Question

for each triangle, check all that apply.
triangle a
30°
60°
☑ scalene
□ isosceles
□ equilateral
triangle b
45°
45°
□ scalene
☑ isosceles
□ equilateral
triangle c
6
10
6
☑ scalene
□ isosceles
□ equilateral
triangle d
9
9
9
□ scalene
☑ isosceles
☑ equilateral

Explanation:

Step1: Analyze Triangle A

A scalene triangle has all sides of different lengths. A right - angled triangle with angles \(30^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\) has sides in the ratio \(1:\sqrt{3}:2\), so all sides are different. So it is scalene. An isosceles triangle has at least two equal sides (or angles), and an equilateral triangle has all sides equal (all angles \(60^{\circ}\)). Since it doesn't meet the criteria for isosceles or equilateral, the check for scalene is correct, isosceles is incorrect, equilateral is incorrect.

Step2: Analyze Triangle B

A right - angled triangle with two angles of \(45^{\circ}\) means the two legs are equal (by the property that in a triangle, equal angles correspond to equal sides). So it is isosceles (has two equal sides). A scalene triangle has all sides different, so it is not scalene. An equilateral triangle has all angles \(60^{\circ}\), so it is not equilateral. So the check for scalene is incorrect, isosceles is correct, equilateral is incorrect.

Step3: Analyze Triangle C

The triangle has two sides of length 6 and one side of length 10. So it has two equal sides, which means it is isosceles. A scalene triangle has all sides different, so the check for scalene is incorrect. An equilateral triangle has all sides equal, so it is not equilateral. So the check for scalene is wrong, isosceles should be checked, equilateral is wrong.

Step4: Analyze Triangle D

All three sides are of length 9. An equilateral triangle has all sides equal (and all angles \(60^{\circ}\)), and an equilateral triangle is also a special case of an isosceles triangle (since it has at least two equal sides). So it is both isosceles and equilateral. A scalene triangle has all sides different, so the check for scalene is incorrect, isosceles is correct, equilateral is correct.

Answer:

• Triangle A: Correct - Scalene; Incorrect - Isosceles, Equilateral
• Triangle B: Correct - Isosceles; Incorrect - Scalene, Equilateral
• Triangle C: Correct - Isosceles; Incorrect - Scalene, Equilateral
• Triangle D: Correct - Isosceles, Equilateral; Incorrect - Scalene

(To correct the given checks:

• Triangle C: Uncheck Scalene, Check Isosceles
• Triangle D: Uncheck Scalene, Check Equilateral (it is already isosceles))