Question

1. are the following triangles similar? t 60° 16 34 p 30° 30 n r 15 k 30° 17 60° 8 g

Explanation:

Step1: Analyze angles of first triangle

Triangle \( PTN \) is right - angled at \( P \), with \( \angle N = 30^{\circ} \), \( \angle T=60^{\circ} \) (since sum of angles in triangle is \( 180^{\circ} \), \( 90^{\circ}+30^{\circ}+\angle T = 180^{\circ}\Rightarrow\angle T = 60^{\circ} \)).

Step2: Analyze angles of second triangle

Triangle \( RKG \) (and the triangle with \( R \)): Triangle \( RKG \) is right - angled at \( K \), \( \angle G = 60^{\circ} \), so \( \angle R\) (the non - right, non - 60° angle) is \( 30^{\circ} \) (since \( 90^{\circ}+60^{\circ}+\angle R=180^{\circ}\Rightarrow\angle R = 30^{\circ} \)). Also, check the ratios of sides. For triangle \( PTN \), sides are \( PT = 16 \), \( PN = 30 \), \( TN = 34 \). For the other triangle, sides are \( RK = 15 \), \( KG = 8 \), \( RG = 17 \). Now, check the ratios: \( \frac{PT}{KG}=\frac{16}{8} = 2 \), \( \frac{PN}{RK}=\frac{30}{15}=2 \), \( \frac{TN}{RG}=\frac{34}{17} = 2 \). Also, the corresponding angles are equal (\( 90^{\circ} \), \( 30^{\circ} \), \( 60^{\circ} \)). By AA (Angle - Angle) similarity criterion (if two angles of one triangle are equal to two angles of another triangle, the triangles are similar) or by SSS (Side - Side - Side) similarity criterion (if the ratios of corresponding sides are equal, the triangles are similar), the triangles are similar.

Answer:

Yes, the triangles are similar.