Question

(4) the distance from / to c and the distance from c to /

1. in the figure shown below, \\( \triangle q r s \\) is the image of \\( \triangle m n p \\) after a clockwise rotation about point \\( c \\) by an angle

of \\( 120 ^ { \circ } \\). the angles of \\( \triangle m n p \\) are shown. which of the following must be the measure of \\( \angle r \\) ?
(1) \\( 51 ^ { \circ } \\)
(2) \\( 59 ^ { \circ } \\)
(3) \\( 70 ^ { \circ } \\)
(4) \\( 120 ^ { \circ } \\)

Explanation:

Step1: Propiedad de la rotación

La rotación es una transformación rígida. Las figuras rígidas conservan la forma y el tamaño. Entonces, \(\triangle MNP\cong\triangle QRS\).

Step2: Ángulos congruentes

Si \(\triangle MNP\cong\triangle QRS\), entonces \(\angle N\cong\angle Q\), \(\angle M\cong\angle S\) y \(\angle P\cong\angle R\).

Step3: Identificar el ángulo

En \(\triangle MNP\), \(\angle P = 51^{\circ}\). Como \(\angle P\cong\angle R\), entonces \(\angle R=51^{\circ}\).

Answer:

(1) \(51^{\circ}\)