Lecture · 16 min · intermediate

Triangles

Congruence, similarity, and Pythagoras

Euclid, Elements · Book I, Props. 4, 8, 26, 47

Two triangles are congruent when one can be placed on the other and match exactly — a rigid motion. They are similar when a single scale factor does the rest. Pythagoras is not a slogan about $3$ - $4$ - $5$ ; it is a statement about areas on the sides of a right triangle, and Euclid's proof never mentions coordinates.

On the board today

—Use SAS, ASA, SSS as congruence tests (not AAA).
—Scale similar triangles and transfer ratios.
—State Pythagoras as an equality of areas.

Pythagoras (Elements I.47)

a^{2} + b^{2} = c^{2}

Squares on the legs, taken together, equal the square on the hypotenuse.

AAA does not give congruence: all equilateral triangles have the same angles, but not the same size. AAA does give similarity. SAS, ASA, SSS freeze the size as well as the shape. SSA is the awkward cousin — it can produce two non-congruent triangles (the ambiguous case) unless the given angle is right or obtuse.

Worked example · A $6$-$8$-hypotenuse

A right triangle has legs $6$ and $8$ . Find the hypotenuse, then the altitude to the hypotenuse.

1.Hypotenuse $c = 36 + 64 = 10$ .
2.Area is also $\frac{1}{2} ab = 24$ , and $\frac{1}{2} c h = 24$ , so $h = 4.8$ .

Hypotenuse $10$ ; altitude to hypotenuse $24/5$ .

Pause the lecture

Why is AAA not a congruence theorem?

Common mistake. Writing “the triangles are the same” without naming the correspondence of vertices. Congruence is $A B C ≅ D E F$ , which says $A \leftrightarrow D$ , $B \leftrightarrow E$ , $C \leftrightarrow F$ . Shuffle the letters and you shuffle the sides.

Practice this lecture

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