# Lesson Explainer: Geometric Constructions: Angle Bisectors Mathematics

In this explainer, we will learn how to construct angle bisectors using rulers and compasses without protractors.

We first note that bisecting is a common term in geometry, and it means to split into two equal parts (or two congruent parts). We can extend this idea to trisecting and beyond. Therefore, bisecting an angle means to split the angle into two congruent angles.

We can define this formally as follows.

### Definition: Angle Bisector

The bisector of an angle is the line through the vertex of the angle that splits the angle into two congruent angles.

For example, if we wanted to bisect an angle of , we would need to split it into two angles, each with a measure of .

Before we discuss how to bisect an angle in general, letβs see an example of an angle bisector. We recall that a kite is a quadrilateral with two pairs of congruent adjacent sides. We can note that one of the diagonals of any kite splits the shape into two congruent triangles, here and .

Indeed, we can see that both triangles have three congruent sides; hence, the triangles are congruent by the SSS criterion. Therefore, the triangles have congruent angles, so and .

We have shown that the longer diagonal of a kite bisects two of the kiteβs internal angles. The same result and proof hold true for rhombuses and actually for both diagonals in a rhombus since they split the rhombus into four congruent triangles.

We can use a similar idea to bisect any angle using a compass and straightedge. Letβs say we have , shown below.

We can trace a circle centered at point that intersects both and at points we will label and as shown.

We note that since these are radii of the same circle. We now want to trace two circles of the same radius centered at and that intersect at a point on the same side as the angle. We can call this point and sketch the lines , , and .

We note that since they are both radii of circles of the same radius and that since they are radii of the same circle. Thus, and have all sides congruent, so by the SSS criterion. In particular, this means that their corresponding angles are congruent, and so . Hence, bisects .

It is worth noting that this same construction also works for obtuse, reflex, and straight angles. The construction for all three is the same, so we will only show the reflex and straight angle cases.

First, if were a reflex angle, we once again would trace a circle centered at to find points and , which are the points of intersection between the circle and and , as shown.

We now want to trace two circles centered at and that intersect on the same side as the angle we want to bisect. To do this, we increase the radius of our compass. We call the point of intersection as shown.

We now add in the lines , , and and note that and .

Once again, we see that and have all sides congruent, so by the SSS criterion. In particular, this means that their corresponding angles are congruent, and so . Hence, bisects the reflex angle .

It is worth noting here that the angle bisector of any angle and the angle bisector of the reflex angle are the same straight line. So, we can use either construction.

Similarly, if is a straight angle, then we trace a circle centered at that intersects and to find the points of intersection and as shown.

We then trace two circles of the same radius centered at and that intersect on the same side as the angle we want to bisect. We call this point of intersection .

We now add in the lines , , and and note that and .

We see that and have all sides congruent, so by the SSS criterion. In particular, this means that their corresponding angles are congruent, and so . Hence, bisects the straight angle .

Therefore, we can follow this process to bisect any angle.

### How To: Bisecting a Given Angle with a Compass and Straightedge

We can bisect any given angle with a compass and straightedge using the following construction:

1. Trace a circle centered at that intersects and at distinct points that we will name and respectively.
2. Trace circles of the same radius centered at and that intersect at a point on the same side as the angle we want to bisect. We will name this point .
3. Sketch line that bisects .

Letβs now see an example of using this construction to correctly identify what a given geometric construction illustrates.

### Example 1: Identifying the Construction Used to Find an Angle Bisector

What does the following figure illustrate?

1. A bisector of an angle
2. A perpendicular from a point lying outside a straight line
3. A bisector of a line segment
4. A straight line parallel to another line
5. An angle congruent to another angle

We start by recalling that tracing a circle at the vertex of an angle and then tracing circles of equal radii centered at those points of intersection such that they intersect at a point on the same side as the angle allows us to bisect the angle by connecting its vertex to the intersection of these circles. Since this is the construction illustrated, the answer is choice A: a bisector of an angle.

We can show this directly from the diagram by labeling the points as follows.

Then, we note that , as they are radii of the same circle, and , as they are radii of congruent circles. Sketching and gives us the following.

We note that and are congruent by the SSS criterion and hence .

In our next example, we will use this construction of an angle bisector to estimate the length of a section of a line in a triangle.

### Example 2: Constructing an Angle Bisector to Find a Length in a Triangle

Given that is a triangle, use a ruler and the compass to draw the triangle and bisect by the bisector that intersects at . Use the ruler to measure the length of to the nearest decimal.

1. 14.2 cm
2. 23.7 cm
3. 33.2 cm
4. 45.2 cm
5. 4.7 cm

We first need to draw triangle using a compass and ruler. We will do this by first measuring a straight line of length 5 cm and labeling the endpoints and . We then measure the compass to have a radius of 7 cm and trace a circle of radius 7 cm centered at . Then, we trace a circle of radius 8 cm centered at . The point of intersection between these two circles above is our point . This gives us the following.

We now need to bisect . We do this by first tracing a circle centered at that intersects both and . We will call these points and as shown.

We now need to trace arcs of two circles of the same radius centered at and that intersect at a point on the same side as . We will label this point as shown.

We now recall that bisects . We can extend this line to intersect and label this point as shown.

We can then measure the length of using a ruler; we get 4.7 cm to the nearest tenth of a centimetre.

In our next example, we will estimate the perimeter of a triangle constructed using angle bisectors.

