Question

decide whether each statement about the example is true true false $overline{pq}$ is congruent to $overline{pq}$. true false $moverline{pj}
eq moverline{jp}$ true false $moverline{qt}
eq moverline{tq}$ true false $overline{qq}perpoverleftrightarrow{jt}$ true false $overline{pp}perpoverleftrightarrow{jt}$

Explanation:

Step1: Recall congruence in geometry

In a rigid - transformation (like translation, rotation, reflection), lengths of line - segments are preserved. So, if $\overline{P'Q'}$ is the image of $\overline{PQ}$ under a rigid - transformation, $\overline{P'Q'}\cong\overline{PQ}$. So the first statement is True.

Step2: Consider arc - measures

Arc - measure is related to the central angle subtended by the arc. If the transformation is a rigid - transformation, the measure of an arc between two points and its image arc between the corresponding points are equal. So, $m\overline{PJ}=m\overline{JP'}$ is False (should be equal), and $m\overline{QT}=m\overline{TQ'}$ is False (should be equal).

Step3: Recall properties of rigid - transformations and perpendicularity

In a reflection (a type of rigid - transformation), the line joining a point and its image is perpendicular to the line of reflection. If $\overleftrightarrow{JT}$ is the line of reflection, then $\overline{QQ'}\perp\overleftrightarrow{JT}$ and $\overline{PP'}\perp\overleftrightarrow{JT}$. So the fourth and fifth statements are True.

Answer:

True. $\overline{P'Q'}$ is congruent to $\overline{PQ}$.
False. $m\overline{PJ}
eq m\overline{JP'}$ (should be equal).
False. $m\overline{QT}
eq m\overline{TQ'}$ (should be equal).
True. $\overline{QQ'}\perp\overleftrightarrow{JT}$.
True. $\overline{PP'}\perp\overleftrightarrow{JT}$.