Question

complete with > or <. explain your reasoning. pt______qr select the correct choice below and fill in the answer box within your choice. a. the converse of the hinge theorem states that if two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the smaller included angle is opposite the longer third side. thus, pt□qr. b. the hinge theorem states that if two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle. thus, pt□qr.

Explanation:

Step1: Identify Triangles and Sides

In triangles \( \triangle PQT \) and \( \triangle TRS \) (or \( \triangle QTR \) and \( \triangle PQT \)? Wait, better: Look at \( \triangle PQT \) and \( \triangle QTR \)? No, actually, triangles \( \triangle PQT \) and \( \triangle RTS \)? Wait, no, let's check the sides. \( PQ = RT = 14 \), \( QT = TS = 12 \). The included angles: in \( \triangle PQT \), the included angle at \( Q \) is \( 90^\circ \), in \( \triangle RTS \), the included angle at \( T \) (wait, no, the two triangles with two congruent sides: \( \triangle PQT \) and \( \triangle RQT \)? Wait, \( PQ = RT = 14 \), \( QT = QT \)? No, wait, \( PQ = RT = 14 \), \( QT = TS = 12 \)? Wait, the two triangles are \( \triangle PQT \) and \( \triangle RTS \)? No, let's see the angles. The included angle in \( \triangle PQT \) is \( \angle PQT = 90^\circ \), and in \( \triangle RQT \)? Wait, no, the Hinge Theorem: two sides of one triangle congruent to two sides of another triangle. So \( PQ = RT = 14 \), \( QT = TS = 12 \)? Wait, no, \( QT = 12 \), \( TS = 12 \), \( PQ = 14 \), \( RT = 14 \). The included angles: \( \angle PQT = 90^\circ \), \( \angle RTS \)? Wait, no, the angle at \( T \) in \( \triangle QTR \) is \( 85^\circ \), and the angle at \( Q \) in \( \triangle PQT \) is \( 90^\circ \). So triangles \( \triangle PQT \) and \( \triangle RQT \)? Wait, \( PQ = RT = 14 \), \( QT = QT \) (common side), and \( PT \) and \( QR \) are the third sides. The included angles: \( \angle PQT = 90^\circ \), \( \angle RTQ = 85^\circ \). So by Hinge Theorem: if two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle. So \( \angle PQT = 90^\circ > 85^\circ = \angle RTQ \), so \( PT > QR \)? Wait, no, wait: the two triangles are \( \triangle PQT \) and \( \triangle RQT \)? Wait, \( PQ = RT = 14 \), \( QT = QT \), included angles \( \angle PQT = 90^\circ \) and \( \angle RTQ = 85^\circ \). So by Hinge Theorem, the side opposite the larger angle is longer. So \( PT \) is opposite \( \angle PQT = 90^\circ \), \( QR \) is opposite \( \angle RTQ = 85^\circ \). So since \( 90^\circ > 85^\circ \), \( PT > QR \). Now, the options: Option B says "The Hinge Theorem states that if two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle. Thus, \( PT > QR \)." Option A is converse, which is about third sides not congruent implying smaller angle opposite longer side, but here we use the Hinge Theorem (not converse) to compare angles and then sides. So we use Option B.

Step2: Apply Hinge Theorem

The Hinge Theorem (also called SAS Inequality Theorem) states that if two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle. Here, in \( \triangle PQT \) and \( \triangle RQT \) (wait, actually \( \triangle PQT \) and \( \triangle RTQ \)): \( PQ = RT = 14 \), \( QT = QT \) (common side), included angles \( \angle PQT = 90^\circ \) and \( \angle RTQ = 85^\circ \). Since \( 90^\circ > 85^\circ \), the third side \( PT \) (opposite \( 90^\circ \)) is longer than \( QR \) (opposite \( 85^\circ \)). So \( PT > QR \), and the correct option is B, with \( PT > QR \).

Answer:

B. The Hinge Theorem states that if two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle. Thus, \( PT > QR \).