Solved Exercises on Understanding Angles in Grade 4

Master angles: types of angles, measuring angles, angle relationships, and geometric applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Identifying angle types
Exercise 1
Classify these angles as acute, right, or obtuse:
a) 35°
b) 90°
c) 120°
d) 60°
Definition:

Angle types: Acute (less than 90°), Right (exactly 90°), Obtuse (greater than 90° but less than 180°)

Classification method:
  1. Compare the angle measure to 90°
  2. If measure < 90° → Acute angle
  3. If measure = 90° → Right angle
  4. If measure > 90° and < 180° → Obtuse angle
📐
Acute Angle
Less than 90°
Example: 35°, 60°
∟
📏
Right Angle
Exactly 90°
Example: 90°
⦜
⦝
Obtuse Angle
Greater than 90°
Example: 120°
Step 1: Compare angle a) 35° to 90°

Since 35° < 90°, it's an acute angle

Step 2: Compare angle b) 90° to 90°

Since 90° = 90°, it's a right angle

Step 3: Compare angle c) 120° to 90°

Since 120° > 90° and 120° < 180°, it's an obtuse angle

Step 4: Compare angle d) 60° to 90°

Since 60° < 90°, it's an acute angle

a) 35° = Acute angle
b) 90° = Right angle
c) 120° = Obtuse angle
d) 60° = Acute angle
Final answer:

Acute angles: 35°, 60° | Right angle: 90° | Obtuse angle: 120°

Applied rules:

• Acute angle: Measures between 0° and 90°

• Right angle: Measures exactly 90°

• Obtuse angle: Measures between 90° and 180°

2 Measuring angles with a protractor
Exercise 2
If angle ABC measures 45° and angle CBD measures 35°,
what is the measure of angle ABD?
Definition:

Adjacent angles: Two angles that share a common vertex and side, but do not overlap

∠ABC
45°
+
∠CBD
35°
=
∠ABD
?°
Step 1: Understand the relationship

Angles ABC and CBD are adjacent angles that together form angle ABD

Step 2: Apply the angle addition postulate

∠ABD = ∠ABC + ∠CBD

Step 3: Substitute known values

∠ABD = 45° + 35°

Step 4: Calculate the sum

∠ABD = 80°

∠ABD = 80°
Final answer:

The measure of angle ABD is 80°

Applied rules:

• Angle Addition Postulate: If point C is in the interior of angle ABD, then ∠ABD = ∠ABC + ∠CBD

• Adjacent angles: Share a common vertex and side

3 Finding missing angle measures
Exercise 3
In a triangle, if one angle measures 50° and another measures 60°,
what is the measure of the third angle?
Definition:

Triangle Angle Sum: The sum of all three angles in a triangle is always 180°

∠A = 50°
+
∠B = 60°
+
∠C = ?°
= 180°
Step 1: Apply the Triangle Angle Sum Theorem

∠A + ∠B + ∠C = 180°

Step 2: Substitute known values

50° + 60° + ∠C = 180°

Step 3: Simplify the equation

110° + ∠C = 180°

Step 4: Solve for the unknown angle

∠C = 180° - 110° = 70°

Step 5: Verify the answer

50° + 60° + 70° = 180° ✓

The third angle measures 70°
Final answer:

The measure of the third angle is 70°

Applied rules:

• Triangle Angle Sum Theorem: The sum of all angles in a triangle equals 180°

• Algebraic reasoning: Use subtraction to find the unknown angle

Angle Types, Rules and Properties
∠A + ∠B + ∠C = 180°
Triangle Angle Sum
Acute Angle
0° < θ < 90°
Less than 90°
Right Angle
θ = 90°
Exactly 90°
Obtuse Angle
90° < θ < 180°
Greater than 90°
Key definitions:

Angle: The space between two rays that meet at a common endpoint (vertex)

Vertex: The common endpoint where two rays meet to form an angle

Ray: A part of a line that starts at a point and extends infinitely in one direction

Angle measurement methods:
  1. Using a protractor: Align the baseline with one ray and read the measure where the other ray intersects
  2. Estimation: Compare to known angles (90° corner, half of 90°, etc.)
  3. Calculation: Use angle relationships to find unknown measures
Key Property: Adjacent angles that form a straight line sum to 180° (supplementary angles).
Key Property: Adjacent angles that form a right angle sum to 90° (complementary angles).
Solution: Exercises 4 to 5
4 Finding angles in geometric shapes
Exercise 4
A rectangle has four right angles. If one angle is divided into two parts,
where one part measures 35°, what is the measure of the other part?
Definition:

Rectangle: A quadrilateral with four right angles (each 90°)

∟
90° total
=
35°
+
?°
Step 1: Understand the problem

A right angle (90°) is divided into two parts. One part is 35°.

Step 2: Set up the equation

Part 1 + Part 2 = Total angle

35° + Part 2 = 90°

Step 3: Solve for the unknown part

Part 2 = 90° - 35° = 55°

Step 4: Verify the answer

35° + 55° = 90° ✓

The other part measures 55°
Final answer:

The measure of the other part is 55°

Applied rules:

• Rectangle property: All angles in a rectangle are right angles (90°)

• Angle partition: Parts of an angle sum to the whole angle

5 Real-world angle application
Exercise 5
A ladder leans against a wall forming a 75° angle with the ground.
What is the angle between the ladder and the wall?
Definition:

Perpendicular lines: Lines that intersect at a 90° angle (right angle)

∟
90° (wall-ground)
=
75°
(ground-ladder)
+
?°
(wall-ladder)
Step 1: Visualize the situation

The wall and ground form a right angle (90°). The ladder creates two angles with these surfaces.

