Angle Bisector Theorem - Proof, Converse, Formula, Examples

Edited By Team Careers360 | Updated on Jul 02, 2025 05:13 PM IST

In geometry, the angle bisector of a triangle is a line or ray that divides the angle formed by two sides of the triangle into two equal angles. The angle bisector theorem states that the line bisecting an angle of a triangle also divides the opposite side into segments that are proportional to the other two sides.

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This Story also Contains

The exterior angle bisector theorem
The internal angle bisector theorem
Angle bisector example
Show that the angle bisectors of a triangle are concurrent
Prove that the angle bisectors of a triangle are concurrent
External Bisector
Angle bisector real life example

To find the angle bisector of a triangle, you will need to know the lengths of all three sides of the triangle. You can then use the following steps:

Choose any two sides of the triangle and use the angle bisector theorem to find the length of the angle bisector for the angle between those two sides.
Draw the angle bisector for that angle.
Repeat the process for the other two angles of the triangle.
The three angle bisectors will intersect at a single point, which is called the incenter of the triangle.

The angle bisector theorem is often used to solve problems involving triangles, such as finding the length of a side or the measure of an angle. It can also be used to prove that certain triangles are similar.

The exterior angle bisector theorem

The exterior angle bisector theorem states that the angle bisector of an exterior angle of a triangle is also an angle bisector of the opposite interior angle. In other words, the angle bisector of an exterior angle of a triangle bisects the opposite interior angle into two equal angles.

To understand this theorem, it may be helpful to first review the definitions of the interior and exterior angles of a triangle. The interior angles of a triangle are the angles formed by two sides of the triangle and are always inside the triangle. The exterior angles of a triangle are the angles formed by one side of the triangle and an extension of another side and are always outside the triangle.

Here is an example of how the exterior angle bisector theorem works:

Imagine that you have a triangle with sides of lengths a, b, and c and interior angles A, B, and C. If you draw the angle bisector of the exterior angle C, it will also bisect the opposite interior angle A into two equal angles.

The exterior angle bisector theorem is often used in geometry to solve problems involving triangles, such as finding the length of a side or the measure of an angle. It can also be used to prove that certain triangles are similar.

The internal angle bisector theorem

The internal angle bisector theorem, also known as the triangle angle bisector theorem or the angle bisector theorem, states that the line bisecting an angle of a triangle also divides the opposite side into segments that are proportional to the other two sides. In other words, if you draw an angle bisector for an angle of a triangle, the line will also bisect the opposite side into segments that are proportional to the lengths of the other two sides.

To understand this theorem, it may be helpful to first review the definitions of the sides and angles of a triangle. In any triangle, there are three sides and three angles. The sides are labeled a, b, and c, and the angles are labeled A, B, and C.

Here is an example of how the internal angle bisector theorem works:

Imagine that you have a triangle with sides of lengths a, b, and c and interior angles A, B, and C. If you draw the angle bisector of angle A, it will also bisect the opposite side c into two segments, one of which is proportional to side a and the other of which is proportional to side b.

The internal angle bisector theorem is often used in geometry to solve problems involving triangles, such as finding the length of a side or the measure of an angle. It can also be used to prove that certain triangles are similar.

Angle bisector example

Here is an example of how to use the angle bisector theorem to find the length of a side of a triangle:

Imagine that you have a triangle with sides of lengths a = 6, b = 8, and c = 10 and interior angles A, B, and C. You are asked to find the length of side c.

To solve this problem, you can use the angle bisector theorem to find the ratio of the length of side c to the length of one of the other sides, a or b. For example, you could use the following steps:

Draw a diagram to represent the triangle.
Choose one of the angles of the triangle and draw the angle bisector for that angle.
Use the angle bisector theorem to find the ratio of the length of side c to the length of the other side. For example, if you choose angle B, you can use the following equation:

c/b = a/c

Solve the equation for the unknown side length. In this case, you can rearrange the equation to solve for c:

c = (a*b)/c

Substitute the known values for a and b into the equation and solve for c:

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c = (6*8)/c

c = 48/c

c = 48/10

c = 4.8

Round the answer to the nearest whole number, since side lengths must be whole numbers:

c = 5

Therefore, the length of side c is 5.

