Proof
Apply I.3 to mark equal segments on the produced sides, then I.4 to
two pairs of congruent triangles. Common Notion 3 gives equality of
the remaining angles.
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Full neighborhood
Depends on (3)
- I.3Proposition I.3Given two unequal straight lines, to cut off from the greater a straight line equal to the less.
- I.4Proposition I.4If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight…
- 3Common notion 3If equals be subtracted from equals, the remainders are equal.
Required by (dependents) (14)
- I.7Proposition I.7Given two straight lines constructed on a straight line and meeting in a point, there cannot be constructed on the same…
- I.18Proposition I.18In any triangle the greater side subtends the greater angle.
- I.19Proposition I.19In any triangle the greater angle is subtended by the greater side.
- I.20Proposition I.20In any triangle two sides taken together in any manner are greater than the remaining one.
- I.24Proposition I.24If two triangles have the two sides equal to two sides respectively, but have the one of the angles contained by the…
- II.4Proposition II.4If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the…
- II.9Proposition II.9If a straight line be cut into equal and unequal segments, the squares on the unequal segments of the whole are double…
- II.10Proposition II.10If a straight line be bisected and a straight line be added to it in a straight line, the square on the whole with the…
- III.2Proposition III.2If on the circumference of a circle two points be taken at random, the straight line joining the points will fall…
- III.7Proposition III.7If on the diameter of a circle a point be taken which is not the centre, and from the point straight lines fall upon…
- III.20Proposition III.20In a circle the angle at the centre is double of the angle at the circumference, when the angles have the same…
- III.31Proposition III.31In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less…
- IV.10Proposition IV.10To construct an isosceles triangle having each of the angles at the base double of the remaining one.
- XIII.7Proposition XIII.7If three angles of an equilateral pentagon, taken either in order or not in order, be equal, the pentagon will be…
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