Question

for each triangle, check all that apply.
triangle a\ttriangle b\ttriangle c\ttriangle d
\t\t\t
\t\t\t
scalene
isosceles
equilateral\tscalene
isosceles
equilateral\tscalene
isosceles
equilateral\tscalene
isosceles
equilateral

Explanation:

Step1: Recall triangle definitions

• Scalene triangle: All sides are of different lengths.
• Isosceles triangle: At least two sides are of equal length.
• Equilateral triangle: All three sides are of equal length and all angles are \(60^{\circ}\).

Step2: Analyze Triangle A

• Sides are \(4\), \(7\), \(10\). All lengths are different. So, it is a scalene triangle.

Step3: Analyze Triangle B

• Angles are \(45^{\circ}\), \(60^{\circ}\), \(75^{\circ}\). No two angles (and thus no two sides) are equal. So, it is a scalene triangle.

Step4: Analyze Triangle C

• All angles are \(60^{\circ}\). By the property of triangles, if all angles are equal, all sides are equal. So, it is an equilateral triangle (and also an isosceles triangle since an equilateral triangle satisfies the isosceles triangle condition).

Step5: Analyze Triangle D

• Sides are \(6\), \(6\), \(10\). Two sides (\(6\) and \(6\)) are equal. So, it is an isosceles triangle.

Answer:

• Triangle A: Scalene
• Triangle B: Scalene
• Triangle C: Isosceles, Equilateral
• Triangle D: Isosceles