Proving Theorems Using Similarity

HSG.SRT.B.4

High School Geometry

Learning Objectives

• State and prove the Side-Splitter Theorem.
• State and prove the Converse of the Side-Splitter Theorem.
• Prove the Pythagorean Theorem using similar right triangles.
• Explain how similarity provides a proof strategy for geometric relationships.
• Solve complex problems involving triangles and parallel lines.

The Side-Splitter Theorem

If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.

If , then:

Proving Side-Splitter: Step 1

Identify Similar Triangles:

• (Small)
• (Large)

Why?

1. is shared.
2. (Corresponding angles).
3. by AA.

Proving Side-Splitter: Step 2

From Similarity:

Substitute segments:

Proving Side-Splitter: Step 3

Algebraic Manipulation:

The Converse

If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

If , then .

Pythagorean Theorem: A New Proof

Recall:

We usually prove this using area or squares.
Today, we prove it using Similarity.

The Setup: Altitude to Hypotenuse

Draw the altitude from the right angle to the hypotenuse .

This creates three similar right triangles:

1. The Large triangle (Original)
2. The Medium triangle (Left side)
3. The Small triangle (Right side)

Proving Similarity (AA)

• Large & Medium: Both have a right angle; both share .
• Large & Small: Both have a right angle; both share .

Similarity to Proportions (1)

From Large Medium:
(leg / hypotenuse ) = (leg / hypotenuse )

Similarity to Proportions (2)

From Large Small:
(leg / hypotenuse ) = (leg / hypotenuse )

The Final Step: Combine

Since :

Strategy: Similarity as Proof

1. Identify similar figures (usually via AA).
2. Write proportionality statements.
3. Algebraically manipulate to find the proof.

Application: Solving Segments

Problem: . . Find .

Application: Geometric Mean

In a right triangle with altitude :

The altitude is the geometric mean of the two hypotenuse segments.

Watch Out: Which Ratios?

Side-Splitter:
Full Similarity:

DON'T use:

Summary

1. Side-Splitter: proportional segments.
2. Converse: Proportional segments .
3. Pythagorean Proof: Altitude to hypotenuse creates 3 similar triangles.
4. Strategy: Use AA similarity to build the algebra for your proof.

Next Steps

Similarity Applications

Using these theorems to solve complex geometric modeling problems.

Right Triangle Trigonometry

How similarity ratios lead to Sine, Cosine, and Tangent.