Euclidean Geometry Exam 1

I.1

To make an equilateral triangle on a segment

I.2/3

To copy a given segment at a point or from a longer segment

I.4

SAS congruence; given two triangles, triangle ABC and triangle DEF, if AB=DE and AC=DF

I.5

Pons Asinorum; in an isosceles triangle the angles at the base will be equal and if the equal sides be produced the angles under the base will be equal

I.6

converse of I.5, in a triangle in which we have equal sides then the sides opposite the equal angles will be equal

I.8

SSS congruence; given triangle ABC and triangle DEF if AB=DE, AC=DF, BC=EF then angle A=angle D, angle B=angle E, and angle C=angle F

I.9

Bisect the given angle

I.10

bisect a segment

I.11/12

Make a perpendicular angle to the line through the point

I.13

If a line stands on another line, then the adjacent angles are either both right or the sum of the two angles is 180 degrees, aka supplementary angles add up to two right angles

I.14

Converse of I.13; If adjacent angles add to two right angles, ten the sides not in common lie in a straight line

I.15

Vertical angles are equal

I.16

exterior angle theorem; given triangle ABC if we extend AB by BD, the exterior angle formed CBD will be greater than each of the remote or non-adjacent interior angles (angle A and C)

I.17

Converse of postulate 5; in a triangle the sum of any two of its interior angles will be strictly less than two right angles

I.18/19

In any triangle, the greater angle or side will be opposite the greater side or angle

I.20

the triangle inequality; in any triangle, the sum of any two sides will be greater than the third side

I.22

converse of triangle inequality; given 3 segments AB, CD, EF for which the sum of any two is greater than the third, then there is a triangle whose sides equal the three segments

I.23

copy angles; given angle A and line BC to make an angle at B with BC as a side to angle A

Definition 10

When a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the line standing on the other is called a perpendicular to that on which it stands

Definition 15

A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another

Definition 20

Of Trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides acute

Postulate 1

To draw a straight line from any point to any point

Postulate 2

To produce a finite straight line continuously in a straight line

Postulate 3

To describe a circle with any center and distance

Postulate 4

That all right angles are equal to one another

Common Notion 1

Things which are equal to the same thing are also equal to one another

Common Notion 2

If equals be added to equals, the wholes are equal

Common Notion 3

If equals be subtracted from equals, the remainders are equal

Common Notion 5

The whole is greater than the part

Proposition 1

given a finite straight line to construct an equilateral triangle

Proposition 2

To draw at a given point (as extremity) a line equal to a given line

Proposition 3

To cut off from a greater segment a portion equal to a lesser segment

Proposition 4

SAS

Proposition 5

In an isosceles triangle the angles at the base are equal to each other, and, if the equal straight lines be produced further, the angles under the base will be equal to one another

Proposition 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

Proposition 7

Given two straight lines constructed on a straight line, this is not true

Proposition 8

SSS

Proposition 9

To bisect a given rectilineal angle

Proposition 10

To bisect a given finite straight line

Proposition 11

To draw a straight line at right angles to a given straight line from a given point on it

Proposition 12

To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line

Proposition 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

Proposition 14

If with any straight line, and at a point 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

Proposition 15

If two straight lines cut one another, they make the vertical angles equal to one another

Proposition 16

In any triangle, if one of the sides be produced, the exterior angle is greater than either of the interior and opposite angles

Proposition 17

In any triangle two angles taken together in any manner are less than two right angles

Proposition 18

In any triangle the greater side subtends the greater angle

Proposition 19

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

Proposition 20

In any triangle two sides taken together in any manner are greater than the remaining one

Definition 1

A point is that which has no part.