# Book:Euclid/The Elements/Book I

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## Euclid:

## Euclid: *The Elements: Book I*

Published $\text {c. 300 B.C.E}$

### Contents

Book $\text{I}$: Straight Line Geometry

- Proposition $1$: Construction of Equilateral Triangle
- Proposition $2$: Construction of Equal Straight Line
- Proposition $3$: Construction of Equal Straight Lines from Unequal
- Proposition $4$: Triangle Side-Angle-Side Congruence
- Proposition $5$: Isosceles Triangle has Two Equal Angles
- Proposition $6$: Triangle with Two Equal Angles is Isosceles
- Proposition $7$: Two Lines Meet at Unique Point
- Proposition $8$: Triangle Side-Side-Side Congruence
- Proposition $9$: Bisection of Angle
- Proposition $10$: Bisection of Straight Line
- Proposition $11$: Construction of Perpendicular Line
- Proposition $12$: Perpendicular through Given Point
- Proposition $13$: Two Angles on Straight Line make Two Right Angles
- Proposition $14$: Two Angles making Two Right Angles make Straight Line
- Proposition $15$: Two Straight Lines make Equal Opposite Angles
- Proposition $16$: External Angle of Triangle is Greater than Internal Opposite
- Proposition $17$: Two Angles of Triangle are Less than Two Right Angles
- Proposition $18$: Greater Side of Triangle Subtends Greater Angle
- Proposition $19$: Greater Angle of Triangle Subtended by Greater Side
- Proposition $20$: Sum of Two Sides of Triangle Greater than Third Side
- Proposition $21$: Lines Through Endpoints of One Side of Triangle to Point Inside Triangle is Less than Sum of Other Sides
- Proposition $22$: Construction of Triangle from Given Lengths
- Proposition $23$: Construction of Equal Angle
- Proposition $24$: Hinge Theorem
- Proposition $25$: Converse Hinge Theorem
- Proposition $26$: Triangle Angle-Side-Angle and Side-Angle-Angle Congruence
- Proposition $27$: Equal Alternate Angles implies Parallel Lines
- Proposition $28$: Equal Corresponding Angles or Supplementary Interior Angles implies Parallel Lines
- Proposition $29$: Parallelism implies Equal Alternate Angles, Corresponding Angles, and Supplementary Interior Angles
- Proposition $30$: Parallelism is Transitive Relation
- Proposition $31$: Construction of Parallel Line
- Proposition $32: \text{ Part } 1$: External Angle of Triangle equals Sum of other Internal Angles
- Proposition $32: \text{ Part } 2$: Sum of Angles of Triangle equals Two Right Angles
- Proposition $33$: Lines Joining Equal and Parallel Straight Lines are Parallel
- Proposition $34$: Opposite Sides and Angles of Parallelogram are Equal
- Proposition $35$: Parallelograms with Same Base and Same Height have Equal Area
- Proposition $36$: Parallelograms with Equal Base and Same Height have Equal Area
- Proposition $37$: Triangles with Same Base and Same Height have Equal Area
- Proposition $38$: Triangles with Equal Base and Same Height have Equal Area
- Proposition $39$: Equal Sized Triangles on Same Base have Same Height
- Proposition $40$: Equal Sized Triangles on Equal Base have Same Height
- Proposition $41$: Parallelogram on Same Base as Triangle has Twice its Area
- Proposition $42$: Construction of Parallelogram equal to Triangle in Given Angle
- Proposition $43$: Complements of Parallelograms are Equal
- Proposition $44$: Construction of Parallelogram on Given Line equal to Triangle in Given Angle
- Proposition $45$: Construction of Parallelogram in Given Angle equal to Given Polygon
- Proposition $46$: Construction of Square on Given Straight Line
- Proposition $47$: Pythagoras's Theorem
- Proposition $48$: Square equals Sum of Squares implies Right Triangle