# Angle Bisector

🏆Practice bisector

A bisector is a line segment that passes through the vertex of an angle and divides it into two equal angles.

The bisector can appear in a triangle, parallelogram, rhombus and in other geometric figures.

For example, a bisector that passes through an angle of $120°$ degrees will create two angles of $60°$ degrees each.

## Test yourself on bisector!

Given:

$$∢\text{ABC}=90$$

$$∢DBC=45$$

Is BD a bisector?

## Examples of Bisector

### Example 1: Bisector of an equilateral triangle

In the following example, there is an equilateral triangle $\triangle ABC$.

Note the bisector $BD$ that goes from point $B$ to point $D$ and divides the angle $∡ABC$ into $2$.

That is, angle$∡ABD = 30°$ and angle $∡CBD = 30°$ therefore are equal to each other.

Bisector within an equilateral triangle

### Example 2: Bisector inside a square

In the following example, a square $ABCD$ is presented.

Note that the bisector $BD$ that goes from point $D$ to point $B$ and divides the angle $∡ADC$ into $2$.

That is, the angle $∡ADB = 45°$ and the angle $∡CDB = 45°$.

Bisector inside a square

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### Example 3: Bisector within a Rhombus

In the following example, a rhombus $ACBD$ is presented.

Note that the bisector $CD$ that goes from point $C$ to point $D$ and divides the angle ∡ACB into $2$.

That is, the angle $∡ACD = 45°$ and the angle $∡DCB = 45°$ which are equal.

Bisector inside a Rhombus

### Example 4: Bisector in a graph with parallel lines

In this example, a graph is presented with two parallel lines $A$ and $B$.

Note that the bisector $DE$ that goes from point $D$ to point $E$ and divides the angle $∡ADF$ into $2$.

That is, the angle $∡ADE = 25°$ and the angle $∡EDF = 25°$ which are equal.

Bisector in a graph with parallel lines

### Example 5: Bisector inside acircle

Bisector inside a circle

The line $DB$ intersects with the line $AC$ at point $O$ and forms the angle $∡AOD = 90°$

The bisector $FO$ splits the angle $∡AOD = 90°$ into $2$ equal angles of $45$ degrees.

Having that the angle $∡AOF = 45°$ and the angle $∡FOD = 45°$ are equal.

If you are interested in learning more about other angle topics, you can enter one of the following articles:

In the blog of Tutorela you will find a variety of articles about mathematics.

Do you know what the answer is?

## Exercises from the previous examples

### Exercise 1: (Bisector within an equilateral triangle)

In the following example, there is an equilateral triangle $ABC$.

Bisector within an equilateral triangle

A. Try to draw a new bisector, that divides the angle $∡ABD$ into $2$.

B. Specify the size of the two newly formed angles

Solution to exercise 1:

A. The new bisector $BE$ splits the angle $∡ABD$ into $2$ equal angles of $15°$ degrees each.

B. The size of the formed angles is $∡ABE = 15°$ which is equal to angle $∡EBD = 15°$.

### Exercise 2: (Bisector inside a square)

In the following example, a square $ABCD$ is presented.

A. The angle $∡ABC$ is equal to the angle of $∡ADC$. Can it be said that $BD$ serves as the bisector of the angle $∡ABC$?

Bisector inside a square

Solution to exercise 2:

The line $BD$ created $2$ points where the angle was divided into $2$ equal angles.

Therefore, $DB$ is a bisector of the two angles $∡ADC$ and $∡ABC$

### Exercise 3: (Bisector within arhombus )

In the following example, a rhombus $ACBD$ is presented.

Note the bisector $CD$ that goes from point $C$ to point $D$ and divides the angle $∡ACB$ by $2$.

If a bisector is drawn between points $A$ and $B$, and the intersection point between the two lines in the center of the rhombus is $O$.

What type of triangle would triangle $∡AOC$ be?

Solution to exercise 3:

If we draw the bisector from point $A$ to point $B$ when the intersection point between the lines $AB$ and $CD$ will be $O$.