This is just one example of how the angle bisector theorem can be used to find the length of a side of a triangle. You can use similar steps to solve other problems involving triangles.

Show that the angle bisectors of a triangle are concurrent

To show that the angle bisectors of a triangle are concurrent, you can use the angle bisector theorem and the fact that the sum of the interior angles of a triangle is 180 degrees.

Here are the steps to follow:

Draw a diagram to represent the triangle.
Choose any two angles of the triangle and draw the angle bisector for each angle.
Use the angle bisector theorem to find the ratio of the lengths of the sides opposite the two angles to the lengths of the other two sides. For example, if you choose angles A and B, you can use the following equations:

c/b = a/c

a/c = b/a

Solve the equations for the unknown side lengths. In this case, you can rearrange the equations to solve for a and b:

a = (c*b)/a

b = (a*c)/b

Substitute the known values for c and the angle measures into the equations and solve for a and b.
Use the fact that the sum of the interior angles of a triangle is 180 degrees to find the third angle measure.
Show that the three angle measures add up to 180 degrees, which proves that the angle bisectors are concurrent.

For example, if the measures of angles A and B are 45 degrees and 60 degrees, respectively, you can use the following steps:

A = 45 degrees

B = 60 degrees

a = (c*b)/a

a= (c*60)/45

a = (4/3)c

b = (a*c)/b

b = (45*c)/60

b = (3/4)c

A + B + C = 180 degrees

45 + 60 + C = 180 degrees

C = 75 degrees

A + B + C = 180 degrees

45 + 60 + 75 = 180 degrees,

Therefore, the angle bisectors of the triangle are concurrent.

This is just one example of how to show that the angle bisectors of a triangle are concurrent. You can use similar steps to solve other problems involving triangles.

Prove that the angle bisectors of a triangle are concurrent

To prove that the angle bisectors of a triangle are concurrent, you can use the angle bisector theorem and the fact that the sum of the interior angles of a triangle is 180 degrees.

Here are the steps to follow:

Draw a diagram to represent the triangle.
Choose any two angles of the triangle and draw the angle bisector for each angle.
Use the angle bisector theorem to find the ratio of the lengths of the sides opposite the two angles to the lengths of the other two sides. For example, if you choose angles A and B, you can use the following equations:

c/b = a/c

a/c = b/a

Solve the equations for the unknown side lengths. In this case, you can rearrange the equations to solve for a and b:

a = (c*b)/a

b = (a*c)/b

Substitute the known values for c and the angle measures into the equations and solve for a and b.
Use the fact that the sum of the interior angles of a triangle is 180 degrees to find the third angle measure.
Show that the three angle measures add up to 180 degrees, which proves that the angle bisectors are concurrent.

For example, if the measures of angles A and B are 45 degrees and 60 degrees, respectively, you can use the following steps:

A = 45 degrees

B = 60 degrees

a = (c*b)/a

a= (c*60)/45

a = (4/3)c

b = (a*c)/b

b = (45*c)/60

b = (3/4)c

A + B + C = 180 degrees

45 + 60 + C = 180 degrees

C = 75 degrees

A + B + C = 180 degrees

45 + 60 + 75 = 180 degrees,

Therefore, the angle bisectors of the triangle are concurrent.

This is just one example of how to show that the angle bisectors of a triangle are concurrent. You can use similar steps to solve other problems involving triangles.

External Bisector

The external bisector of a triangle is a line or ray that divides an exterior angle of a triangle into two equal angles. It is often used in geometry to solve problems involving triangles, such as finding the length of a side or the measure of an angle.

