruence
3. Fourth blank: ASA (Angle - Side - Angle) Congruence Criterion