Step-by-Step Method to Bisect Any Angle with Compass and Ruler
Angle bisectors can be seen quite often in our daily life. They are defined as a ray or a line that splits an angle into two equal halves. It is important to know their use and methods to construct them.
What Are Angle Bisectors and Their Use
Before we learn what angle bisectors are, Let’s look at what an angle is and what bisectors are. An angle is a combination of two rays having a common endpoint. This common endpoint is known as the vertex of the angle and the spacing between the rays is called the measure of an angle. A bisector is simply a line (a ray in our case) that divides the angle into two equal parts.
Scissor Showing an Angle
The scissor pivot is called the angle's vertex. The line passing through the vertex divides the scissor into two symmetrical portions and is called the bisector. If a bisector bisects an angle, it is called an angle bisector.
Real-world applications for angle bisectors include quilting, baking, and architecture. Numerous games, including rugby, football, and pool, use angle bisectors.
How to Draw an Angle Bisector?
Let’s take a fun approach in trying to make an angle bisector. Take a paper and make an angle on it. Now fold the paper so that the first line completely overlaps the second line. Make a crease. Now unfold the paper and draw a line along the crease.
Angle Bisector Using Paper Fold
What are the Properties of Angle Bisectors?
It is essential to know the properties of angle bisectors. Since we cannot call all lines angle bisectors, we need to know a few distinct properties of angle bisectors. These are:
It must pass through the vertex of the angle.
It must be equally inclined[spaced] between the rays of the angle.
The angle bisector is also the line of symmetry of the angle.
In the case of a straight line, the angle bisector is perpendicular to the line.
For example, the first image shows a slice of pizza unevenly portioned by the line. Thus the line is not the angle bisector. However, the pizza is evenly portioned by the line in the second image. Hence it can be called an angle bisector.
Pizza Showing Angle Bisector
Construction of Angle Bisectors
Construction of Angle Bisectors using Protractor
We have to bisect ∠PQR. We measure the angle using a protractor. Let’s say the measure is 110°. The half of 110° is 55°. So we make an angle of measure 55°, called ∠PQS. QS is the required angle bisector. And ∠SQR \[ = \] ∠PQS \[ = \] 55°.
Drawing Angle Bisector using a Protractor
Let us now move on to learn how to construct an angle bisector using a compass.
Construction of Angle Bisectors using a compass
We have to bisect ∠ABC. To do that,
We first place the needle end of the compass on vertex B.
Draw an arc, letting the arc intersect AB and BC at R and P, respectively.
Now we place the needle end of the compass on P and draw a big enough arc.
Do the same at point R and let these two arcs meet at point S.
Join BS.
We have our required angle bisector for ∠ABC.
Constructing Angle Bisector using Compass.
Great! You have now understood what angle bisectors are, their main properties, and their construction. Now, let us take a look at some solved examples below.
Solved Examples
1. Given an ∠ABC \[ = \] 40°. Find the measure of the angle after an angle bisector bisects it.
Using the property that the angle bisector divides the angle into two equal parts, angle bisectors, we can find the measure of the angle. Let this measure be x. We have \[{\rm{x + x}} = {40^ \circ }\]. This gives \[{\rm{x}} = {20^ \circ }\]. Hence, the measure of the angle after its bisection is 20°.
2. Consider an angle ∠AOB. Write the steps to construct its angle bisector using a compass.
Steps to construct an angle bisector using a compass. First, draw the angle using a protractor.
Steps to Construct an Angle Bisector using a Compass
1. Make an arc of any radius using the compass with the needle on point O. Let’s name the points where the arc cuts the rays of angle AOB
Steps to Construct an Angle Bisector using a Compass
2. Now with the needle on Point C, draw an arc with a greater radius. Now put the needle on Point D and draw an arc of the same length. And let these arcs intersect at Point E.
Steps to Construct an Angle Bisector using a Compass
3. Join OE. This completes the construction of the angle bisector.
Steps to Construct an Angle Bisector using a Compass
Conclusion
The angle bisectors are very important in proving complicated proofs. Great! We now know how to identify angles, how to construct angle bisectors, and their construction and mathematical implementation of angle bisectors.
FAQs on How to Bisect an Angle: A Complete Student Guide
1. What is the basic definition of an angle bisector?
An angle bisector is a ray or a line that divides a given angle into two smaller angles of equal measure. For instance, if you bisect a 60° angle, you create two 30° angles. The fundamental purpose of a bisector is to cut something into two perfectly equal halves.
2. What is the primary importance of understanding angle bisectors in geometry?
Understanding angle bisectors is crucial as they are a fundamental concept in geometry, especially for triangles. They are used to:
- Find the incenter of a triangle, which is the center of its inscribed circle.
- Prove theorems related to triangle similarity and congruence.
- Solve for unknown angles and side lengths using the Angle Bisector Theorem.
3. How do you construct an angle bisector using only a compass and a ruler?
To construct an angle bisector, follow these steps:
- Place the compass point on the angle's vertex and draw an arc that intersects both rays of the angle.
- From each of those intersection points, draw two more arcs of the same radius inside the angle so they intersect each other.
- Draw a straight line from the vertex to the point where the two inner arcs cross.
4. What is a real-world example of an angle bisector?
A classic real-world example can be seen in the structure of a kite. The main vertical support of a kite often acts as an angle bisector for the pair of angles at the top and bottom vertices. This ensures the kite is symmetrical, which is essential for stable flight.
5. What is the main difference between an angle bisector and a perpendicular bisector?
The main difference lies in what they divide. An angle bisector divides an angle into two equal angles. In contrast, a perpendicular bisector divides a line segment into two equal lengths at a right angle (90°). One acts on angles, the other on line segments.
6. What is the significance of the point where the three angle bisectors of a triangle meet?
The point where the three angle bisectors of a triangle intersect is called the incenter. This point is unique because it is equidistant from all three sides of the triangle. Because of this property, the incenter is the center of the triangle's incircle—the largest possible circle that can be drawn inside the triangle.
7. Can an angle have more than one bisector?
Yes, an angle formed by two intersecting lines technically has two bisectors. There is an internal angle bisector that divides the interior angle (less than 180°), and an external angle bisector that divides the exterior (reflex) angle. These two bisector lines are always perpendicular to each other.
8. What is the Angle Bisector Theorem and why is it useful?
The Angle Bisector Theorem states that if an interior angle of a triangle is bisected, the bisector divides the opposite side into two segments that are proportional to the other two sides of the triangle. This theorem is extremely useful for calculating unknown side lengths in a triangle without needing to know any angles.
9. Is there a relationship between the angle bisector and the sides of an isosceles triangle?
Yes, there is a special relationship. In an isosceles triangle, the angle bisector of the vertex angle (the angle between the two equal sides) is also the perpendicular bisector of the opposite base. This means it not only splits the angle equally but also splits the base into two equal lengths at a 90-degree angle and acts as the triangle's altitude and median.
