Proposition 4
If two triangles have the two sides equal to two sides, respectively, and have the angles contained by the 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.
Suppose you have two triangles ABC and DEF with side AB equal to side DE and side AC equal to DF, and angle BAC equal to angle EDF, as shown below.
We are trying to prove the following four statements:
- Side BC is equal to side EF
- Triangle ABC is equal to triangle DEF
- Angle ABC is equal to angle DEF
- Angle ACB is equal to angle DFE
If we apply triangle ABC to triangle DEF by placing point A on point D and the straight line AB on the straight line DE, then point B will also coincide with point E because AB is equal to DE.
The straight line AC will also coincide with the straight line DF because AB coincides with DE and angle BAC is equal to angle EDF.
Thus, point C coincides with the point F because AC is equal to DF.
So, base BC coincides with the base EF and is equal to it because the point B coincides with the point E and the point C coincides with the point F.
(Common Notion 4: Things which coincide with one another are equal to one another.)
Thus the triangle ABC coincides with the triangle DEF, and the two triangles are equal by Common Notion 4.
Now we know that angle ABC coincides with angle DEF, so those two angles are equal. By similar reasoning angles ACB and DFE are equal as well.
Q.E.D.
(Note: Q.E.D. stands for the Latin “quod erat demonstradum” and means that which was to have been demonstrated. It is common to write Q.E.D. to signify the end of a proof.)
Proposition 5
In isosceles triangles the angles at 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.
The first part of the proof is to prove that the angles that are opposite of the equal sides in an isosceles triangle are equal to one another. This proof relies on Proposition 3 (If two line segments have different lengths, a segment of the longer segment can be cut off that is equal to the length of the shorter segment) and Proposition 4 (If two triangles have two equal sides and an equal angle between the two sides, then the two triangles are equal.)