And note that the sum of the angles of a triangle must equal $180°$.

Therefore the triangle $AOC$ will be a right triangle.

This is because:

The angle $∡CAO = 30°$ and the angle $∡OCA = 60°$.

And the sum of the angles of a triangle is equal to $180°$.

Therefore $180° - 90° = 90°$.

And a triangle whose one of its angles is equal to $90°$ is a right triangle.

### Exercise 4: (Bisector of an Angle)

In this example, a graph is presented with two parallel lines $A$ and $B$.

Note that the bisector $DE$ that goes from point $D$ to point $E$ divides the angle $∡ADF$ by $2$.

That is, in an angle $∡ADE = 25°$ and in an angle $∡EDF = 25°$ which are equal.

In this exercise, we will ask you to draw another line parallel to line $ED$.

Solution to exercise 4

Keep in mind that line $A$ and line $B$ are parallel lines, and are intersected by line $CF$.

Drawing a line $GH$ that is the bisector of angle $∡BKF$.

And to be sure that line $GH$ is parallel to line $ED$, it is enough to see that the angle of $∡GKF$ is equal to $25°$.

### Exercise 5

If a line is drawn between point $A$ and point $B$, will the new line created $AB$ be parallel to the line $FE$?

Solution to exercise 5:

Given that the two lines $AC$ and $DB$ intersect perpendicularly forming a $90°$ angle between them.

It can be concluded that if we draw a line between the 4 points $ABCD$ we will form a square that is divided in the middle by the line $FE$.

Then the line $FE$ forms a rectangle $ABHG$.

One of the properties of a rectangle is that the opposite sides of the rectangle are parallel to each other.

Therefore, the line $AB$ is parallel to the line $FE$.

Do you think you will be able to solve it?

## Questions on the topic

What is a bisector?

It is a line segment that passes through the vertex of an angle and divides it into two equal parts.

What is known as a bisector in a triangle?

It is the line segment that divides an interior angle of the triangle into two equal angles.

How many bisectors does a triangle have?

Remember that a triangle has three vertices, therefore, it has three bisectors.

What is the measure of the equal angles generated by the bisectors of an equilateral triangle?

Recalling that the measure of the internal angles of any equilateral triangle is $60°$, then the bisectors will divide these angles into two equal angles of $30°$ each.

## Examples with solutions for Bisector

### Exercise #1

Which of the following figures has a bisector?

### Step-by-Step Solution

The answer is C because the angle bisector divides the angle into two equal angles. In diagram C, the angle bisector divides the right angle, which is equal to 90 degrees, into 2 angles that are equal to each other. $45=45$

### Exercise #2

BD is a bisector.

What is the size of angle ABC?

### Step-by-Step Solution

Since we are given that the value of angle DBC is 65 degrees, and we know that the angle bisector divides angle ABC into two equal angles, we can calculate the value of angle ABC:

$65+65=130$

130

### Exercise #3

Calculate angle $\alpha$ given that it is a bisector.

### Step-by-Step Solution

Since an angle bisector divides the angle into two equal angles, and we are given that one angle is equal to 60 degrees. Angle $\alpha$ is also equal to 60 degrees

60

### Exercise #4

ABCD is a square.

$∢\text{ABC}=\text{?}$

### Step-by-Step Solution

Since in a square all angles are equal to 90 degrees, and BC bisects an angle, we can calculate angle ABC:

$90:2=45$

45

### Exercise #5

ABCD is a deltoid.

$∢DAC=\text{?}$

### Step-by-Step Solution

As we know that ABCD is a deltoid, and AC is the bisector of an angle and therefore:

$BAC=CAD=2X$

Now we focus on the triangle BAD and calculate the sum of the angles since we know that the sum of the angles in a triangle is 180 degrees:

$2X+2X+2X+60=180$

$6X+60=180$

$180-60=6X$

$120=6X$

We divide the two sections by 6:$\frac{120}{6}=\frac{6x}{6}$

$20=x$

Now we can calculate the angle DAC:

$20\times2=40$