Question

2 - 84 determine which of the following pairs of triangles are similar. justify your answer.

Explanation:

Step1: Recall similarity criteria

Two triangles are similar if their corresponding angles are equal or corresponding - sides are in proportion.

Step2: Analyze pair a

In the first triangle of pair a, angles are \(40^{\circ}\) and \(60^{\circ}\), so the third angle is \(180-(40 + 60)=80^{\circ}\). In the second triangle of pair a, angles are \(85^{\circ}\) and \(60^{\circ}\), so the third angle is \(180-(85 + 60)=35^{\circ}\). Since the angles are not equal, the triangles in pair a are not similar.

Step3: Analyze pair b

In the first triangle of pair b, angles are \(50^{\circ}\) and \(45^{\circ}\), so the third angle is \(180-(50 + 45)=85^{\circ}\). In the second triangle of pair b, angles are \(50^{\circ}\) and \(85^{\circ}\), so the third angle is \(180-(50 + 85)=45^{\circ}\). Since the corresponding angles are equal (\(50^{\circ}=50^{\circ}\), \(45^{\circ}=45^{\circ}\), \(85^{\circ}=85^{\circ}\)), the triangles in pair b are similar by the AA (angle - angle) similarity criterion.

Step4: Analyze pair c

Check the ratio of corresponding sides. \(\frac{6}{12}=\frac{1}{2}\), \(\frac{7}{14}=\frac{1}{2}\), \(\frac{8}{16}=\frac{1}{2}\). Since the ratios of the corresponding sides are equal (\(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}=\frac{1}{2}\)), the triangles in pair c are similar by the SSS (side - side - side) similarity criterion.

Answer:

The triangles in pair b and pair c are similar.