To find the external bisector of a triangle, you will need to know the lengths of all three sides of the triangle. You can then use the following steps:

Choose an exterior angle of the triangle and draw the angle bisector for that angle.
Use the angle bisector theorem to find the ratio of the length of the side opposite the exterior angle to the length of the other side.
Solve the equation for the unknown side length.
Substitute the known values for the other two side lengths into the equation and solve for the unknown side length. Round the answer to the nearest whole number, since side lengths must be whole numbers.

Here is an example of how to use the external bisector of a triangle to find the length of a side:

Imagine that you have a triangle with sides of lengths a = 6, b = 8, and c = 10 and exterior angle C. You are asked to find the length of side a.

To solve this problem, you can use the following steps:

Draw a diagram to represent the triangle.
Draw the external bisector for exterior angle C.
Use the angle bisector theorem to find the ratio of the length of side a to the length of side b: a/b = c/a
Solve the equation for the unknown side length: a = (c*b)/a
Substitute the known values for b and c into the equation and solve for a: a = (10*8)/6 a = 16/2 a = 8 Therefore, the length of side is 8.

Angle bisector real life example

Here is an example of how the angle bisector theorem can be used in real life:

Imagine that you are a surveyor and you are asked to measure the height of a tall building. You are standing 100 meters away from the base of the building, and you have a surveyor's level, which allows you to measure the angle between the ground and the top of the building. The angle measure is 45 degrees.

To find the height of the building, you can use the angle bisector theorem as follows:

Draw a diagram to represent the situation.
The angle of elevation is formed by a horizontal line and a line of sight from a lower point to a higher point. Measure the distance between the two points.
In this case, the distance is 100 meters.
Measure the angle of elevation. In this case, the angle is 45 degrees.
Use the tangent function to find the height of the building.
You can use the following formula:
height = horizontal distance * tan(angle)
Substitute the known values into the equation and solve for the height: height = 100 * tan(45) height = 100 * 1 height = 100

Therefore, the height of the building is 100 meters.

This is just one example of how the angle bisector theorem can be used in real life. The theorem is often used in surveying and other fields to measure distances and angles.

Frequently Asked Questions (FAQs)

1. How can you verify the Angle Bisector Theorem experimentally?

You can verify the Angle Bisector Theorem experimentally by drawing various triangles, constructing their angle bisectors, measuring the segments created on the opposite side and the lengths of the other two sides, and then checking if the ratios are equal as stated in the theorem.

2. Is there a special case of the Angle Bisector Theorem for isosceles triangles?

In isosceles triangles, where two sides are equal, the angle bisector of the vertex angle is also the perpendicular bisector of the base. This is a special case where the angle bisector divides the opposite side into two equal parts.

3. What is the significance of the ratio in the Angle Bisector Theorem?

The ratio in the Angle Bisector Theorem (AD/DB = AC/BC) is significant because it shows that the angle bisector divides the opposite side proportionally to the lengths of the other two sides. This ratio is constant for a given triangle and angle bisector.

4. What is a common misconception about the Angle Bisector Theorem?

A common misconception is that the angle bisector divides the opposite side into two equal parts. This is not true in general; it only divides the side proportionally to the lengths of the other two sides, which may or may not be equal.

5. What is the relationship between the Angle Bisector Theorem and the concept of similarity in triangles?

The Angle Bisector Theorem creates similar triangles within the original triangle. This similarity is key to understanding and proving the theorem. It demonstrates how the concept of similarity is fundamental to many geometric relationships.

6. Why is the Angle Bisector Theorem important in geometry?

The Angle Bisector Theorem is important because it establishes a relationship between angle bisectors and side lengths in triangles. This relationship is useful in solving various geometric problems, proving other theorems, and in practical applications like surveying and engineering.

7. How does the Angle Bisector Theorem help in solving triangle problems?

The Angle Bisector Theorem provides an additional relationship in triangles, which can be used to solve problems where side lengths or segment lengths are unknown. It's particularly useful when angle bisectors are involved in the problem setup.

8. How can you apply the Angle Bisector Theorem in real-world situations?

The Angle Bisector Theorem can be applied in surveying, architecture, and engineering. For example, it can be used to determine distances or positions when direct measurements are not possible, by utilizing known angles and ratios.

