Question

for each triangle, check all that apply.
triangle a
40°
40° 100°
☐ scalene
☐ isosceles
☐ equilateral
triangle b
4
8
8
☐ scalene
☐ isosceles
☐ equilateral
triangle c
60°
60° 60°
☐ scalene
☐ isosceles
☐ equilateral
triangle d
7
10
4
☐ scalene
☐ isosceles
☐ equilateral

Explanation:

Triangle A

• Step1: Check angle - side relationship

Two angles (\(40^{\circ}\)) are equal. In a triangle, equal angles correspond to equal sides. So it has two equal sides.

• Step2: Classify the triangle

Since it has two equal sides, it is isosceles.

Triangle B

• Step1: Check side lengths

Two sides (both \(8\)) are equal.

• Step2: Classify the triangle

Since it has two equal sides, it is isosceles.

Triangle C

• Step1: Check angles

All angles (\(60^{\circ}\)) are equal. In a triangle, equal angles correspond to equal sides.

• Step2: Classify the triangle

Since all angles are equal (\(60^{\circ}\)) and all sides will be equal (by the property of triangles: if \(A = B=C\), then \(a = b = c\) using the law of sines \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)), it is equilateral (and also isosceles as a special case).

Triangle D

• Step1: Check side lengths

All side lengths (\(4\), \(7\), \(10\)) are different.

• Step2: Classify the triangle

Since all sides are of different lengths, it is scalene.

Answer:

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