Question

1. in △lmn below, an angle bisector for each of its angles has been drawn. it is known that overline{lm} ≅ overline{ln}. based on this information, which of the following must be a right angle? (1) ∠nem (2) ∠nmg (3) ∠mgn (4) ∠lfm

Explanation:

Step1: Identificar el triángulo isósceles

Como \( \overline{LM}\cong\overline{LN} \), \( \triangle LMN \) es isósceles con \( \angle LMN=\angle LNM \). Las bisectrices de ángulos dividen \( \angle LMN \) y \( \angle LNM \) en mitades iguales.

Step2: Analizar la suma de ángulos en \( \triangle MGN \)

Sea \( \angle MLN = 2\alpha \), \( \angle LMN=\angle LNM = 2\beta \). Entonces \( 2\alpha + 2\beta+ 2\beta=180^{\circ}\) (suma de ángulos en un triángulo), \( \alpha + 2\beta=90^{\circ}\).
En \( \triangle MGN \), \( \angle MGN = 180^{\circ}-(\beta+\beta+\alpha)\). Sustituyendo \( \alpha + 2\beta = 90^{\circ}\), \( \angle MGN=90^{\circ}\).

Answer:

(3) \( \angle MGN \)