Proofs Using Similarity

In geometry, proving that two triangles are similar is a powerful tool. Once you establish that triangles are similar, you can prove that their corresponding angles are congruent and their corresponding sides are proportional. This forms the foundation for many important geometric proofs.

Core Similarity Theorems

To use similarity in a proof, you first need to prove that the triangles are similar using one of these three methods:

• AA (Angle-Angle) Similarity: If two angles of one triangle are congruent to two angles of another, the triangles are similar.
• SAS (Side-Angle-Side) Similarity: If two sides are proportional and their included angle is congruent, the triangles are similar.
• SSS (Side-Side-Side) Similarity: If all three corresponding sides are proportional, the triangles are similar.

Proving Proportional Relationships

When you prove $\triangle ABC \sim \triangle DEF$, you instantly know that their side lengths form equal ratios: $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$ Cross-multiplying these ratios allows you to prove segment products.

Example: Intersecting Chords Theorem If two chords $AB$ and $CD$ intersect inside a circle at point $E$, we can prove that $AE \cdot EB = CE \cdot ED$.

1. Draw line segments $AC$ and $BD$ to form $\triangle AEC$ and $\triangle DEB$.
2. $\angle A \cong \angle D$ because they are inscribed angles intercepting the same arc $CB$.
3. $\angle C \cong \angle B$ because they intercept the same arc $AD$.
4. By AA Similarity, $\triangle AEC \sim \triangle DEB$.
5. Because corresponding sides are proportional: $\frac{AE}{DE} = \frac{CE}{BE}$.
6. Cross-multiply to get the final proof: $AE \cdot BE = CE \cdot DE$.

Proving the Pythagorean Theorem via Similarity

Similarity provides one of the most elegant proofs of the Pythagorean Theorem ($a^2 + b^2 = c^2$).

Consider a right triangle $\triangle ABC$ with the right angle at $C$. Draw an altitude from $C$ to the hypotenuse $AB$ at point $D$. This altitude splits the hypotenuse $c$ into two segments, $x$ (adjacent to side $b$) and $y$ (adjacent to side $a$).

1. Identify Similar Triangles: The altitude to the hypotenuse of a right triangle divides it into two smaller triangles that are similar to the original triangle and to each other. Therefore, $\triangle ABC \sim \triangle ACD \sim \triangle CBD$.
2. Set Up Proportions:
   • From $\triangle ABC \sim \triangle ACD$, the ratio of the hypotenuse to the shorter leg is constant: $\frac{c}{b} = \frac{b}{x}$. Cross-multiplying gives $b^2 = cx$.
   • From $\triangle ABC \sim \triangle CBD$, the ratio of the hypotenuse to the longer leg is constant: $\frac{c}{a} = \frac{a}{y}$. Cross-multiplying gives $a^2 = cy$.
3. Combine the Equations: Add the two equations together: $a^2 + b^2 = cy + cx$
4. Factor and Substitute: Factor out $c$ on the right side: $a^2 + b^2 = c(y + x)$ Since the segments $x$ and $y$ make up the entire hypotenuse $c$ ($x + y = c$), we substitute $c$ back in: $a^2 + b^2 = c(c)$ $a^2 + b^2 = c^2$