The Angle Bisector Theorem helps you find unknown lengths of sides of triangles, because an angle bisector divides the side opposite that angle into two segments that are proportional to the triangle's other two sides.
Draw on a piece of paper. Interior angles have opposite sides .
Get some linear object and put one endpoint at . Allow its other end to cross side . Divide into two equal angles. You bisected .
The linear object is an angle bisector. When it crossed side opposite , it divided into two smaller triangles and divided side in two.
Replace your object with a drawn line segment or ray. Where the angle bisector crosses side , label that point . The angle bisector is now line segment and creates two smaller triangles, and . Side is now two line segments, and .
Here is one version of the Angle Bisector Theorem:
An angle bisector of a triangle divides the interior angle's opposite side into two segments that are proportional to the other two sides of the triangle.
Angle bisector cuts side into two line segments, and . and relate to sides () and () in the same proportion as and relate to each other. and , created by the angle bisector, are also similarly proportional.
Ratios compare values. You can establish ratios between sides and , and line segments and . Proportions compare ratios; you can learn if two ratios are equal.
For with angle bisector , sides and , and side divided into and , we can set up ratios between the sides and line segments and compare them:
Line segment (from angle bisector ) has the same ratio to line segment as the triangle's side has to side .
Be careful to set up the ratios correctly. is the smaller of the two line segments, so it is the numerator of our ratio. is the longer line segment. So is to as the triangle's shorter side , is to the triangle's longer side .
Other ratios emerge, too. Compare line segments of the side divided by the angle bisector to the remaining sides:
The relationship of the two sides of the new, smaller is the same relationship as the two sides of new, smaller . They are in the same proportions.
Suppose we are told a line segment divides side into and , of lengths 10 cm and 30 cm. We are also told side is 30 cm and side is 90 cm. See if the ratios are proportional to each other:
We see is the same ratio as , so is an angle bisector.
If we know the length of original sides and , we can use the Angle Bisector Theorem to find the unknown length of side . The angle bisector divides side into and (the total length of side , ).
Assume these lengths:
Recall our ratios and substitute the values:
Using cross multiplication () and then division () , we get = 50 meters.
You are now able to define the Angle Bisector Theorem, use ratios and proportions to verify an angle is a bisector, use the Angle Bisector Theorem to find the unknown lengths of sides of triangles, and identify an angle bisector by evaluating the lengths of the sides of the triangle.
Once you view the video, examine the graphics, and read the lesson, you will be able to:
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