Angle Bisector - Definition, Construction, Properties, Examples

Edited By Team Careers360 | Updated on Jul 02, 2025 05:32 PM IST

An angle bisector can be defined as a ray, line segment or line that divides an angle into two equal parts. In other words, a ray, line segment or line that cuts an angle in two congruent parts is termed an angle bisector.

Properties Of Angle Bisector

An angle is divided into two equal pieces by an angle bisector.
Any point on an angle's bisector is equally distanced from the angle's sides or arms.
It divides the opposite side of a triangle in proportion to the other two sides

This Story also Contains

Properties Of Angle Bisector
Some Important terms
Construction Of An Angle Bisector
An Angle Bisector Of A Triangle

Some Important terms

Angle: An angel is a structure created by two rays that share an endpoint and are referred to as the angle’s sides and vertices, respectively. Angles created by two rays are in the plane where rays are located. The meeting point of two planes also creates angles.
Bisector: A bisector in geometry can be termed as a ray, line or line segment that divides a figure into two equal parts. It can divide angles as well as line segments.
Bisect: Bisect in geometry is used for dividing into two exactly equal parts.

Construction Of An Angle Bisector

Step 1: Drawing an angle, \angle ACB .

Step 2: Draw an arc with C as the centre and any suitable radius to intersect the rays AC and CB, say at D and E, respectively.

Step 3: Draw two arcs that will intersect at F by using D and E as the centres and the same radius as in the previous step.

Step 4: Drawing a ray by connecting C and F. The required angle bisector of angle ACB is this ray CF.

An Angle Bisector Of A Triangle

The straight line that divides an angle in a triangle into two equal or congruent angles is known as the angle bisector of a triangle. As a triangle has three vertices, there can be three angle bisectors in it. The incenter of a triangle is the location where these three angle bisectors intersect. All of a triangle's vertices are equally distanced from the incenter. The figure below shows the triangle's angle bisector. The angle bisectors of ABC, BCA and CAB in this case are BD, CE and AF, respectively. G is the incenter, or point of intersection of all three bisectors, and it is located at an equal distance from each vertex.

Frequently Asked Questions (FAQs)

1. What is an angle bisector?

An angle bisector is a ray, line or line segment that divides an angle into exactly equal parts.

2. What is an angle bisector?

An angle bisector is a line that divides an angle into two equal parts. It passes through the vertex of the angle and creates two smaller angles, each exactly half of the original angle's measure.

3. The angle bisector of the vertex angle of an isosceles triangle divides the angle into two equal parts. Is it true or false?

Yes, indeed, the angle bisector of the vertex angle of an isosceles triangle divides the angle into two equal parts.

4. What does an angle bisector theorem state?

According to the angle bisector theorem, The opposite side of a triangle is divided into two parts by the angle bisector in proportion to the other two sides.

5. How to construct an angle bisector?

For constructing an angle bisector, first we will draw the angle and then draw an arc of any radius taking the angle’s vertex as the centre. Then taking the intersecting points of the sides of the angle and the arc as the centre we will further cut two arcs. Further, we will draw a line passing through the intersecting point of the arcs to the vertex of the angle. The line formed is the required angle bisector.

6. How does the angle bisector theorem relate to triangles?

The angle bisector theorem states that the angle bisector of an angle in a triangle divides the opposite side into segments proportional to the lengths of the other two sides. This means the ratio of these segments is equal to the ratio of the lengths of the adjacent sides.

7. What is the relationship between an angle bisector and the incenter of a triangle?

The incenter of a triangle is the point where all three angle bisectors of the triangle intersect. This point is equidistant from all three sides of the triangle and is the center of the triangle's inscribed circle.

8. How does the angle bisector of an exterior angle of a triangle relate to the other sides?

The angle bisector of an exterior angle of a triangle is parallel to the angle bisector of the opposite interior angle. This relationship demonstrates the connection between interior and exterior angles in triangles.

9. How does the length of an angle bisector in a triangle relate to the sides of the triangle?

The length of an angle bisector in a triangle is related to the sides of the triangle through the angle bisector formula: (la)² = bc - (ad)(ae), where la is the length of the angle bisector from vertex A, b and c are the lengths of the sides adjacent to angle A, and d and e are the segments of the opposite side created by the angle bisector.

