Angle Bisector

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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° 120° degrees will create two angles of 60° 60° degrees each.

A1  -  Bisector

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Test yourself on bisector!

einstein

\( ∢\text{ABD}=90 \)

CB bisects \( \sphericalangle\text{ABD} \).

\( \sphericalangle\text{CBD}=\alpha \)

Calculate the size of \( ∢ABC \).

AAABBBDDDCCCα

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Examples of Bisector

Example 1: Bisector of an equilateral triangle

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

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

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

Bisector within an equilateral triangle

A2 - Bisector inside an equilateral triangle


Example 2: Bisector inside a square

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

Note that the bisector BD BD that goes from point D D to point B B  and divides the angle ADC ∡ADC into 22 .

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

Bisector inside a square

A3 - Bisector inside a square


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Example 3: Bisector within a Rhombus

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

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

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

Bisector inside a Rhombus

A4 -  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 A and B B .

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

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

Bisector in a graph with parallel lines

A5 - Bisector in a graph with parallel lines


Example 5: Bisector inside a circle

Bisector inside a circle

A6 -  Bisector inside a circle

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

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

Having that the angle AOF=45° ∡AOF = 45° and the angle FOD=45° ∡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 ABC .

Bisector within an equilateral triangle

A7 - Bisector within an equilateral triangle

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

B. Specify the size of the two newly formed angles

Solution to exercise 1:

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

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

A -Solution to exercise 1


Exercise 2: (Bisector inside a square)

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

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

Bisector inside a square

A8 - Bisector inside a square

Solution to exercise 2:

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

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


Exercise 3: (Bisector within a rhombus )

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

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

Clean exercise solution

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

What type of triangle would triangle AOC ∡AOC be?

A9 - Solution to exercise 3

Solution to exercise 3:

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

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

Therefore the triangle AOC AOC will be a right triangle.

This is because:

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

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

Therefore 180°90°=90° 180° - 90° = 90° .

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


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Exercise 4: (Bisector of an Angle)

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

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

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

Bisector in a graph with parallel lines 22

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

Solution to exercise 4

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

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

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


Exercise 5

Bisector inside a circle1

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

Solution to exercise 5:

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

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

Then the line FE FE forms a rectangle ABHG 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 AB is parallel to the line FE FE.

Exercise solution 5


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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° 60°, then the bisectors will divide these angles into two equal angles of 30° 30° each.


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Examples with solutions for Bisector

Exercise #1

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

ααα606060AAAaaa

Video Solution

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 α \alpha is also equal to 60 degrees

Answer

60

Exercise #2

BD is a bisector.

What is the size of angle ABC?

656565AAABBBCCCDDD

Video Solution

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 65+65=130

Answer

130

Exercise #3

Which of the following figures has a bisector?

Video Solution

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 45=45

Answer

4545

Exercise #4

ABCD is a square.

ABC=? ∢\text{ABC}=\text{?}

AAABBBDDDCCC

Video Solution

Step-by-Step Solution

Due to the fact that all angles in a square are equal to 90 degrees, and BC bisects an angle, we can calculate angle ABC accordingly:

90:2=45 90:2=45

Answer

45

Exercise #5

ABCD is a deltoid.

DAC=? ∢DAC=\text{?}

AAABBBCCCDDD2x602x

Video Solution

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

6X+60=180 6X+60=180

18060=6X 180-60=6X

120=6X 120=6X

We divide the two sections by 6:1206=6x6 \frac{120}{6}=\frac{6x}{6}

20=x 20=x

Now we can calculate the angle DAC:

20×2=40 20\times2=40

Answer

30

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