I have just started reading the propositions in book one of Euclid. I hope to be able to read through about 10 per week while I am about my daily activities. Below I listed the propositions contained in the first of Euclid’s 13 books along with some of their modern counterparts. The theorems are mostly the same, just their names have changed. You may also notice that the presentation of material is also quite different than what you might find in a typical geometry course.
| Propositions | Euclid (Thomas Heath Translation) | Modern High School Geometry |
| 1 | On a given finite straight line construct an equilateral triangle. | Construct an equilateral triangle using a given segment. |
| 2 | Place a point(as an extremity) equal to a given straight line. | Construct congruent segments. |
| 3 | Given two unequal straight lines, to cut off from the greater a straight line equal to the less. | |
| 4 | If two triangles have the two sides equal to two sides respectively, and have the angles contained by equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend. | SAS congruence: If two sides and the included angle of one triangle are congruent to two sides and the included of another triangle, then the triangles are congruent. |
| 5 | In an isosceles triangle the angles of the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another. | Isosceles Triangle Theorem: If a triangle has two congruent sides, then the angles opposite to those sides are congruent. |
| 6 | If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another. | Base Angles Theorem: If two angles in a triangle are congruent, then the sides opposite those angles are also congruent. |
| 7 | Given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there cannot be constructed on the same straight line (from its extremities), and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which has the same extremity with it. | |
| 8 | If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the equal straight lines. | SSS congruence: If the corresponding sides of two triangles are congruent, then the triangles are congruent. |
| 9 | To bisect a given rectilinear angle | Construct the bisector of an angle. |
| 10 | To bisect a given finite straight line. | Construct the bisector of an segment. |
| 11 | To draw a straight line at right angles to a given straight line from a given point on it. | Construct a perpendicular line to a given line passing through a point. |
| 12 | To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line. | |
| 13 | If a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles. | Linear Pair Theorem: Linear pairs are supplementary. |
| 14 | If with any straight line, and at a point not on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the two straight lines will be in a straight line with one another. | |
| 15 | If two straight lines cut one another, they make the vertical angles equal to one another. | Vertical Angles Theorem: Vertical angles are congruent. |
| 16 | In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles. | |
| 17 | In any triangle, two angles taken together in any manner are less than two right angles. | |
| 18 | In any triangle, the greater side subtends the greater angle. | |
| 19 | In any triangle, the greater angle is subtended by the greater side. | |
| 20 | In any triangle, two sides taken together in any manner, are greater than the remaining one. | Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the measure of the third side. |
| 21 | If on one of the sides of a triangle from its extremities there be constructed two straight lines meeting within the triangle, the straight lines so constructed will be less than the remaining two sides of the triangle, but will contain a greater angle. | |
| 22 | Out of three straight lines which are equal to the three given straight lines, to construct a triangle: thus it is necessary that two of the straight lines taken together in any manner should be greater than the remaining one. | Construct a triangle with given side lengths, given any two lengths together are greater than the third. |
| 23 | On a given straight line and at a point on it, to construct a rectilinear angle equal to a given rectilinear angle. | Construct congruent angles. |
| 24 | If two triangles have the two sides equal to two sides respectively, but have the one of the angles contained by the equal straight lines greater than the other, they will also have the base greater than the base. | |
| 25 | If two triangles have the two sides equal, to two sides respectively, but have the base greater than the base, they will also have the one of the angles contained by the equal straight lines greater than the other. | |
26 | ASA congruence: If two angles of one triangle and the included side are congruent to two corresponding angles and the included side of another triangle, then the triangles are congruent. | |
| 27 | If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another. | |
28 | If a Straight line falling on two straight lines make the exterior angle equal to the interior and opposite angle on the same side, or the interior angles on the same side equal to two right angles, the straight lines will be parallel to one another. | Converse Corresponding Angles Theorem: If two lines are cut by a transversal such that corresponding angles are congruent, then the two lines are parallel. |
| Converse Consecutive Interior Angles Theorem: If to lines are cut by a transversal such that consecutive interior angles are supplementary, then the two lines are parallel. | ||
29 | A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle equal to the interior and opposite angle, and the interior angles on the same side equal to two right angles. | Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then alternate interior angles are congruent. |
| Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then alternate interior angles are congruent. | ||
| Corresponding Angles Theorem: if two parallel lines are cut by a transversal, then alternate interior angles are congruent. | ||
| Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary. | ||
| 31 | Through a given point, two draw a straight line parallel to a given straight line. | Construct a line parallel to a given line passing through a point. |
32 | In any triangle, if one side is produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles are equal to two right angles. | Triangle Sum Theorem: The sum of the interior angles of a triangle is 180 degrees. |
| Exterior Angle Theorem: The measure of the exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. | ||
| 33 | The straight lines joining equal and parallel straight lines(at their extremities which are) in the same directions (respectively) are themselves equal and parallel. | If two sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram. |
| 34 | In parallelogramic areas the opposite sides and angles are equal to one another, and diameter bisects the areas. | If a quadrilateral is a parallelogram, then its opposite sides are congruent, opposite angles are congruent, and the diagonals bisect each other. |
| 35 | Parallelograms which are on the same base and in the same parallels are equal to each other. | |
| 36 | Parallelograms which are on equal bases and in the same parallels are are equal to one another. | |
| 37 | Triangles which are on the same base and in the same parallels are equal to one another. | |
| 38 | Triangles which are on equal bases and in the same parallels are equal to one another. | |
| 39 | Equal triangles which are on the same base and on the same side are also in the same parallels. | |
| 40 | Equal triangles which are on equal bases and on the same side are also in the same parallels. | |
| 41 | If a parallelogram have the same base with a triangle and be in the same parallels, the parallelogram is double of the triangle. | |
| 42 | To construct, in a given rectilinear angle, a parallelogram equal to a given triangle. | |
| 43 | In any parallelogram the complements of the parallelograms about the diameter are equal to one another. | |
| 44 | To a given straight line to apply, in a given rectilinear angle, a parallelogram equal to a given triangle. | |
| 45 | To construct a given rectilinear angle, a parallelogram equal to a given rectilinear figure. | |
| 46 | On a given straight line to describe a square. | Construct a square whose sides are a given length. |
| 47 | In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides of the right angle. | Pythagorean Theorem: The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse. |
| 48 | If in a triangle the square of one of the sides be equal to the squares on the remaining two sides of the triangle, the angle contained by the remaining two sides of the triangle is right. | Converse Pythagorean Theorem: If the sum of the squares of the lengths of two sides of a triangle equals the square of the length of the third side, then the triangle is a right triangle. |