10. How does an angle bisector affect the area of a triangle?

An angle bisector divides a triangle into two smaller triangles. The areas of these two triangles are proportional to the lengths of the sides adjacent to the bisected angle. This property is known as the angle bisector theorem for areas.

11. Can an angle bisector be constructed for any angle?

Yes, an angle bisector can be constructed for any angle, regardless of its measure. The construction process remains the same whether the angle is acute, right, obtuse, or even a reflex angle.

12. What are the steps to construct an angle bisector using a compass and straightedge?

To construct an angle bisector: 1) Draw an arc centered at the angle's vertex, intersecting both sides of the angle. 2) With the same compass width, draw arcs centered at these intersection points. 3) Draw a line from the vertex through the point where these new arcs intersect. This line is the angle bisector.

13. Can an angle bisector be used to trisect an angle?

No, an angle bisector alone cannot be used to trisect an angle. Angle trisection using only a compass and straightedge is famously impossible for all angles, as proven by Pierre Wantzel in 1837. Angle bisection, however, is always possible with these tools.

14. What is the locus of points equidistant from the sides of an angle?

The locus of points equidistant from the sides of an angle is the angle bisector. Every point on the angle bisector is equidistant from both sides of the angle, making it a useful concept in geometry and real-world applications like positioning objects equidistant from two intersecting lines.

15. How does the concept of angle bisectors extend to polygons with more than three sides?

In polygons with more than three sides, angle bisectors can still be constructed for each interior angle. However, unlike in triangles, these bisectors may not all intersect at a single point. The properties and theorems related to angle bisectors in triangles don't necessarily extend to all polygons.

16. How does an angle bisector relate to the concept of symmetry?

An angle bisector creates line symmetry within the angle. If you were to fold the angle along its bisector, the two halves would perfectly overlap, demonstrating that the angle bisector is a line of symmetry for the angle.

17. Can two angle bisectors in a triangle be perpendicular to each other?

Yes, two angle bisectors in a triangle can be perpendicular to each other. This occurs in isosceles right triangles, where the angle bisectors of the two equal angles are perpendicular and meet at the midpoint of the hypotenuse.

18. What is the concurrency property of angle bisectors in a triangle?

The concurrency property of angle bisectors states that the three angle bisectors of any triangle always intersect at a single point. This point of intersection is called the incenter of the triangle and is equidistant from all three sides of the triangle.

19. How does the angle bisector theorem apply to similar triangles?

The angle bisector theorem applies similarly to all triangles, regardless of their size. In similar triangles, the ratios created by the angle bisectors will be the same, preserving the proportional relationships between corresponding parts of the similar triangles.

20. What is the relationship between angle bisectors and the inscribed circle of a triangle?

The angle bisectors of a triangle intersect at the incenter, which is the center of the triangle's inscribed circle. The inscribed circle touches all three sides of the triangle, and the points of tangency are where the angle bisectors intersect the sides of the triangle.

21. What is the difference between an angle bisector and a median in a triangle?

An angle bisector divides an angle into two equal parts and may or may not divide the opposite side equally. A median, on the other hand, always connects a vertex to the midpoint of the opposite side but doesn't necessarily bisect the angle at the vertex.

22. What is the relationship between the incenter and the centroid of a triangle?

The incenter (intersection of angle bisectors) and centroid (intersection of medians) are two of the four triangle centers. While they are distinct points, they are always located inside the triangle. The centroid divides each median in a 2:1 ratio, while the incenter's position depends on the angles of the triangle.

23. What is the difference between an internal and external angle bisector of a triangle?

An internal angle bisector divides an interior angle of a triangle into two equal parts. An external angle bisector divides an exterior angle (formed by one side of the triangle and the extension of another side) into two equal parts. Both types of bisectors have unique properties and relationships within the triangle.

24. How does the angle bisector of a right angle relate to the coordinate plane?

The angle bisector of a right angle (90°) creates two 45° angles. In the coordinate plane, if this right angle is at the origin with sides along the x and y axes, its angle bisector would be the line y = x, which makes a 45° angle with both axes.

