How to construct a 45 degree angle

How to Construct a 45 Degree Angle is a simple geometry topic for students. How To Construct 45 Degree Angle helps us to draw an exact 45 degree angle using easy steps and basic tools. This topic is important because it builds a strong base in maths and improves drawing skills.

What Is a 45 Degree Angle?

A 45 degree angle is an acute angle that measures exactly 45°. It is half of a right angle (90°) and is commonly constructed by bisecting a 90° angle using a compass and ruler.

Instruments Required to Construct a 45 Degree Angle

• A pencil

• A ruler or straightedge (for drawing straight lines only)

• A pair of compasses that can hold its width steady

How to Construct a 45 Degree Angle Step by Step

Step 1: Draw a ray BA.

Step 2: With B as the vertex, construct ∠ABE = 90° using a compass and ruler.

Step 3: Place the compass on point F, where the arc intersects BA, and draw an arc inside the 90° angle. Without changing the compass width, place the compass on point G, where the arc intersects AC, and draw another arc so that the two arcs intersect at point H.

Step 4: Join point H to vertex B. Line AF bisects ∠BAC, forming ∠ABH = 45°. This is the required 45° angle.

Constructing Related Angles From a 45 Degree Angle

Once you know how to construct a 45° angle, you can use it to create several other common angles.

• 22.5°: Bisect the 45° angle. Half of 45° is 22.5°.

• 90°: Construct a 90° angle directly before bisecting it.

• 135°: Construct a 45° angle and extend one arm to form a straight line. Since a straight angle is 180°, the supplementary angle is 135° (180° − 45°).

• 180°: Extend a ray in the opposite direction to form a straight angle.

• 225° (Reflex Angle): Construct a 45° angle and measure the remaining angle around the point. Since a full angle is 360°, the reflex angle is 225° (360° − 135°).

Practice Questions on Construct a 45 Degree Angle

1. Which angle should you construct first before creating a 45° angle using a compass and ruler?

2. What geometric construction is used to divide a 90° angle into two equal angles?

3. Draw a ray AB and construct a 45° angle at point A using only a ruler and compass.

4. If you bisect a 90° angle, what is the measure of each resulting angle?

5. Which type of angle is a 45° angle acute, right, obtuse, or straight?

6. Using a constructed 45° angle, explain how you can construct a 135° angle.

7. True or False: A 45° angle is half of a right angle.

8. Which of the following constructions results in a 45° angle?

• a) Bisecting a 60° angle

• b) Bisecting a 90° angle

• c) Bisecting a 180° angle

• d) Constructing two 30° angles

9. Construct a 45° angle, then bisect it. What is the measure of the new angle formed?

10. Explain why constructing a 90° angle is an important first step in constructing a 45° angle.

Answers:

1. Construct a 90° angle first.

2. An angle bisector is used to divide a 90° angle into two equal parts.

3. Draw ray AB, construct a 90° angle at A, and then bisect it to obtain a 45° angle.

4. 45°

5. Acute angle

6. Construct a 90° angle and add a 45° angle next to it. Together they form 135°.

7. True

8. b) Bisecting a 90° angle

9. 22.5°

10. A 45° angle is obtained by bisecting a 90° angle, so constructing a right angle is the first and essential step.