# Proposition: 1.05: Isosceles Triangles I

### (Proposition 5 from Book 1 of Euclid's “Elements”)

For isosceles triangles, the angles at the base are equal to one another, and if the equal sides are produced then the angles under the base will be equal to one another.

• Let $ABC$ be an isosceles triangle having the side $AB$ equal to the side $AC$, and let the straight lines $BD$ and $CE$ have been produced in a straight line with $AB$ and $AC$ (respectively) [Post. 2] .
• I say that the angle $ABC$ is equal to $ACB$, and (angle) $CBD$ to $BCE$.

### Modern Formulation

Suppose a given triangle is isosceles, with the sides $$\overline{AB}$$ and $$\overline{BC}$$ equal to each other. Then

1. the angles $$\alpha=\angle{ABC}$$ and $$\beta=\angle{BCA}$$ (at the side $$\overline{BC}$$) are equal to each other,
2. the angles $$\alpha=\angle{ABC}$$ and $$\beta=\angle{BCA}$$ (at the side $$\overline{BC}$$) are equal to each other,

Proofs: 1 Corollaries: 1

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### References

#### Adapted from CC BY-SA 3.0 Sources:

1. Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014