Warning: This name is used in different way in different textbooks. In part textbooks, it refers to the organize which says that any suggest on an angle bisector is equidistant from the two sides that the angle.

What many textbooks contact the angle Bisector to organize is this:

one angle bisector in a triangle divides opposing side into two segments which space in the very same proportion together the various other two political parties of the triangle. In the figure above, ns l ¯ bisects ∠ R p Q , therefore R l together Q = p R p Q .

## Triangle edge Bisector organize

one edge bisector of an angle of a triangle divides the opposite side in 2 segments that are proportional come the other two political parties of the triangle. by the edge Bisector Theorem,

B D D C = A B A C

Proof:

draw B E ↔ ∥ A D ↔ .

prolong C A ¯ to accomplish B E ↔ at point E . through the Side-Splitter Theorem,

C D D B = C A A E ---------(1)

The angles ∠ 4 and also ∠ 1 are corresponding angles.

So, ∠ 4 ≅ ∠ 1

because A D ¯ is a edge bisector of the edge ∠ C A B , ∠ 1 ≅ ∠ 2 .

by the alternating Interior edge Theorem , ∠ 2 ≅ ∠ 3 .

Therefore, by leg property, ∠ 4 ≅ ∠ 3 .

due to the fact that the angles ∠ 3 and ∠ 4 are congruent , the triangle Δ A B E is an isosceles triangle with A E = A B .

replacing A E by A B in equation (1),

C D D B = C A A B

You are watching: Which type of triangle will always have a perpendicular bisector that is also an angle bisector?

Example: discover the worth of x .

by Triangle-Angle-Bisector Theorem,

A B B C = A D D C .

Substitute.

5 12 = 3.5 x

overcome multiply.

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5 x = 42

divide both sides by 5 .

5 x 5 = 42 5     x = 8.4