Solved Exercises on Types of Angle Triangles in Grade 4

Master types of angle triangles: acute, right, and obtuse triangles through these 5 detailed exercises with visual diagrams.

Solution: Exercises 1 to 3
1 Acute Triangle
Exercise 1
A triangle has angles measuring 60°, 70°, and 50°. What type of triangle is this? Explain your answer.
Definition:

Acute triangle: A triangle where ALL three angles are less than 90°

Method:
  1. Identify all three angles of the triangle
  2. Check if each angle is less than 90°
  3. If all angles are less than 90°, it's an acute triangle
60° 70° 50°
Given
60°, 70°, 50°
Check
All < 90°
Result
Acute triangle
Step 1: List all angles

The triangle has angles: 60°, 70°, and 50°

Step 2: Compare each angle to 90°

60° < 90° ✓, 70° < 90° ✓, 50° < 90° ✓

Step 3: Apply the definition

Since ALL angles are less than 90°, this is an acute triangle

Step 4: Verify the sum

60° + 70° + 50° = 180° ✓ (Triangle angle sum property)

This is an acute triangle
Final answer:

This is an acute triangle because all three angles are less than 90°

Applied rules:

• Acute triangle: All angles less than 90°

• Triangle angle sum: All three angles sum to 180°

• Classification: Based on largest angle in the triangle

2 Right Triangle
Exercise 2
A triangle has angles measuring 90°, 45°, and 45°. What type of triangle is this? Explain your answer.
Definition:

Right triangle: A triangle that has ONE angle equal to exactly 90°

90° 45° 45°
Given
90°, 45°, 45°
Check
Has 90° angle
Result
Right triangle
Step 1: List all angles

The triangle has angles: 90°, 45°, and 45°

Step 2: Look for the 90° angle

One of the angles is exactly 90°

Step 3: Apply the definition

Since it has one angle equal to 90°, this is a right triangle

Step 4: Verify the sum

90° + 45° + 45° = 180° ✓ (Triangle angle sum property)

This is a right triangle
Final answer:

This is a right triangle because it has one angle equal to 90°

Applied rules:

• Right triangle: Has exactly one 90° angle

• Triangle angle sum: All three angles sum to 180°

• Special case: This is also an isosceles right triangle

3 Obtuse Triangle
Exercise 3
A triangle has angles measuring 100°, 40°, and 40°. What type of triangle is this? Explain your answer.
Definition:

Obtuse triangle: A triangle that has ONE angle greater than 90°

100° 40° 40°
Given
100°, 40°, 40°
Check
Has >90° angle
Result
Obtuse triangle
Step 1: List all angles

The triangle has angles: 100°, 40°, and 40°

Step 2: Identify the largest angle

The largest angle is 100°, which is greater than 90°

Step 3: Apply the definition

Since it has one angle greater than 90°, this is an obtuse triangle

Step 4: Verify the sum

100° + 40° + 40° = 180° ✓ (Triangle angle sum property)

This is an obtuse triangle
Final answer:

This is an obtuse triangle because it has one angle greater than 90°

Applied rules:

• Obtuse triangle: Has exactly one angle greater than 90°

• Triangle angle sum: All three angles sum to 180°

• Impossibility: Cannot have more than one obtuse angle

Triangle Classification by Angles
∠A + ∠B + ∠C = 180°
Triangle Angle Sum
Acute Triangle
All angles < 90°
All angles are acute
Right Triangle
One angle = 90°
Has one right angle
Obtuse Triangle
One angle > 90°
Has one obtuse angle
Triangle angle sum: The sum of all three angles in any triangle is always 180°.
Impossible combinations: A triangle cannot have two right angles or two obtuse angles.
Maximum one: A triangle can have at most one right or obtuse angle.
Solution: Exercises 4 to 5
4 Triangle Identification
Exercise 4
In triangle ABC, angle A = 30° and angle B = 60°. What is the measure of angle C? What type of triangle is this?
Definition:

Triangle angle sum property: The sum of all three angles in a triangle is always 180°

30° 60° ?
Given
∠A = 30°, ∠B = 60°
Sum
30° + 60° = 90°
Find ∠C
∠C = 180° - 90° = 90°
Step 1: Apply triangle angle sum property

