Moderate Level Worksheet Questions

Triangles are fundamental geometric shapes with fascinating properties. Understanding their angles, sides, classifications, and theorems is essential for advanced geometry.

First, let's explore the basic properties and classifications of triangles.

1. Define an equilateral triangle. Triangle with all sides equalTriangle with two sides equalTriangle with no sides equalTriangle with right angle

Awesome! An equilateral triangle has all three sides equal and all angles equal to 60°.

2. Write the angle sum property of a triangle. Sum of angles is 180°Sum of angles is 360°Sum of angles is 90°Sum of angles is 270°

Great job! The sum of all interior angles in any triangle is always 180°.

3. Find the value of x if the angles of a triangle are 60°, 60°, and x°. 60°30°90°120°

Perfect! Since 60° + 60° + x° = 180°, we get x = 60°.

4. What is the measure of each angle of an equilateral triangle? 60°90°45°30°

Excellent! In an equilateral triangle, all angles are equal: 180° ÷ 3 = 60°.

5. Name the triangle whose all three sides are unequal. Scalene triangleIsosceles triangleEquilateral triangleRight triangle

Super! A scalene triangle has all three sides of different lengths.

6. The sum of two angles of a triangle is 110°. Find the third angle. 70°80°60°90°

That's correct! Third angle = 180° - 110° = 70°.

7. Which property of a triangle states that the sum of any two sides is greater than the third side? Triangle inequality propertyAngle sum propertyPythagoras theoremCongruence property

Well done! The triangle inequality property ensures triangle existence.

8. In a right-angled triangle, if one acute angle is 35°, find the other acute angle. 55°45°65°35°

Brilliant! Since 90° + 35° + x° = 180°, we get x = 55°.

9. State the Pythagoras Theorem. a² + b² = c²a + b = ca² = b² + c²a = b + c

You nailed it! In a right triangle, a² + b² = c² where c is the hypotenuse.

10. In a triangle, one angle is 50° and the other is 60°. Name the type of triangle. Acute triangleRight triangleObtuse triangleIsosceles triangle

Perfect! Since all angles (50°, 60°, 70°) are less than 90°, it's an acute triangle.

Drag each classification to its correct category:

Part B: Short Answer Questions (2 Marks Each)

1. Find the third angle of a triangle if two angles are 70° and 50°.

Step 1: Apply angle sum property

Sum of all angles in a triangle =

Sum of given angles = 70° + 50° =

Step 2: Calculate third angle

Third angle = 180° - 120° =

Excellent! The third angle is 60°.

2. The three angles of a triangle are in the ratio 24. Find all three angles.

Step 1: Set up variables

Let the angles be , , and

Step 2: Apply angle sum property

2x + 3x + 4x =

= 180°

x = 180° ÷ 9 =

Step 3: Find all angles

First angle = 2x = 2 × 20° =

Second angle = 3x = 3 × 20° =

Third angle = 4x = 4 × 20° =

Perfect! The angles are 40°, 60°, and 80°.

3. The sides of a triangle are 7 cm, 24 cm, and 25 cm. Check whether it is a right-angled triangle.

Step 1: Apply Pythagoras theorem

For a right triangle: (smaller side)² + (medium side)² = (largest side)²

Check: 7² + 24² = + =

Step 2: Check largest side

25² =

Step 3: Compare results

Since 7² + 24² = 25² (625 = 625), it is a triangle

Great work! The triangle satisfies Pythagoras theorem.

4. The two acute angles of a right-angled triangle are equal. Find their measures.

Step 1: Set up equation

Let each acute angle =

Sum of angles: x + x + 90° =

Step 2: Solve for x

2x + 90° = 180°

2x =

x =

Step 3: Verify

Each acute angle =

Excellent! Both acute angles are 45° each.

5. Find the value of x in these cases:

(a) If angles are 2x°, 3x°, and 4x°

2x + 3x + 4x =

9x = 180°, so x =

(b) If angles are 5x°, 3x°, and (x + 10)°

5x + 3x + (x + 10) =

9x + 10 = 180°, so 9x =

x = 170° ÷ 9 = (approximately)

Outstanding! You've mastered variable angle calculations.