9. How does the Angle Bisector Theorem differ from the Perpendicular Bisector Theorem?

The Angle Bisector Theorem relates to angle bisectors and creates a ratio on the opposite side, while the Perpendicular Bisector Theorem deals with the perpendicular bisector of a side and its relationship to the vertices. They serve different purposes in triangle geometry.

10. Can you use the Angle Bisector Theorem to find the length of an angle bisector?

While the Angle Bisector Theorem doesn't directly give the length of the angle bisector, it can be used in conjunction with other theorems and formulas to calculate the length. The theorem provides a crucial relationship that can be part of the calculation process.

11. Can the Angle Bisector Theorem be extended to other polygons?

The Angle Bisector Theorem as stated is specific to triangles. While angle bisectors exist in other polygons, the specific ratio relationship described by this theorem doesn't directly extend to polygons with more than three sides.

12. How does the Angle Bisector Theorem compare to other triangle theorems like the Pythagorean Theorem?

While both are important triangle theorems, they serve different purposes. The Pythagorean Theorem relates the sides of a right triangle, while the Angle Bisector Theorem relates angle bisectors to side lengths in any triangle. They are used in different types of problems and proofs.

13. How can the Angle Bisector Theorem be used in trigonometry?

In trigonometry, the Angle Bisector Theorem can be used to derive relationships involving trigonometric ratios. It provides a geometric basis for some trigonometric identities and can be applied in solving trigonometric problems involving triangles.

14. What are some common errors students make when applying the Angle Bisector Theorem?

Common errors include confusing the Angle Bisector Theorem with the Perpendicular Bisector Theorem, incorrectly identifying which segments correspond to which sides in the ratio, and forgetting that the theorem applies to the internal angle bisector, not external angle bisectors.

15. How does the Angle Bisector Theorem relate to the concept of congruence in triangles?

While the Angle Bisector Theorem doesn't directly deal with congruence, it can be used in proofs involving congruent triangles. Understanding how angle bisectors divide sides can help in establishing congruence in certain triangle configurations.

16. Can you explain the proof of the Angle Bisector Theorem?

The proof of the Angle Bisector Theorem involves constructing an auxiliary line parallel to one side of the triangle, creating similar triangles, and then using the properties of similar triangles to establish the ratio relationship. The proof demonstrates why the theorem holds true for all triangles.

17. What is the relationship between the Angle Bisector Theorem and the concept of concurrent lines in a triangle?

The Angle Bisector Theorem is related to concurrent lines in a triangle because all three angle bisectors of a triangle are concurrent, meeting at a point called the incenter. This property, along with the theorem, helps in understanding the geometry of triangles more deeply.

18. How is the Angle Bisector Theorem different from the Angle Bisector Postulate?

The Angle Bisector Theorem relates to triangles specifically, while the Angle Bisector Postulate is a general statement about angle bisectors. The theorem provides a ratio relationship in triangles, whereas the postulate simply states that an angle bisector divides an angle into two equal parts.

19. How can you prove the converse of the Angle Bisector Theorem?

The proof of the converse involves assuming that a line divides the opposite side in the ratio of the other two sides, then constructing the actual angle bisector and showing that it must coincide with the given line, thus proving that the given line is indeed the angle bisector.

20. How does the Angle Bisector Theorem relate to the area of a triangle?

While the Angle Bisector Theorem doesn't directly relate to the area of a triangle, it can be used in conjunction with area formulas. For instance, knowing the ratio of segments created by an angle bisector can help in calculating areas of smaller triangles within the original triangle.

21. How does the Angle Bisector Theorem relate to the concept of triangle centers?

The Angle Bisector Theorem is related to one of the important triangle centers - the incenter. The incenter is the point where all three angle bisectors of a triangle intersect, and understanding the theorem helps in locating and working with this center.

22. Can the Angle Bisector Theorem be used to prove other geometric theorems?

Yes, the Angle Bisector Theorem can be used as a tool to prove other geometric theorems, especially those involving triangles and their properties. It provides a useful relationship that can be applied in various proofs and problem-solving scenarios.

