# Definition 3

A straight line is at right angles to a plane when it makes right angles with all the straight lines which meet it and are in the plane.

# Definition 4

A plane is at right angles to a plane when the straight lines drawn in one of the planes at right angles to the intersection of the planes are at right angles to the remaining plane.

# Definition 5

The inclination of a straight line to a plane is, assuming a perpendicular drawn from the end of the straight line which is elevated above the plane to the plane, and a straight line joined from the point thus arising to the end of the straight line which is in the plane, the angle contained by the straight line so drawn and the straight line standing up.

## Guide

Although definition 3 states that a line needs to be at right angles with all of the straight lines which meet it and lie in the plane, proposition XI.4 states that it is only necessary that a straight line be at right angles to two lines in the plane in order that it be at right angles to all the rest.

There is an implicit assumption in definition 3 as it speaks of a straight line making right angles with straight lines which meet it and are in the plane. The concept of two lines making a right angle assumes that the two sides of the angles lie in one plane, that is, that two intersecting lines lie in a plane, a statement that is supposedly verified in proposition XI.2.

The concept of a line being perpendicular to a plane is central to solid geometry. It is developed and used in many propositions in Book XI, starting with XI.4.

There is also an implicit assumption in definition 4, namely that the intersection of the two planes is a straight line, a statement that is supposedly verified in proposition XI.3. The concept of planes perpendicular to planes first appears inproposition XI.18 which states that if one straight line drawn in one of the planes is at right angles to the other plane, then the two planes are at perpendicular.

Definition 5 is meant to define the inclination (angle) between a line and a plane as the angle between that line and the projection of it in the plane. This requires that there is a line at right angles to a plane from a point not on the plane which is assured by proposition XI.11. It also requires that the angle constructed in the definition is independent of the construction.