# Two Angles of Triangle Less than Two Right Angles

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## Contents

## Theorem

In the words of Euclid:

*In any triangle two angles taken together in any manner are less than two right angles.*

(*The Elements*: Book $\text{I}$: Proposition $17$)

## Proof

Let $\triangle ABC$ be a triangle.

Let the side $BC$ be extended to $D$.

Since the angle $\angle ACD$ is an external angle of $\triangle ABC$, it follows that it is greater than both $\angle BAC$ and $\angle ABC$.

We add $\angle ACB$ to both, so that $\angle ACD + \angle ACB$ is greater than $\angle ABC + \angle ACB$.

But $\angle ACD + \angle ACB$ is equal to two right angles.

Therefore $\angle ABC + \angle ACB$ is less than two right angles.

In a similar manner we show that the same applies to the other two pairs of internal angles of $\triangle ABC$.

$\blacksquare$

## Historical Note

This theorem is Proposition $17$ of Book $\text{I}$ of Euclid's *The Elements*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 1*(2nd ed.) ... (previous) ... (next): Book $\text{I}$. Propositions - 1968: M.N. Aref and William Wernick:
*Problems & Solutions in Euclidean Geometry*... (previous) ... (next): Chapter $1$: Triangles and Polygons: Theorems and Corollaries $1.8$: Corollary $3$