Step 4: Join the point of intersection with the vertex of the angle.Step 3: With the intersection points as the center, mentioned in step 2, and without any change in the radius, draw two arcs such that they intersect each other and lie between the intersection points on the legs of the angle.Step 2: With one end of the horizontal ray which makes the angle as the center and measuring any width (less than the length of the ray drawn) in the compass, draw an arc that intersects the two rays of the angle at any two points.Here, we can observe that one ray is horizontal just to ease out our constructions. To construct an angle bisector for an angle, follow the steps given below. Topics Related to Constructing Angle BisectorsĬheck out some interesting topics related to constructing angle bisectors.įAQs on Constructing Angle Bisectors How Do You Construct an Angle Bisector?Īn angle bisector divides an angle into two congruent angles. If in other cases we know the measurement of the angle on which angle bisector is to be constructed, then we can simply use a protractor to construct an angle with half of the measurement of the given angle. It is to be noted that no angle measurements were required for this construction. Thus, ray OE is the angle bisector of ∠COD or ∠AOB. Let us see how equal angles are made using the angle bisector with proof.ġ. OC = OD (radii of the same circular arc)īy the SSS criterion, the two triangles are congruent, which means that ∠COE = ∠DOE. The constructed angle bisector has created two similar triangles. The proof of constructing an angle bisector is given below.įrom the above figure, we see that the angle bisector is constructed for the ∠AOB. This is the required angle bisector of angle AOB. Note that CE = DE, since the two arcs were drawn in this step was of the same radius. Step 2: Without changing the distance between the legs of the compass, draw two arcs with C and D as centers, such that these two arcs intersect at a point named E (in the image). Note that OC = OD, since these are radii of the same circle. Step 1: Span any width of radius in a compass and with O as the center, draw two arcs such that it cut the rays OA and OB at points C and D respectively. This challenge is a fundamental idea behind geometrical constructions.įollow the sequence of steps mentioned below to construct an angle bisector. When no angle measurements have been asked for, we must avoid using a protractor, and use only a ruler and a compass. So, we do not need a protractor in constructing the angle bisector. Note that the measure of the angle is not mentioned here. Let us consider the angle AOB shown below. Any angle can be bisected using an angle bisector. To geometrically construct an angle bisector, we would need a ruler, a pencil, and a compass, and a protractor if the measure of the angle is given. D is such a point on the angle bisector.Construct an Angle Bisector With a CompassĪn angle bisector is a line that bisects or divides an angle into two equal halves. (This is easily proven with congruent triangles and the angle-side-angle postulate). Now, as we've seen, all points on the angle bisector are equidistant from the two sides of the angle. That means that the ratio of their areas will be the same as the ratio of their bases: Let's look at the two triangles formed by the angle bisector, △ABD and △ADC. Instead, we will use one of the other tools already in our pocket for comparing ratios of line segments- triangles with the same height. But here, we will provide a proof that does not rely on such advanced knowledge. Many proofs of this theorem use trigonometry and the law of sines. ProblemĪD is the angle bisector of angle ∠BAC in triangle △ABC (∠BAD≅ ∠CAD). This is another useful tool in problems that require you to compare lengths of different line segments. The angle bisector theorem states that in a triangle, the angle bisector partitions the opposite side of the triangle into two segments, with a ratio that is the same as the ratio between the two sides forming the angle it bisects: The angle bisector is a line that divides an angle into two equal halves, each with the same angle measure. In today's lesson, we will show a straightforward way of proving the Angle Bisector Theorem.
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