Question

2 in \\( \triangle l m n \\) below, an angle bisector for each of its angles has been drawn. it is known that \\( \overline{l m} \cong \overline{l n} \\). based on this information, which of the following must be a right angle? (1) \\( \angle n e m \\) (2) \\( \angle n m g \\) (3) \\( \angle m g n \\) (4) \\( \angle l f m \\) 3. if the vertex angle of an isosceles triangle has a measure of \\( 52^{circ} \\), then which of the following is the measure of one of its base angles? (1) \\( 64^{circ} \\) (2) \\( 68^{circ} \\) (3) \\( 74^{circ} \\) (4) \\( 128^{circ} \\)

Explanation:

Step1: Propiedades de triángulo isósceles

En un triángulo isósceles, los ángulos de la base son iguales. La suma de los ángulos internos de un triángulo es \(180^{\circ}\). Sea \(x\) el ángulo de la base. Entonces \(2x + 52^{\circ}=180^{\circ}\).

Step2: Resolver la ecuación

Restar \(52^{\circ}\) de ambos lados: \(2x=180^{\circ}- 52^{\circ}=128^{\circ}\). Dividir por \(2\): \(x = \frac{128^{\circ}}{2}=64^{\circ}\).

Answer:

1. \(64^{\circ}\)