Part C: Long Answer Questions (4 Marks Each)

1. Two angles of a triangle are in the ratio 3:5. The third angle is 40°. Find the angles of the triangle and name the type.

Step 1: Set up variables

Let the two angles be and

Third angle = (given)

Step 2: Apply angle sum property

3x + 5x + 40° =

8x + 40° = 180°

8x =

x =

Step 3: Find all angles

First angle = 3x = 3 × 17.5° =

Second angle = 5x = 5 × 17.5° =

Third angle =

Step 4: Classify triangle

Since all angles (52.5°, 87.5°, 40°) are < 90°, it's an triangle

Perfect! The triangle has angles 52.5°, 87.5°, 40° and is acute.

2. The sides of a triangle are 15 cm, 20 cm, and 25 cm. Check whether it is a right-angled triangle using Pythagoras theorem.

Step 1: Identify sides

Smallest side = cm, Medium side = cm, Largest side = cm

Step 2: Apply Pythagoras theorem

Check: 15² + 20² = + =

Step 3: Calculate hypotenuse square

25² =

Step 4: Verify theorem

Since 15² + 20² = 25² (625 = 625), theorem is

Therefore, it is a triangle

Excellent! The triangle satisfies Pythagoras theorem perfectly.

3. In triangle XYZ, ∠X = 70°, ∠Y = 50°. Find ∠Z and state the type of triangle.

Step 1: Apply angle sum property

∠X + ∠Y + ∠Z =

70° + 50° + ∠Z = 180°

120° + ∠Z = 180°

∠Z =

Step 2: Classify triangle

Check all angles: 70°, 50°, 60°

Since all angles are , it's an triangle

Great! ∠Z = 60° and the triangle is acute.

4. The sides of a triangle are 5 cm, 12 cm, and 13 cm. Verify Pythagoras theorem and identify the triangle.

Step 1: Apply Pythagoras theorem

Check: 5² + 12² = + =

Step 2: Check largest side

13² =

Step 3: Verify theorem

Since 5² + 12² = 13² (169 = 169), theorem is

Step 4: Identify triangle type

Since Pythagoras theorem is satisfied, it's a triangle

Outstanding! This is a perfect right triangle (5-12-13).

5. The perimeter of a triangle is 45 cm. The sides are in the ratio 5 : 6 : 7. Find the sides and verify triangle inequality.

Step 1: Set up equation

Let sides be , , and

Perimeter = 5x + 6x + 7x = cm

Step 2: Solve for x

18x = 45

x =

Step 3: Find all sides

First side = 5x = 5 × 2.5 = cm

Second side = 6x = 6 × 2.5 = cm

Third side = 7x = 7 × 2.5 = cm

Step 4: Verify triangle inequality

Check: 12.5 + 15 = > 17.5

Check: 12.5 + 17.5 = > 15

Check: 15 + 17.5 = > 12.5

Triangle inequality is

Fantastic! All sides are 12.5 cm, 15 cm, 17.5 cm and triangle inequality holds.

Test your understanding with these multiple choice questions:

For each question, click on the correct answer:

1. The sum of all angles of a triangle is:

(a) 90° (b) 120° (c) 180° (d) 360°

Super job! The angle sum property states that all angles sum to 180°.

2. In a right triangle, if one acute angle is 40°, the other is:

(a) 60° (b) 50° (c) 40° (d) 45°

Well done! 90° + 40° + 50° = 180°.

3. The side opposite to the right angle is called:

(a) Base (b) Height (c) Hypotenuse (d) Median

That's right! The hypotenuse is always the longest side in a right triangle.

4. In an equilateral triangle, each angle measures:

(a) 45° (b) 60° (c) 90° (d) 120°

Correct! 180° ÷ 3 = 60° for each angle in equilateral triangle.

5. The triangle whose all angles are less than 90° is called:

(a) Right triangle (b) Acute triangle (c) Obtuse triangle (d) Equilateral triangle

Fantastic! Acute triangles have all angles less than 90°.