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Congruence & Similarity
to Solve Problems

Master theorems about triangles: a parallel line divides sides proportionally, and prove the Pythagorean Theorem using triangle similarity.

Geometry

What Are Congruence & Similarity?

Congruent Figures
Same shape and same size. All corresponding sides and angles are equal: \(\triangle ABC \cong \triangle DEF\).
Similar Figures
Same shape, different size. Corresponding angles are equal and sides are proportional: \(\triangle ABC \sim \triangle DEF\).

Similar Triangles Have Proportional Sides

\(A\)
\(B\)
\(C\)
\(12\)
\(10\)
\(8\)
\(\sim\)
\(D\)
\(E\)
\(F\)
\(6\)
\(5\)
\(4\)

Why Does This Matter?

These theorems let you find unknown lengths and prove geometric relationships without measuring directly. They appear in architecture, engineering, map-making, and standardized tests.

Parallel-Line Theorem
A line parallel to one side of a triangle divides the other two sides proportionally.
Pythagorean Proof
The Pythagorean Theorem can be proved using triangle similarity: a clean, elegant argument.

Triangle Proportionality Theorem

\(A\)
\(B\)
\(C\)
\(D\)
\(E\)
\(AD\)
\(AE\)
\(DB\)
\(EC\)
\(\overline{DE} \parallel \overline{BC}\)

Why Proportionality Works

1
AA Similarity: Since \(\overline{DE} \parallel \overline{BC}\), corresponding angles \(\angle ADE = \angle ABC\) and \(\angle AED = \angle ACB\).
2
Similar Triangles: By AA, \(\triangle ADE \sim \triangle ABC\).
3
Proportional Sides: From similarity: \(\frac{AD}{AB} = \frac{AE}{AC}\). Rewrite using \(AB = AD + DB\) and \(AC = AE + EC\).
4
Result: Cross-multiply and simplify to get \(\frac{AD}{DB} = \frac{AE}{EC}\).

Proving \(a^2 + b^2 = c^2\) with Similarity

\(A\)
\(B\)
\(C\)
\(H\)
\(b\)
\(a\)
\(c\)
\(h\)

Completing the Proof

1
From \(\triangle ACH \sim \triangle ABC\): \(\frac{b}{c} = \frac{AH}{b}\), so \(b^2 = c \cdot AH\).
2
From \(\triangle CBH \sim \triangle ABC\): \(\frac{a}{c} = \frac{BH}{a}\), so \(a^2 = c \cdot BH\).
3
Add the two equations: \(a^2 + b^2 = c \cdot BH + c \cdot AH = c(AH + BH)\).
4
Since \(AH + BH = c\): \(a^2 + b^2 = c \cdot c = c^2\). ✓

Congruence vs. Similarity

Congruence (\(\cong\)) Similarity (\(\sim\))
Shape Same Same
Size Same Can differ
Angles All equal All equal
Sides Equal Proportional
Scale Factor \(k = 1\) \(k\) can be any positive number
Tests SSS, SAS, ASA, AAS AA, SAS~, SSS~

How to Apply These Theorems

1
Identify parallel lines or right angles in the figure. Look for altitude lines or midsegments.
2
Establish similarity using AA, SAS~, or SSS~. Write the similarity statement with vertices in corresponding order.
3
Set up a proportion using corresponding sides. Keep the same triangle on top (or bottom) throughout.
4
Cross-multiply and solve for the unknown length.
5
Verify: check your answer makes sense: the longer side should belong to the larger triangle.

Example 1: Parallel Line Problem

\(A\)
\(B\)
\(C\)
\(D\)
\(E\)
\(6\)
\(8\)
\(9\)
\(x\)
\(\overline{DE} \parallel \overline{BC}\)

Example 2: Altitude to the Hypotenuse

\(A\)
\(B\)
\(C\)
\(H\)
\(AH = 4\)
\(HB = 9\)
\(h = ?\)

Example 3: Finding a Missing Side

\(P\)
\(Q\)
\(R\)
\(8\)
\(10\)
\(6\)
\(\sim\)
\(X\)
\(Y\)
\(Z\)
\(12\)
\(x\)
\(y\)

The Converse: Proving Lines Parallel

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

Common Mistakes

✕ Wrong Correspondence
Writing \(\frac{AD}{AE} = \frac{DB}{EC}\) instead of \(\frac{AD}{DB} = \frac{AE}{EC}\). Match the same side in numerator and denominator.
✕ Mixing Similarity Tests
Using SSA to prove similarity: SSA is not a valid test. Use AA, SAS~, or SSS~ only.
✕ Forgetting Vertex Order
Writing \(\triangle ABC \sim \triangle EDF\) when the correct order is \(\triangle ABC \sim \triangle DEF\). Order encodes correspondence.

Key Theorems Summary

Theorem Statement Key Condition
Proportionality \(\frac{AD}{DB} = \frac{AE}{EC}\) \(\overline{DE} \parallel \overline{BC}\)
Converse \(\overline{DE} \parallel \overline{BC}\) Sides divided proportionally
Pythagorean (via ~) \(a^2 + b^2 = c^2\) Altitude to hypotenuse
Geometric Mean \(h^2 = p \cdot q\) Altitude creates two ~ triangles

Question 1 of 5

In \(\triangle ABC\), \(D\) is on \(\overline{AB}\) and \(E\) is on \(\overline{AC}\) with \(\overline{DE} \parallel \overline{BC}\). If \(AD = 5\), \(DB = 10\), and \(AE = 7\), what is \(EC\)?

Question 2 of 5

The altitude to the hypotenuse of a right triangle divides the hypotenuse into segments of length 3 and 12. What is the length of the altitude?

Question 3 of 5

\(\triangle PQR \sim \triangle XYZ\) with a side ratio of \(\frac{PQ}{XY} = \frac{3}{5}\). If \(QR = 9\), what is \(YZ\)?

Question 4 of 5

In \(\triangle ABC\), \(D\) is on \(\overline{AB}\) and \(E\) is on \(\overline{AC}\). If \(\frac{AD}{DB} = \frac{AE}{EC} = \frac{2}{3}\), what can you conclude?

Question 5 of 5

In the similarity proof of the Pythagorean Theorem, the altitude from the right angle to the hypotenuse creates how many similar triangles in total?