∠A + ∠B + ∠C = 180°

Step 2: Substitute known values

30° + 60° + ∠C = 180°

Step 3: Solve for ∠C

90° + ∠C = 180°

∠C = 180° - 90° = 90°

Step 4: Classify the triangle

Since one angle is exactly 90°, this is a right triangle

∠C = 90°, This is a right triangle
Final answer:

Angle C measures 90°, so this is a right triangle

Applied rules:

• Triangle angle sum: ∠A + ∠B + ∠C = 180°

• Right triangle: Has one angle equal to 90°

• Algebra: Solve for unknown angle using addition/subtraction

5 Triangle Angle Relationship
Exercise 5
In triangle XYZ, angle X is twice as large as angle Y, and angle Z is 30°. What are the measures of all angles? What type of triangle is this?
Definition:

Algebraic relationships: Use variables to represent unknown angles and set up equations based on given relationships

2x° x° 30°
Setup
∠Y = x, ∠X = 2x, ∠Z = 30°
Equation
2x + x + 30° = 180°
Solve
3x = 150°, x = 50°
Step 1: Define variables

Let ∠Y = x°, then ∠X = 2x° (since it's twice as large)

∠Z = 30° (given)

Step 2: Apply triangle angle sum property

∠X + ∠Y + ∠Z = 180°

2x + x + 30° = 180°

Step 3: Solve the equation

3x + 30° = 180°

3x = 150°

x = 50°

Step 4: Find all angles

∠Y = x° = 50°

∠X = 2x° = 2 × 50° = 100°

∠Z = 30° (given)

Step 5: Classify the triangle

Since one angle (100°) is greater than 90°, this is an obtuse triangle

∠X = 100°, ∠Y = 50°, ∠Z = 30°, This is an obtuse triangle
Final answer:

The angles are 100°, 50°, and 30°, so this is an obtuse triangle

Applied rules:

• Triangle angle sum: ∠X + ∠Y + ∠Z = 180°

• Algebra: Use variables to represent unknowns

• Obtuse triangle: Has one angle greater than 90°

Detailed Summary: Types of Angle Triangles
∠A + ∠B + ∠C = 180°
Triangle Angle Sum Property
Key definitions:

Acute triangle: A triangle where ALL three angles are less than 90° (all angles are acute angles)

Right triangle: A triangle that has exactly ONE angle equal to 90° (one right angle)

Obtuse triangle: A triangle that has exactly ONE angle greater than 90° (one obtuse angle)

Acute angle: An angle that measures less than 90°

Right angle: An angle that measures exactly 90°

Obtuse angle: An angle that measures greater than 90° but less than 180°

Triangle classification method:
  1. Measure all three angles: Determine the measure of each angle in the triangle
  2. Compare to 90°: Check each angle against 90° to see if it's less than, equal to, or greater than 90°
  3. Count special angles: Count how many angles are 90° or greater than 90°
  4. Classify: Based on the count, determine if it's acute, right, or obtuse
Tip 1: Remember the mnemonic: "All Acute, One Right, One Obtuse" - this tells you how many angles of each type define each triangle category.
Tip 2: A triangle can only have one right angle or one obtuse angle at most, because the sum of angles in a triangle is always 180°.
Tip 3: Always verify that your angle measurements add up to 180° to check your work.
Tip 4: If you know two angles, you can always find the third using: Third angle = 180° - (first angle + second angle).
Common errors: Misidentifying the largest angle, forgetting that triangle angles sum to 180°, thinking a triangle can have two obtuse angles.
Memory aid: "A ROA" - Acute (All less than 90°), Right (One equals 90°), Obtuse (One greater than 90°).
Essential facts to remember:

• Triangle angle sum: ∠A + ∠B + ∠C = 180°

• Acute triangle: All angles < 90°

• Right triangle: One angle = 90°

• Obtuse triangle: One angle > 90°

• Impossible: A triangle cannot have two right angles or two obtuse angles

Visual Guide: Triangle Classification by Angles
Acute Triangle
70° 60° 50°
All angles < 90°
Right Triangle
90° 45° 45°
One angle = 90°
Obtuse Triangle
110° 35° 35°
One angle > 90°
Angle Sum
60° 70° 50°
60° + 70° + 50° = 180°

