Question

triangles separating overlapping triangles and identifying common features consider \\( \triangle p q r \\) and \\( \triangle t s r \\) in the figure below. use the figure above to complete the following. (a) \\( \triangle p q r \\) and \\( \triangle t s r \\) have been separated. fill in the missing vertex labels. (b) choose the correct statement below about \\( \triangle p q r \\) and \\( \triangle t s r \\). then fill in the additional information as necessary. the triangles have a common side. common side: (choose one) the triangles have a common angle. common angle: (choose one) the triangles have neither a common side nor a common angle.

Explanation:

Step1: Identify vertices for \( \triangle PQR \)

In the original figure, \( \triangle PQR \) has vertices \( P \), \( Q \), and \( R \). The side with length \( 10 \) is \( PS \) (but in \( \triangle PQR \), the vertex corresponding to the start of the side with length \( 10 \) is \( P \), the side with length \( 3 \) is adjacent to \( Q \), and the common vertex for both triangles is \( R \). So the triangle with side lengths \( 10 \), \( 3 \) has vertices \( P \), \( Q \), \( R \).

Step2: Identify vertices for \( \triangle TSR \)

In the original figure, \( \triangle TSR \) has vertices \( T \), \( S \), and \( R \). The side with length \( 6 \) is \( TQ \) (but in \( \triangle TSR \), the vertex corresponding to the start of the side with length \( 6 \) is \( T \), the side with length \( 4 \) is adjacent to \( S \), and the common vertex for both triangles is \( R \). So the triangle with side lengths \( 6 \), \( 4 \) has vertices \( T \), \( S \), \( R \).

Step3: Check for common elements

For angles, in \( \triangle PQR \) and \( \triangle TSR \), \( \angle PRQ\) (angle at \( R \) for \( \triangle PQR \)) and \( \angle TRS\) (angle at \( R \) for \( \triangle TSR \)) are the same angle. So the triangles have a common angle \( \angle R\).

Answer:

(a) Left - hand triangle ( \( \triangle PQR \)): vertices \( P \), \( Q \), \( R \); Right - hand triangle (\( \triangle TSR \)): vertices \( T \), \( S \), \( R \)
(b) The triangles have a common angle. Common angle: \( \angle R\)