Step 2: Apply the concept of complementary angles

Since the wall and ground are perpendicular, the two angles formed by the ladder sum to 90°

Step 3: Set up the equation

Angle with ground + Angle with wall = 90°

75° + Angle with wall = 90°

Step 4: Solve for the unknown angle

Angle with wall = 90° - 75° = 15°

Step 5: Verify the answer

75° + 15° = 90° ✓

The angle between the ladder and the wall is 15°
Final answer:

The angle between the ladder and the wall is 15°

Applied rules:

• Complementary angles: Two angles that sum to 90°

• Perpendicular lines: Form right angles (90°)

• Real-world application: Use geometric principles to solve practical problems

Angle Types, Properties and Measurement Techniques
Acute: 0° < θ < 90°
Angle Classification
Key definitions:

Acute angle: An angle that measures greater than 0° but less than 90°

Right angle: An angle that measures exactly 90°

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

Complementary angles: Two angles whose measures sum to 90°

Supplementary angles: Two angles whose measures sum to 180°

Complete methodology:
  1. Identify the problem: Determine what type of angle relationship is involved
  2. Recall relevant properties: Apply the appropriate angle theorem or rule
  3. Set up equations: Express relationships mathematically
  4. Solve systematically: Use arithmetic to find the unknown measure
  5. Verify the answer: Check if the solution makes sense in context
Tip 1: Remember that the corner of a piece of paper is a perfect 90° angle for comparison.
Tip 2: An acute angle is "cute" because it's small (less than 90°).
Tip 3: The sum of angles in any triangle is always 180°.
Tip 4: Adjacent angles forming a straight line always sum to 180°.
Common errors: Confusing acute and obtuse angles, forgetting to subtract from 90° or 180° when needed.
Exam preparation: Practice identifying angle types, memorize angle sum properties, understand complementary/supplementary relationships.
Properties to know by heart:

• Acute angle: 0° < θ < 90°

• Right angle: θ = 90°

• Obtuse angle: 90° < θ < 180°

• Complementary angles: θ₁ + θ₂ = 90°

• Supplementary angles: θ₁ + θ₂ = 180°

• Triangle angle sum: θ₁ + θ₂ + θ₃ = 180°

Questions & Answers

Question: I sometimes get confused between acute and obtuse angles. Can you explain the difference in a way that's easy to remember?

Answer: Great question! Here are some memory tricks:

  • Acute angle: Think of "acute" as meaning "cute" - cute things are small, so acute angles are small (less than 90°)
  • Obtuse angle: Think of "obtuse" as meaning "obvious" - it's obviously bigger than a right angle (between 90° and 180°)
  • Visual comparison: An acute angle looks sharp like a pencil point, while an obtuse angle looks wide like the open mouth of a crocodile

Examples: The corner of a book is 90° (right angle). An angle smaller than that is acute (like 45°). An angle larger than 90° but less than 180° is obtuse (like 120°).

Question: Why do all triangles have angles that add up to 180°? It seems like different triangles would have different sums.

Answer: That's a wonderful observation! The fact that all triangles have interior angles that sum to 180° is a fundamental geometric property. Here's why:

  • Imagine cutting out a triangle and tearing off its three corners
  • If you place these corners together at a point, they will form a straight line
  • A straight line represents 180°, which is why the three angles of any triangle always sum to 180°
  • This works for ALL triangles: equilateral (60°+60°+60°=180°), isosceles, scalene, right triangles, etc.

This property is so important that it helps us find missing angles in triangles when we know one or two of the angles!

Question: What are complementary and supplementary angles? How are they different?

Answer: These are special angle pairs based on their sum:

  • Complementary angles: Two angles that add up to 90° (form a right angle together)
  • Supplementary angles: Two angles that add up to 180° (form a straight line together)

Memory trick: "Co" in complementary sounds like "corner" (90° corner), and "Su" in supplementary sounds like "straight" (180° straight line).

Examples: 30° and 60° are complementary (30+60=90). 110° and 70° are supplementary (110+70=180).

Question: How can I estimate angle measurements without a protractor?

Answer: Estimating angles is a valuable skill! Here are reference points:

  • Right angle (90°): Corner of a piece of paper, corner of a book, L-shaped object
  • 45°: Half of a right angle (bisect a corner of paper)
  • 30°: About one-third of a right angle
  • 60°: About two-thirds of a right angle
  • 180°: Straight line, completely flat

Practice comparing unknown angles to these benchmarks. Over time, you'll develop a good intuition for angle sizes!

Question: Can angles be bigger than 180°? We learned about acute, right, and obtuse angles, but what comes after obtuse?

Answer: Yes, angles can be larger than 180°! The classification we're learning in 4th grade focuses on angles up to 180°, but here are the extended classifications:

  • Acute: 0° to 90°
  • Right: Exactly 90°
  • Obtuse: 90° to 180°
  • Straight: Exactly 180°
  • Reflex: 180° to 360° (larger than a straight line)
  • Full rotation: Exactly 360°

For now, focus on acute, right, and obtuse angles. As you advance in math, you'll learn about reflex and other larger angles!