Question

1. in the diagram of quadrilateral acbd, diagonal \\( \overline{cd} \\) is the perpendicular bisector of diagonal \\( \overline{ab} \\). which of the following congruence statements is not always true?

(1) \\( \overline{ad} \cong \overline{bd} \\)
(2) \\( \overline{bc} \cong \overline{ac} \\)
(3) \\( \overline{ce} \cong \overline{de} \\)
(4) \\( \overline{ae} \cong \overline{be} \\)

1. the vertex angle of an isosceles triangle has a measure of \\( 38 ^ { \circ } \\). which of the following is the measure of one of its base angles?

(1) \\( 38 ^ { \circ } \\)
(2) \\( 49 ^ { \circ } \\)
(3) \\( 64 ^ { \circ } \\)
(4) \\( 71 ^ { \circ } \\)

Explanation:

Step1: Propiedades de triángulos isósceles

En un triángulo isósceles, los ángulos de la base son iguales. La suma de los ángulos internos de un triángulo es \(180^{\circ}\).

Step2: Cálculo del ángulo de la base

Sea \(x\) el ángulo de la base. Sabemos que \(2x + 38^{\circ}=180^{\circ}\). Resolviendo para \(x\):
\(2x=180^{\circ}- 38^{\circ}\)
\(2x = 142^{\circ}\)
\(x=\frac{142^{\circ}}{2}=71^{\circ}\)

Answer:

(4) \(71^{\circ}\)