Angle Bisector Theorem - MathBitsNotebook(Geo - CCSS Math)

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The "Angle Bisector" Theorem says that an angle bisector of a triangle will divide the opposite side into two segments that are proportional to the other two sides of the triangle.

beware Be sure to set up the proportion correctly. While proportions can be re-written into various forms, be sure to start with a correct arrangement. For example, in the diagram shown below, a correct proportion may be: angbis2angbis3
Find x. angbis1 It may be helpful to remember that the segments formed by the angle bisector will form a correct proportion when paired with their adjacent triangle sides, such as:

angbis4

ang6

 angbis7

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Theorem2 An angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle. (proof below)

angbispgiven

Auxiliary lines will be needed to create similar triangles.

 whiteone1
Proof:
Statements Reasons
angp1 1. Given
angp2 2. An angle bisector is a ray in the interior of an angle forming two congruent angles.
angp3 3. Parallel Postulate: through a point not on a line there is only one line parallel to the given line.
angp5 4. If 2 || lines are cut by a transversal, the corresponding angles are congruent.
angp55 5. If 2 || lines are cut by a transversal, the alternate interior angles are congruent.
angp66 6. Substitution
angp7 7. Side Splitter Theorem: If a line is || to one side of a triangle and intersects the other 2 sides, it divides the sides proportionally.
angp8 8. If 2 angles of a triangle are congruent, the sides opposite the angles are congruent (Isosceles triangle).
angp9 9. Congruent segments have = lengths.
angp10 10. Substitution

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Topical Outline | Geometry Outline | MathBitsNotebook.com | MathBits' Teacher Resources Terms of Use Contact Person: Donna Roberts

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