Deep Dive: Angle Bisector Proportionality Theorem and Application of Congruence

In this lesson, some interesting properties of quadrilaterals, trapezoids, and angle bisectors of triangles will be explored. Each of these will be proven using congruence and similarity.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Properties of congruence.
Properties of similarity.

Explore

Investigating Properties of a Quadrilateral

In the net, a quadrilateral, the segments divide the sides into eight congruent segments.

Use the measuring tool to investigate how the segments divide each other inside the quadrilateral.
Explore what happens when the vertices are moved!

Example

Investigating the Midpoints of a Quadrilateral

The Triangle Midsegment Theorem gives a relationship between a midsegment and a side of a triangle. There too, is an exciting result for quadrilaterals, formed by the midpoints of the sides of a quadrilateral. Illustrated in the diagram are $P,$ $Q,$ $R,$ and $S$ which are midpoints of the sides of the quadrilateral $A B C D .$

Show that $P Q R S$ is a parallelogram, and that $\overline{P R}$ and $\overline{Q S}$ bisect each other.

Hint

Draw a diagonal in quadrilateral $A B C D$ and focus on the two triangles.

Solution

Parallelogram

Draw diagonal $\overline{A C}$ of quadrilateral $A B C D$ and focus on the two triangles $△ A B C$ and $△ A D C .$

According to the Triangle Midsegment Theorem, both

\overline{P Q}

and

\overline{S R}

are parallel to the diagonal

\overline{A C},

and they are half the length of

\overline{A C} .

That means these midsegments are parallel to each other, and they have the same length.

\overline{P Q} P Q ∥ \overline{S R} = S R

These relationships can be plotted on the diagram.

Similarly, $\overline{P S}$ and $\overline{Q R}$ are also parallel and have the same length.

By definition, when the opposite sides of a quadrilateral are parallel, then it is a parallelogram. Therefore, the quadrilateral $P Q R S$ is a parallelogram.

Bisecting Diagonals

To show that the diagonals $\overline{P R}$ and $\overline{Q S}$ bisect each other, focus on two of the triangles formed by these diagonals.

These triangles contain the following properties.

Claim	Justification
$\overline{P Q} ≅ \overline{S R}$	Proved previously
$∠ R P Q ≅ ∠ P R S$	Alternate Interior Angles Theorem
$∠ P Q S ≅ ∠ R S Q$	Alternate Interior Angles Theorem

These claims can be shown in the diagram.

It can be seen that triangles

△ P Q M

and

△ R S M

have two pairs of congruent angles, and the included sides are also congruent. According to the Angle-Side-Angle (ASA) Congruence Theorem, the triangles are congruent.

△ P Q M ≅ △ R S M

Corresponding parts of congruent triangles are congruent.

\overline{P M} \overline{Q M} ≅ \overline{R M} ≅ \overline{S M}

This completes the proof that

\overline{P R}

and

\overline{Q S}

bisect each other.

Explore

Investigating Angle Bisectors of Triangles

The next part of this lesson focuses on triangles. The diagram shows a triangle with one of its angle bisectors drawn. Move the vertices of the triangle and find a relationship between the displayed segment measures.

Discussion

Triangle Angle Bisector Theorem

The relationship stated in the following theorem can be checked on the previous applet for different triangles.

The angle bisector of an interior angle of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle.

In the figure, if $ℓ$ is an angle bisector, then the following equation holds true.

$\frac{A D}{D C} = \frac{A B}{B C}$

Proof

In $△ A B C,$ consider the angle bisector $ℓ$ that divides $∠ A B C$ into two congruent angles. Let $∠ 1$ and $∠ 2$ be these congruent angles.

By the Parallel Postulate, a parallel line to $ℓ$ can be drawn through $A .$ Additionally, if $\overline{B C}$ is extended, it will intersect this line. Let $E$ be their point of intersection.

Let $∠ 3$ be the alternate interior angle to $∠ 1$ formed at $A .$ Also, let $∠ 4$ be the corresponding angle to $∠ 2$ formed at $E .$

By the Corresponding Angles Theorem,

∠ 2

is congruent to

∠ 4 .

Remember that it is also known that

∠ 1

is congruent to

∠ 2 .

By the Transitive Property of Congruence,

∠ 1

and

∠ 4

are congruent angles.

{∠ 1 ≅ ∠ 2 ∠ 2 ≅ ∠ 4 \Rightarrow ∠ 1 ≅ ∠ 4

Additionally, by the Alternate Interior Angles Theorem,

∠ 1

is congruent to

∠ 3 .

Using the Transitive Property of Congruence one more time, it can be said that

∠ 3

and

∠ 4

are also congruent angles.

{∠ 1 ≅ ∠ 3 ∠ 1 ≅ ∠ 4 \Rightarrow ∠ 3 ≅ ∠ 4

This can be shown in the diagram.

Note that

△ A C E

is divided by

ℓ,

which is parallel to

\overline{A E} .

Therefore, by the Triangle Proportionality Theorem,

ℓ

divides the other two sides of this triangle proportionally.

\frac{A D}{D C} = \frac{E B}{B C}

The Converse Isosceles Triangle Theorem states that if two angles in a triangle are congruent, the sides opposite them are congruent. This means that

\overline{E B}

is congruent to

\overline{A B} .

Therefore, by the definition of congruent segments, they have the same length.

A B

can be substituted for

E B

in the above proportion.

\frac{A D}{D C} = \frac{E B}{B C} substitute \frac{A D}{D C} = \frac{A B}{B C}

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