25. How do angle bisectors relate to the concept of power of a point with respect to a circle?

The power of a point with respect to a circle is a measure of how far the point is from the circle. While not directly related to angle bisectors, both concepts are important in the study of circles and their properties. The radical axis of two circles, which is the locus of points with equal power with respect to both circles, is perpendicular to the line joining the centers of the circles, similar to how an angle bisector is perpendicular to the base of an isosceles triangle.

26. How do angle bisectors relate to the concept of pedal triangles?

A pedal triangle is formed by the feet of perpendiculars from a point to the sides of a given triangle. While not directly related to angle bisectors, both concepts are important in studying triangle geometry. Interestingly, if the pedal point is the incenter (intersection of angle bisectors), the resulting pedal triangle has the maximum area for all possible pedal triangles.

27. What is the significance of angle bisectors in the study of Morley's theorem?

Morley's theorem states that the three points of intersection of the adjacent angle trisectors of a triangle form an equilateral triangle. While this theorem deals with angle trisectors rather than bisectors, it highlights the importance of angle division in revealing surprising geometric properties.

28. How can the angle bisector theorem be used to solve problems involving triangles?

The angle bisector theorem can be used to find unknown side lengths or segment lengths in triangles. By setting up a proportion based on the theorem (the ratio of the segments created on one side equals the ratio of the lengths of the other two sides), we can solve for missing values in triangle problems.

29. Can an angle bisector be used to prove the congruence of triangles?

Yes, angle bisectors can be used in triangle congruence proofs. For example, if two triangles have two pairs of congruent sides and the included angle bisector congruent, the triangles are congruent. This is known as the Side-Angle Bisector-Side (SAS) congruence theorem.

30. What is the role of angle bisectors in the construction of regular polygons?

Angle bisectors play a crucial role in constructing regular polygons. For example, to construct a regular hexagon, you can start with a circle and use angle bisectors to divide it into six equal 60° angles. This principle extends to other regular polygons as well.

31. How do angle bisectors relate to the concept of equidistant lines?

An angle bisector represents a set of points that are equidistant from the two sides of the angle. This property makes angle bisectors useful in situations where we need to find a location that's equally distant from two intersecting lines or paths.

32. Can the angle bisector theorem be applied to obtuse triangles?

Yes, the angle bisector theorem applies to all triangles, including obtuse triangles. The ratio of the segments created by the angle bisector on the opposite side will still be equal to the ratio of the lengths of the other two sides, regardless of the triangle's shape.

33. What is the relationship between an angle bisector and the perpendicular bisector of the opposite side in an isosceles triangle?

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 base (the side opposite the vertex angle). This line is also the altitude and median to the base, demonstrating a unique property of isosceles triangles.

34. How does the concept of angle bisectors apply in real-world scenarios?

Angle bisectors have various real-world applications. For example, in architecture, they can be used to position support beams equidistant from walls. In navigation, they can help determine a position equidistant from two landmarks. In optics, they play a role in understanding light reflection.

35. What is the significance of the angle bisector in the study of conic sections?

In conic sections, particularly for parabolas, the angle bisector of the angle between the axis and any tangent line passes through the focus of the parabola. This property is used in the design of parabolic reflectors and antennas.

36. How do angle bisectors relate to the concept of angular velocity in physics?

While angle bisectors themselves don't directly relate to angular velocity, the concept of bisecting angles is important in understanding rotational motion. For instance, in uniform circular motion, the angle bisector of the angle swept in a given time interval points towards the position of the object halfway through that interval.

37. What is the relationship between angle bisectors and the orthocenter of a triangle?

The orthocenter (intersection of altitudes) and the incenter (intersection of angle bisectors) are distinct points in a triangle. However, in an equilateral triangle, these points coincide, along with the centroid and circumcenter, forming the single point where all four triangle centers meet.

38. How does the angle bisector theorem extend to exterior angles of a triangle?

The angle bisector theorem can be extended to exterior angles of a triangle. The external angle bisector divides the opposite side externally in the ratio of the lengths of the other two sides. This is known as the external angle bisector theorem.

