Question

given: \\( \triangle pqr \\) is a dilation of \\( \triangle pqr \\) by a scale factor of 3. choose the statements that are true and necessary for establishing aa similarity of the two triangles \\( \square \overline{pr}\parallel\overline{pr} \\) \\( \square \angle qpr\cong\angle qpr \\) \\( \square \angle prq\cong\angle pqr \\) \\( \square \\) dilated segments are parallel \\( \square \angle pqr\\&\angle pqr \\) are both right angles. \\( \square \overline{qr}\parallel\overline{qr} \\)

Explanation:

Step1: Understand AA (Angle - Angle) similarity

AA similarity states that if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

Step2: Analyze each option

• For \(\overline{PR}\parallel\overline{P'R'}\):

If two lines are parallel, then corresponding angles are congruent. For example, \(\angle QPR\) and \(\angle Q'P'R'\) (corresponding angles) are congruent.

• For \(\angle QPR\cong\angle Q'P'R'\):

This is a direct angle - congruence statement. Since \(\triangle P'Q'R'\) is a dilation of \(\triangle PQR\), the angles of the original triangle and the dilated triangle are congruent.

• For \(\angle PRQ\cong\angle P'Q'R'\):

This is incorrect. \(\angle PRQ\) and \(\angle P'Q'R'\) are not corresponding angles.

• For "Dilated segments are parallel":

This is a general property of dilation. When a figure is dilated, corresponding line segments are parallel. This helps in establishing angle - congruence (corresponding angles) for AA similarity.

• For "\(\angle PQR\) and \(\angle P'Q'R'\) are both right angles":

This gives one pair of congruent right angles.

• For \(\overline{QR}\parallel\overline{Q'R'}\):

If two lines are parallel, then corresponding angles are congruent. For example, \(\angle PRQ\) and \(\angle P'R'Q'\) (corresponding angles) are congruent.

Answer:

\(\overline{PR}\parallel\overline{P'R'}\), \(\angle QPR\cong\angle Q'P'R'\), Dilated segments are parallel, \(\angle PQR\) & \(\angle P'Q'R'\) are both right angles, \(\overline{QR}\parallel\overline{Q'R'}\)