Summary: Understanding triangle classification helps identify geometric properties:

  • Acute triangles have all angles less than 90°, making them appear "sharp" and pointed
  • Right triangles have exactly one 90° angle, forming a perfect corner
  • Obtuse triangles have exactly one angle greater than 90°, giving them a "blunt" appearance
  • All triangles follow the fundamental rule that their three angles sum to 180°

Questions & Answers

Question: Can a triangle have two right angles? Why or why not?

Answer: No, a triangle cannot have two right angles! Here's why:

Remember the fundamental rule: the sum of all three angles in any triangle must equal 180°.

If a triangle had two right angles, that would mean two angles equal 90° each:

90° + 90° = 180°

That would leave no degrees left for the third angle:

180° - 180° = 0°

A triangle needs three angles to exist, so having a 0° angle would mean there's no actual triangle!

This is why a triangle can have at most one right angle (making it a right triangle) or one obtuse angle (making it an obtuse triangle), but never two of either.

Question: If a triangle has one angle that is 100°, what can you say about the other two angles?

Answer: If one angle is 100°, then the triangle is definitely an obtuse triangle (since 100° > 90°).

Now, let's figure out what the other two angles must be:

Total for all three angles: 180°

One angle: 100°

Sum of the other two angles: 180° - 100° = 80°

Since the other two angles together equal 80°, each of them must be less than 80°, which means both are acute angles (less than 90°).

So in an obtuse triangle, the obtuse angle takes up more than half of the total angle measure, leaving less than 90° to be shared between the other two angles. Both remaining angles must be acute!

Question: How can I remember which type of triangle is which? They all seem similar to me.

Answer: Here's a helpful way to remember the three types of triangles based on their angles:

  1. Acute Triangle: "ALL ACUTE" - Think of it as an "A" for "All" - all three angles are acute (less than 90°)
  2. Right Triangle: "ONE RIGHT" - Think of it as an "R" for "Right" - exactly one angle is a right angle (exactly 90°)
  3. Obtuse Triangle: "ONE OBTUSE" - Think of it as an "O" for "Obtuse" - exactly one angle is obtuse (greater than 90°)

Another memory trick:

Acute = "A" for "All" (all angles less than 90°)

Right = "R" for "Right" (one right angle)

Obtuse = "O" for "Only one" (only one obtuse angle)

Also, think about what each word means: "acute" means sharp (small angles), "right" means correct/straight (90° corner), and "obtuse" means blunt (large angle).

Question: What happens if I measure the angles of a triangle and they don't add up to 180°?

Answer: If your angle measurements don't add up to 180°, then one of two things happened:

  1. You made a measurement error: This is the most common cause. Go back and remeasure each angle carefully.
  2. It's not really a triangle: Make sure you're measuring the three interior angles of an actual triangle shape.

The sum of the three interior angles of ANY triangle is always exactly 180° - this is a mathematical law that never changes!

Here's what to do if your angles don't add to 180°:

  • Check your measurements again - were you careful to align the protractor properly?
  • Look for calculation errors when adding the angles
  • Make sure you're measuring interior angles, not exterior angles
  • If you know two angles, calculate the third using: Third angle = 180° - (first angle + second angle)

This angle sum property is a great way to check your work!

Question: Can an equilateral triangle be anything other than acute? Like right or obtuse?

Answer: No, an equilateral triangle can NEVER be right or obtuse! It will always be acute.

Here's why:

An equilateral triangle has all three sides equal, which means all three angles are also equal (this is a property of equilateral triangles).

Since all three angles are equal and they must sum to 180°:

Each angle = 180° ÷ 3 = 60°

Since 60° is less than 90°, all three angles are acute angles.

Therefore, every equilateral triangle is automatically an acute triangle. This is the only type of triangle that is always acute!

In fact, equilateral triangles are quite special because they are not only acute triangles, but also equiangular triangles (all angles equal) and regular polygons (all sides and angles equal).