39. What is the role of angle bisectors in the study of tangent circles?

Angle bisectors play a crucial role in the study of tangent circles. The angle bisector of the angle formed by two tangent lines to a circle passes through the center of the circle. This property is used in various geometric constructions and proofs involving circles.

40. How can the angle bisector theorem be used to solve optimization problems?

The angle bisector theorem can be used in optimization problems, particularly those involving minimizing distances. For example, finding the point on a line segment that minimizes the sum of distances to two fixed points involves using the angle bisector of the angle formed by these points.

41. What is the relationship between angle bisectors and the nine-point circle of a triangle?

While angle bisectors don't directly determine the nine-point circle, they are related through the Euler line. The nine-point circle's center lies on the Euler line, which also contains the orthocenter, centroid, and circumcenter. The incenter (intersection of angle bisectors) generally does not lie on this line, except in equilateral triangles.

42. How do angle bisectors relate to the concept of equiangular spirals?

Equiangular spirals, also known as logarithmic spirals, have the property that any ray from the origin forms the same angle with the spiral at every point of intersection. This constant angle property is related to the concept of angle bisectors, as each turn of the spiral effectively bisects the angle formed by the previous turn.

43. What is the significance of angle bisectors in the study of isogonal conjugates in a triangle?

Isogonal conjugates are pairs of points such that the lines joining them to the vertices of a triangle form equal angles with the respective angle bisectors. This concept is important in advanced triangle geometry and relates angle bisectors to other significant points and lines in a triangle.

44. How does the angle bisector theorem relate to the concept of similarity in geometry?

The angle bisector theorem preserves ratios, which is a key aspect of similarity. When an angle bisector divides the opposite side of a triangle, it creates two smaller triangles that are similar to each other and to the original triangle. This demonstrates how angle bisectors can be used to explore and prove similarity relationships.

45. What is the role of angle bisectors in the construction of the Fermat point of a triangle?

While angle bisectors are not directly used to construct the Fermat point (the point that minimizes the sum of distances to the triangle's vertices), they are related to it. In an acute triangle, the Fermat point is the intersection of the three lines that connect each vertex to the point where the angle trisector of the opposite angle intersects the opposite side.

46. What is the significance of angle bisectors in the study of triangle inequalities?

Angle bisectors play a role in various triangle inequalities. For example, the length of an angle bisector is always less than the arithmetic mean of the lengths of the other two sides of the triangle. Such inequalities help in understanding the relationships between different parts of a triangle and in solving optimization problems.

47. What is the relationship between angle bisectors and the Euler line in a triangle?

The Euler line of a triangle contains several important points including the centroid, orthocenter, and circumcenter. While the incenter (intersection of angle bisectors) generally does not lie on the Euler line, it does in the special case of an isosceles triangle. This relationship highlights the unique properties of isosceles triangles.

48. How can angle bisectors be used in the study of cyclic quadrilaterals?

In a cyclic quadrilateral (a quadrilateral inscribed in a circle), the angle bisectors of two opposite angles intersect on the circumference of the circle. This property can be used to prove whether a quadrilateral is cyclic and to solve problems involving cyclic quadrilaterals.

49. What is the role of angle bisectors in the construction of the Gergonne point of a triangle?

The Gergonne point of a triangle is the point where the lines connecting each vertex to the opposite point of tangency of the incircle intersect. While not directly constructed using angle bisectors, the Gergonne point is closely related to the incenter (intersection of angle bisectors) and the incircle of the triangle.

50. How do angle bisectors relate to the concept of triangle centers?

Angle bisectors define one of the four classical triangle centers: the incenter. The study of triangle centers is a rich area of geometry, and the incenter's properties and relationships with other centers (like the centroid, orthocenter, and circumcenter) provide insights into the structure and properties of triangles.

51. How can angle bisectors be used in solving problems involving the excircles of a triangle?

Excircles are circles tangent to one side of a triangle and the extensions of the other two sides. The centers of the excircles lie on the external angle bisectors of the triangle. Understanding the properties of angle bisectors is crucial in solving problems involving excircles and their relationships to the triangle.

52. What is the relationship between angle bisectors and the symmedian point of a triangle?

The symmedian point is the point where the three symmedians of a triangle intersect