Euclid's Elements Book I

23 definitions · 5 postulates · 5 common notions · 48 propositions

Proposition I.5 · Theorem

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.

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let ABC be an isosceles triangle with AB equal to AC. The claim: the base angles ABC, ACB are equal, and if the equal sides are produced, the angles under the base are equal too. (Dragging C moves it on the circle with centre A, so the two sides stay equal.)
  2. Let AB, AC be produced further in straight lines BD, CE.
    Postulate 2
    Postulate 2: To produce a finite straight line continuously in a straight line.
  3. Let a point F be taken at random on BD (you can drag it), and from the greater AE let AG be cut off equal to AF.
    Proposition I.3
    Proposition I.3: Given two unequal straight lines, to cut off from the greater a straight line equal to the less.
  4. Let FC, GB be joined.
    Postulate 1
    Postulate 1: To draw a straight line from any point to any point.
  5. Since AF is equal to AG and AB to AC, the two sides FA, AC are equal to the two sides GA, AB, and they contain a common angle, FAG; therefore the base FC is equal to the base GB, the triangle AFC is equal to the triangle AGB, and the remaining angles are equal: ACF to ABG, and AFC to AGB.
    Proposition I.4
    Proposition I.4: If two triangles have 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.
  6. Since the whole AF is equal to the whole AG, and in these AB is equal to AC, the remainder BF is equal to the remainder CG.
    Common Notion 3
    Common Notion 3: If equals be subtracted from equals, the remainders are equal.
  7. And FC was proved equal to GB; so the two sides BF, FC are equal to the two sides CG, GB, the angle BFC is equal to the angle CGB, and the base BC is common to them; therefore the triangle BFC is equal to the triangle CGB, and the remaining angles are equal: FBC to GCB, and BCF to CBG.
    Proposition I.4
    Proposition I.4: If two triangles have 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.
  8. The whole angle ABG was proved equal to the whole angle ACF, and in these the angle CBG is equal to BCF; therefore the remaining angle ABC is equal to the remaining angle ACB, and they are at the base. And the angle FBC was proved equal to GCB, and they are under the base.
    Common Notion 3
    Common Notion 3: If equals be subtracted from equals, the remainders are equal.