Classifying And Identifying Triangles

1) A triangle with all angles less than 90 is classified as:

Right

Obtuse

Acute

Scalene

Every angle is small (less than 90°) → acute triangle.

Explanation

Every angle is small (less than 90°) → acute triangle.

2) A triangle has angles of 80°, 50°, and 50°. Which type is it?

Right isosceles

Equilateral

Obtuse scalene

Acute isosceles

The triangle has two equal angles (50° each) and one angle (80°) that is less than 90°. This makes it an isosceles triangle. Since all angles are less than 90°, it is classified as an acute triangle, thus it is an acute isosceles triangle.

Explanation

The triangle has two equal angles (50° each) and one angle (80°) that is less than 90°. This makes it an isosceles triangle. Since all angles are less than 90°, it is classified as an acute triangle, thus it is an acute isosceles triangle.

3) Which triangle has exactly one right angle and two equal sides?

Equilateral

Right isosceles

Right scalene

Obtuse isosceles

A right isosceles triangle is defined by having one right angle (90 degrees) and two sides that are equal in length.

Explanation

A right isosceles triangle is defined by having one right angle (90 degrees) and two sides that are equal in length.

4) Which set of angles forms a valid triangle?

60°, 60°, 70°

90°, 45°, 45°

120°, 60°, 10°

80°, 80°, 30°

For a set of angles to form a valid triangle, the sum of the angles must equal 180°. In option B, the angles sum to 90° + 45° + 45° = 180°, making it a valid triangle. The other options do not satisfy this condition.

Explanation

For a set of angles to form a valid triangle, the sum of the angles must equal 180°. In option B, the angles sum to 90° + 45° + 45° = 180°, making it a valid triangle. The other options do not satisfy this condition.

5) Which triangle can exist?

Sides 5, 5, 10

Sides 7, 8, 15

Sides 9, 9, 5

Sides 2, 2, 5

Triangle inequality:

5,5,10 fails (sum equals largest).

7,8,15 fails (sum equals largest).

9,9,5 works (each pair sums greater).

2,2,5 fails (sum less than largest).

Explanation

Triangle inequality:

5,5,10 fails (sum equals largest).

7,8,15 fails (sum equals largest).

9,9,5 works (each pair sums greater).

2,2,5 fails (sum less than largest).

6) A triangle with angles 90°, 30°, and 60° is:

Right scalene

Right isosceles

Acute scalene

Obtuse isosceles

Right triangle with all angles different means Right scalene.

Explanation

Right triangle with all angles different means Right scalene.

7) A triangle has two angles that measure 45°. What is the measure of the third angle?

45°

60°

90°

120°

In a triangle, the sum of the interior angles is always 180°. If two angles measure 45° each, their sum is 90°. Therefore, the third angle must be 180 - 90 = 90°.

Explanation

In a triangle, the sum of the interior angles is always 180°. If two angles measure 45° each, their sum is 90°. Therefore, the third angle must be 180 - 90 = 90°.

8) You are given two sides of 6 cm and 9 cm and the included angle of 50°. What can you conclude about the triangle?

Unique triangle

More than one triangle

No triangle

Not enough information

In the SAS (side-angle-side) condition, when two sides and their included angle are known, only one unique triangle can be formed.

Explanation

In the SAS (side-angle-side) condition, when two sides and their included angle are known, only one unique triangle can be formed.

9) If a triangle has sides 6 cm, 6 cm, and 6 cm, which statement is true?

It is equilateral and acute.

It is isosceles and right.

It is scalene and obtuse.

It is equilateral and right.

All equal sides means equilateral, which is always acute (all 60°)

Explanation

All equal sides means equilateral, which is always acute (all 60°)

10) A triangle with three equal sides is called:

Scalene

Isosceles

Equilateral

Right

All three sides are the same → all angles are 60°.

So, it’s an equilateral triangle.

Explanation

All three sides are the same → all angles are 60°.

So, it’s an equilateral triangle.

11) If a triangle has angles 30°, 60°, and 90°, what type is it?

Right

Acute

Obtuse

Equilateral

Having a 90° angle makes it a right triangle.

Explanation

Having a 90° angle makes it a right triangle.

12) A triangle has sides 8 cm, 15 cm, and 17 cm. Which type of triangle is it?

Acute

Right

Obtuse

Equilateral

8² + 15² = 64 + 225 = 289 = 17² → right triangle.

Explanation

8² + 15² = 64 + 225 = 289 = 17² → right triangle.

13) A triangle has one angle measuring 120°. Which type of triangle is it?

Acute

Right

Obtuse

Equilateral

An angle greater than 90° makes it obtuse.

Explanation

An angle greater than 90° makes it obtuse.

14) You are given two sides of 10 cm and 7 cm and a nonincluded angle of 30° that is opposite the 7 cm side. What can you conclude about the triangle?

Unique triangle

More than one triangle

No triangle

Not enough information

In the SSA (side-side-angle) condition, when two sides and a nonincluded angle are known, two different triangles can form — this is called the ambiguous case.

Explanation

In the SSA (side-side-angle) condition, when two sides and a nonincluded angle are known, two different triangles can form — this is called the ambiguous case.

15) A triangle with three unequal sides is called:

Scalene

Isosceles

Equilateral

Obtuse

Three different sides means scalene.

Explanation

Three different sides means scalene.

16) You are given two sides of 10 cm and 7 cm and an acute angle that is opposite the 7 cm side, but the angle measure is not specified. What can you conclude about the triangle?

Unique triangle

More than one triangle

No triangle

Not enough information

In the SSA condition, when the angle size isn’t given, we can’t tell if 0, 1, or 2 triangles form → Not enough information.

Explanation

In the SSA condition, when the angle size isn’t given, we can’t tell if 0, 1, or 2 triangles form → Not enough information.

17) If a triangle has side lengths 10, 10, and 14, which type is it?

Isosceles obtuse

Scalene

Equilateral

Isosceles acute

The triangle has two sides of equal length (10 and 10), which classifies it as isosceles. Since the side lengths also satisfy the triangle inequality and the angle opposite the longest side (14) is less than 90 degrees, it is an isosceles acute triangle.

Explanation

The triangle has two sides of equal length (10 and 10), which classifies it as isosceles. Since the side lengths also satisfy the triangle inequality and the angle opposite the longest side (14) is less than 90 degrees, it is an isosceles acute triangle.

18) Which of the following angle measures could not form a triangle?

50°, 60°, 70°

90°, 60°, 30°

120°, 40°, 20°

100°, 40°, 50°

For three angles to form a triangle, their measures must sum up to exactly 180°. In option D, the angles are 100°, 40°, and 50°, which sum to 190° (100° + 40° + 50° = 190°). Since this exceeds 180°, these angles cannot form a triangle. The other options sum to 180° or less, making them valid combinations for triangle formation.

Explanation

For three angles to form a triangle, their measures must sum up to exactly 180°. In option D, the angles are 100°, 40°, and 50°, which sum to 190° (100° + 40° + 50° = 190°). Since this exceeds 180°, these angles cannot form a triangle. The other options sum to 180° or less, making them valid combinations for triangle formation.