These Theorems are ordered so that they build on each other. This means that you shouldn't use any later theorems to prove them but feel free to use any previous ones. Also remember that Euclid didn't have much concept of measurement. He didn't use anything that kept measurement like rulers or stiff compasses so you should try to do the same.

Theorem 1.1:

Theorem 1.2:

Theorem 1.3:

Theorem 1.4(Isosceles Triangle Theorem):

Theorem 1.5:

Theorem 1.6 (SSS):

Theorem 1.7:

Theorem 1.8:

Theorem 1.9:

Theorem 1.10:

Theorem 1.11:

Theorem 1.12:

Theorem 1.13(Vertical Angle Theorem):

Theorem 1.14:Exterior Angle Theorem:

Let ΔABC be a triangle and let D be a point on the ray AB emana ting from A such that B is between A and D. Then the angle CBD is greater than the angle ACB and the angle CAB. (The angle ∟CBD is called an exterior angle of the triangle ΔABC.)

Theorem 1.15:

Any two angles in a triangle are less than two right angles.

Theorem 1.16:

In any triangle the greater side subtends the greater angle.

Theorem 1.17:

In any triangle the greater angle is subtended by the greater side.

Theorem 1.18:Triangle inequality:

Any two sides of a triangle taken together is greater than the remaining side.

Theorem 1.19:

Let ΔABC be given and let D be a point in the interior of the triangle. Construct triangle ΔADC. Then AB and BC together are greater than AD and DC together but ADC is greater than ABC.

Theorem 1.20:

Given three line segments such that any two together is greater than the third one can construct a triangle with sides congruent to the line segments.

Theorem 1.21:

Given an angle ABC and a line L and a point D on L one can construct an angle with vertex D and one side on L that is congruent to ABC.

Theorem 1.22:Hinge theorem:

If triangles ΔABC and ΔDEF have AB congruent to DE and BC congruent to EF and angle ABC greater than DEF then AC is greater than DF.

Theorem 1.23:Converse to Hinge Theorem:

If triangles ΔABC and ΔDEF have AB congruent to DE and BC congruent to EF and AC is greater than DF then the angle ABC is greater than DEF.

Theorem 1.24:ASA:

If a triangle has a pair of angles and the included side congruent to a pair of angles and the included side of a second triangle then the triangles are congruent.

Theorem 1.25:AAS:

If triangles ΔABC and ΔDEF has angles ABC congruent to DEF and angles BCA congruent to EFD and have sides AB and DE congruent then the triangles are congruent.

Theorem 1.26:Converse of Isosceles Triangle Theorem:

If a triangle has a pair of Theorem congruent angles then it is an Isosceles triangle.

Theorem 1.27:Alternate interior angles (AIA):

If a line L transverses lines M and N in such a manner that a pair of alternate interior angles are congruent then M and N are parallel.

Theorem 1.28:

If a line L transverses lines M and N in such a manner that an exterior angle and an opposite and interior angle on the same side are congruent then M and N are parallel.

Theorem 1.29:

If a line L transverses lines M and N in such a manner that a pair of interior angles on the same side are equal to two right angles then M and N are parallel.

Theorem 1.30:

If two lines have a common perpendicular, meaning a line that is perpendicular to both of the lines, then they are parallel.

Theorem 1.31:

A point P is on the perpendicular bisector of a line segment AB if and only if it is equidistant from A and B. That is if AP is congruent to BP.

Theorem 1.32:

A point is on the bisector of an angle is and only if it is equidistant from the sides of the angle.

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