### Example 3: Constructing an Angle Bisector to Find the Perimeter of a Triangle

Given that is a triangle, use a ruler and a compass to draw the triangle shown and bisect and by the bisectors and that intersect at . Use the ruler to measure the perimeter of the triangle to the nearest decimal.

1. 19.1 cm
2. 17.6 cm
3. 18.4 cm
4. 16.9 cm
5. 17.9 cm

We first need to sketch triangle using a ruler and compass. We can do this by first drawing a straight line of length 9 cm and labeling the endpoints and . We then set the compass to have a radius of 6 cm and trace a circle of radius 6 cm centered at . Next, we set the radius of the compass to be 5 cm and trace a circle of radius 5 cm centered at . The point of intersection between these circles above is point .

We now need to find point by bisecting both and using a compass and straightedge.

Letβs start by bisecting . We trace a circle centered at that intersects and at distinct points and as shown.

We then trace the arcs of two circles of equal radii centered at and that intersect on the same side as . We label this intersection .

We then recall that bisects .

We follow the same process to bisect . We trace a circle at to find points on and , which we label and .

We then trace congruent circles centered at and and mark the point of intersection on the same side of . If this point is labeled , we know that bisects .

The point of intersection between the bisectors is then point .

We need to use a ruler to measure the perimeter of triangle to the nearest tenth of a centimetre. We recall that this is the sum of its side lengths, so it is equal to . We note that . If we measure the sides with a ruler, we get and . Adding these together, we get that the perimeter is 18.4 cm to one decimal place.

In the previous example we found the point of intersection between two of the internal angle bisectors of a triangle. Although it is beyond the scope of the explainer to prove the following result, it is worth noting that all of the internal angle bisectors of any triangle intersect at a single point. We can write this formally as follows.

### Property: The Angle Bisectors of the Internal Angles of a Triangle Are Concurrent

The internal angle bisectors of any triangle intersect at a single point.

We can see an example of this in the example above if we also draw the angle bisector of .

Letβs now see an example of using angle bisection and geometric constructions to estimate a length in a trapezoid.

### Example 4: Constructing a Right Trapezoid and Using Angle Bisectors to Find a Missing Length

Given that is a trapezoid, use a ruler and a compass to draw the trapezoid and bisect by the bisector that intersects at . Use the ruler to measure the length of to the nearest decimal.

1. 11.6 cm
2. 12.0 cm
3. 13.0 cm
4. 10.9 cm
5. 14.1 cm

We first need to sketch trapezoid using a ruler and compass. We can do this by first drawing a straight line of length 10 cm and labeling the endpoints and . We can construct a right angle by bisecting a straight angle at . We do this by tracing a circle centered at that intersects at two points, say and . Then, we trace circles of equal radii centered at and that intersect. The points of intersection of these circles bisect the straight angle into two right angles.

If we measure this line to be 12 cm, then we find point .

We now note that is a triangle, so we can construct this using a compass and straightedge since we already have points and . We set our compass to have a radius of 13 cm and trace a circle of radius 13 cm centered at . Then, we set our compass to have a radius of 5 cm and trace a circle of radius 5 cm centered at . The point of intersection of these two circles on the correct side of is then point .

We now follow the bisection process to bisect . We trace a circle centered at and label the points of intersection with and Β  and . Then, we trace circles of equal radii centered at and such that they intersect at a point, , on the same side as as shown.

We then know that is the bisector of . We can sketch this line onto the diagram to find point .

We can then measure the length of using our ruler. To the nearest millimetre, we get 12.0 cm.

In our final example, we will see an example of an interesting geometric property involving angle bisectors in triangles.

### Example 5: Observing Symmetries with Angle Bisectors in an Isosceles Triangle

Use a ruler and a compass to draw a triangle , where and , then bisect and by two bisectors that intersect at .

1. Is equal to ?
2. Is equal to ?

Part 1

To construct triangle , we can start by drawing a line of length 6 cm and labeling the endpoints and . We then trace circles of radius 7 cm centered at and . We can label either point of intersection between these circles .

We now need to bisect and . Letβs start by bisecting . We trace a circle centered at that intersects and at points and as shown.

We then trace circles of equal radii centered at and that intersect on the same side as . We label this intersection point and recall that bisects .

We follow the same process to bisect . It is worth noting that triangle is isosceles, so . This means that the bisected angles will have equal measure. We get the following.

We could then measure the lengths of and and see that they are the same. However, we can prove that they are equal by noting that , so triangle is isosceles.

Hence, we can say that is equal to .

Part 2

We can follow the same process to bisect , or we can recall that the bisections of the internal angles of a triangle meet at a point. Either way, we note that bisects .

We can measure to see that it is approximately 4.5 cm, whereas is approximately 3.5 cm. We can conclude that is not equal to .

In the previous example, we noted that the distances from the vertex to the intersection point of the bisectors are equal if the angles they bisect are equal. We can use this process to prove that the lengths of the bisectors in an equilateral triangle will all be equal.

Letβs finish by recapping some of the important points from this explainer.

### Key Points

• The bisector of an angle is the line through the vertex of the angle that splits the angle into two congruent angles.
• One of the diagonals of a kite bisects two of the kiteβs internal angles.
• We can bisect any given angle with a compass and straightedge using the following construction:
1. Trace a circle centered at that intersects and at distinct points that we will name and respectively.
2. Trace circles of the same radius centered at and that intersect at a point on the same side as the angle we want to bisect. We will name this point .
3. Sketch line that bisects .
• The angle bisectors of the internal angles in a triangle intersect at a point.