23. How does the Angle Bisector Theorem help in understanding the geometry of triangles?

The Angle Bisector Theorem enhances our understanding of triangles by revealing a subtle relationship between angle bisectors and side lengths. It shows how different elements of a triangle are interconnected, contributing to a more comprehensive view of triangle geometry.

24. What is the role of the Angle Bisector Theorem in coordinate geometry?

In coordinate geometry, the Angle Bisector Theorem can be applied to triangles plotted on a coordinate plane. It can be used to find coordinates of points where angle bisectors intersect sides, or to verify if a given line is an angle bisector.

25. How does the Angle Bisector Theorem contribute to the study of triangle inequalities?

The Angle Bisector Theorem provides insights into the relationships between sides of a triangle, which can be useful in understanding and proving triangle inequalities. It shows how the lengths of sides are related to internal divisions of the triangle.

26. Can you use the Angle Bisector Theorem to solve problems involving exterior angles of a triangle?

While the Angle Bisector Theorem primarily deals with interior angles, it can be indirectly used in problems involving exterior angles. By understanding the relationship between interior and exterior angles, you can apply the theorem to solve more complex problems.

27. How does the Angle Bisector Theorem relate to the concept of proportion in geometry?

The Angle Bisector Theorem is a direct application of proportion in geometry. It establishes a proportional relationship between segments of a side and the lengths of the other two sides, demonstrating how proportions can exist within geometric figures.

28. What is the historical significance of the Angle Bisector Theorem?

The Angle Bisector Theorem has been known since ancient times and was included in Euclid's Elements. Its discovery and proof represent an important step in the development of geometry, showcasing how complex relationships in triangles can be expressed mathematically.

29. Can the Angle Bisector Theorem be applied to non-Euclidean geometries?

The Angle Bisector Theorem as we know it is specific to Euclidean geometry. In non-Euclidean geometries (like spherical or hyperbolic geometry), the concept of angle bisectors exists, but the specific relationships may differ due to the different nature of these geometries.

30. How does the Angle Bisector Theorem relate to the concept of symmetry in triangles?

In isosceles triangles, the angle bisector of the vertex angle is also an axis of symmetry. This special case of the Angle Bisector Theorem demonstrates how the theorem can reveal symmetrical properties in certain types of triangles.

31. What is the Angle Bisector Theorem?

The Angle Bisector Theorem states that the angle bisector of an angle of a triangle divides the opposite side in the ratio of the lengths of the other two sides. This theorem provides a relationship between the segments created by the angle bisector and the lengths of the triangle's sides.

32. What is the formula associated with the Angle Bisector Theorem?

The formula for the Angle Bisector Theorem is: AD/DB = AC/BC, where AD and DB are the segments created by the angle bisector on the opposite side, and AC and BC are the lengths of the other two sides of the triangle.

33. What is the converse of the Angle Bisector Theorem?

The converse of the Angle Bisector Theorem states that if a line from a vertex of a triangle divides the opposite side in the ratio of the lengths of the other two sides, then that line is the angle bisector of the angle at that vertex.

34. How does the Angle Bisector Theorem relate to the concept of similarity in triangles?

The Angle Bisector Theorem is closely related to similarity because it creates two triangles within the original triangle that are similar to each other. This similarity is key to understanding why the ratio relationship in the theorem holds true.

35. Can the Angle Bisector Theorem be applied to all types of triangles?

Yes, the Angle Bisector Theorem applies to all types of triangles - acute, right, and obtuse. The ratio relationship holds true regardless of the triangle's shape or size.

36. What are some practical applications of the Angle Bisector Theorem in fields outside of pure mathematics?

The Angle Bisector Theorem has applications in surveying, where it can be used to determine distances indirectly. It's also useful in computer graphics for dividing angles in rendering algorithms, and in engineering for structural analysis of triangular frameworks.

