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Triangle Angle Bisector Theorem

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Triangle Angle Bisector Theorem

Study Guide

Key Definition

The triangle angle bisector theorem states that the angle bisector of a triangle's angle divides the opposite side into two segments that are proportional to the other two sides: $\frac{BD}{DC} = \frac{AB}{AC}$.

Important Notes

The segments divided by the angle bisector are proportional to the opposite sides.
The theorem helps in solving various geometric problems.
Remember the proportional relationships: $\frac{BD}{DC} = \frac{AB}{AC}$
Useful in triangle similarity and congruence problems.
Angle bisectors are not necessarily medians.

Mathematical Notation

$\angle$ represents an angle

$\triangle$ represents a triangle

$\frac{a}{b}$ represents a fraction

$=$ represents equality

$\sim$ represents similarity

Remember to use proper notation when solving problems

Why It Works

The angle bisector theorem works because it relies on the properties of similar triangles and proportionality, which are fundamental in geometry.

Remember

The key formula is $\frac{BD}{DC} = \frac{AB}{AC}$.

Quick Reference

Angle Bisector Theorem:$\frac{BD}{DC} = \frac{AB}{AC}$

Understanding Triangle Angle Bisector Theorem

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

Simple explanation: The angle bisector divides the opposite side into segments that are proportional to the other two sides.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

In $\triangle ABC$, the angle bisector of $\angle A$ divides side $BC$ into segments $BD$ and $DC$. If $AB = 6$ and $AC = 4$, what is $\frac{BD}{DC}$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A teenager is designing a triangular garden. She wants to divide one side proportionally using an angle bisector. If she knows two sides are 8m and 6m, what should be the ratio in which the bisector divides the third side?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Prove the triangle angle bisector theorem using similar triangles.

Challenge Quiz

Single Choice Quiz

Advanced

Given $\triangle ABC$ with angle bisector $AD$, if $AB = 10$, $AC = 5$, and $BD = 3$, find $DC$.

Recap

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Review key concepts and takeaways