37. How can the Angle Bisector Theorem be used in problem-solving strategies?

The Angle Bisector Theorem provides an additional tool in geometric problem-solving. When faced with a triangle problem, considering whether an angle bisector is involved and applying the theorem can often lead to a solution, especially when other methods are not immediately apparent.

38. What is the connection between the Angle Bisector Theorem and the concept of loci in geometry?

The Angle Bisector Theorem is related to the concept of loci because the angle bisector itself is a locus - the set of all points equidistant from the sides of the angle. The theorem provides a specific property of this locus within a triangle.

39. How does understanding the Angle Bisector Theorem contribute to overall geometric reasoning skills?

Understanding the Angle Bisector Theorem enhances geometric reasoning by demonstrating how to derive complex relationships from simpler principles. It encourages students to think about how different parts of a geometric figure relate to each other, fostering deeper analytical skills.

40. Can the Angle Bisector Theorem be generalized or extended in any way?

While the theorem itself is specific to triangles, its underlying principles of proportionality and the properties of angle bisectors can be extended to more complex polygons and even three-dimensional geometry, though not in the same direct form as in triangles.

41. How does the Angle Bisector Theorem compare to other theorems about special lines in triangles?

The Angle Bisector Theorem is one of several theorems about special lines in triangles, along with theorems about medians, altitudes, and perpendicular bisectors. Each of these theorems reveals different properties and relationships within triangles, contributing to a comprehensive understanding of triangle geometry.

42. What role does the Angle Bisector Theorem play in the study of triangle centers?

The Angle Bisector Theorem is crucial in understanding the incenter of a triangle, which is where all three angle bisectors intersect. It helps in calculating the location of the incenter and understanding its properties in relation to the sides of the triangle.

43. How can technology be used to explore and visualize the Angle Bisector Theorem?

Dynamic geometry software like GeoGebra can be used to create interactive visualizations of the Angle Bisector Theorem. Students can drag vertices of triangles and see how the ratio remains constant, helping to build intuition about the theorem.

44. What are some common mistakes in the application of the Angle Bisector Theorem?

Common mistakes include applying the theorem to external angle bisectors, confusing it with the properties of other special lines in triangles, and incorrectly identifying the corresponding sides and segments in the ratio equation.

45. How does the Angle Bisector Theorem contribute to the understanding of triangle inequality?

The Angle Bisector Theorem provides insights into the relationships between sides of a triangle, which indirectly supports the triangle inequality theorem. It shows how the lengths of sides are interconnected, reinforcing the concept that the sum of any two sides must be greater than the third side.

46. Can the Angle Bisector Theorem be used in constructions with compass and straightedge?

Yes, the Angle Bisector Theorem can be used in compass and straightedge constructions. It provides a method for dividing a side of a triangle in a given ratio, which can be useful in various geometric constructions and proofs.

47. How does the Angle Bisector Theorem relate to the concept of triangle similarity?

The Angle Bisector Theorem creates similar triangles within the original triangle. This relationship is fundamental to the proof of the theorem and demonstrates how similarity principles are at work even within a single triangle.

48. What is the significance of the Angle Bisector Theorem in the broader context of Euclidean geometry?

In the broader context of Euclidean geometry, the Angle Bisector Theorem is significant as it demonstrates the interconnectedness of geometric concepts. It shows how properties of angles, lines, and proportions come together in triangles, which are fundamental shapes in Euclidean geometry.

49. How can understanding the Angle Bisector Theorem improve problem-solving skills in geometry?

Understanding the Angle Bisector Theorem improves problem-solving skills by providing an additional tool for analyzing triangles. It encourages students to look for relationships between angles and sides, fostering a more comprehensive approach to geometric problems.

50. What are some advanced applications or extensions of the Angle Bisector Theorem in higher mathematics?

In higher mathematics, the principles behind the Angle Bisector Theorem can be extended to more complex geometric concepts. For example, it plays a role in the study of triangle centers, can be applied in certain aspects of algebraic geometry, and has connections to more advanced topics in trigonometry and analytic geometry.