# MATH SUMMARY

### Orthogonality in space

## Project by Rayan Hamdan

Hariri Highschool II

Grade 11 Sc (B)

*Figures are done using Google SketchUp 8

## Orthogonal & Perpendicular Lines

- When two lines are parallel then every line perpendicular (orthogonal) to one is also perpendicular to the other

Orthogonality of lines

- (d) intersects and is perpendicular to (d')
- (d) is also perpendicular (orthogonal) to (d'') even when (d'') does not intersect with (d)

## Line Perpendicular to a Plane

- A line (d) is perpendicular to a plane (P)
**if and only if**(d) is orthogonal to**two**intersecting lines of (P)

## Properties

## Property 1

- If two lines are parallel, then every plane perpendicular to one is also perpendicular to the other

If (d) // (d') and (P) perp. (d) , then (P) perp. (d')

- If two lines are perpendicular to the same plane, they are parallel

If (d) perp. (P) and (d') perp. (P) , then (d) // (d')

## Property 2

- If two planes are parallel, then every line perpendicular to one is perpendicular to the other

If (P) // (P') and (d) perp. (P) , then (d) perp. (P')

- If two planes are perpendicular to the same line, then they are parallel

If (P) perp. (d) and (P') perp. (d) , then (P) // (P')

## Property 3

- If a line (d) is perpendicular to a plane (P) , then every line orthogonal to (d) is in (P) or perpendicular to (d)

If (d) perp. (P) and (d') perp. (d), then (d') belongs to (P) or (d') // (P)

- If a line is perpendicular to a plane, then every line parallel to the plane is orthogonal to this line

If (d) perp. (P) and (d') // (P), then (d') perp. (d)

- If a line is perpendicular to a plane, then every line contained in (P) is perpendicular to this line

- If a line is parallel to a plane, it is parallel to all lines in the plane

## Propert 4

## Property 5

- (D) belongs to (P)

- (D') belongs to (Q)

===> (D) // (D')

Then, (d) // (D) // (D')

## Mediator Plane of a Segment

## Definition

It is the plane perpendicular to [AB] at its midpoint I

## Property

MA=MB

## Proof

## Angle Between a Line and a Plane

H is the orthogonal projection of A on (P)

(d') = (IH) is orthogonal projection of (d) = (IA) on (P)

The angle between the line and the plane is "a"

## Dihedral

## Definition

## Find the Dihedral Angle

- Find a common line (xy)
- Select a straight line in (P) perpendicular to (xy)
- Select a straight line in (Q) perpendicular to (xy)
- The dihedral angle is the one formed by the 2 perpendiculars

## Perpendicular Planes

## Property 1

- Two planes (Q) and (P) are perpendicular if and only if one of them contains a straight line perpendicular to the other.
- If two planes (Q) and (P) are perpendicular, then every line in one of them perpendicular to their line of intersection ([AB]) is perpendicular to the other.

## Property 2

## Property 3

## Common Perpendicular

The perp. to (delta) in (Q) cut (D') at B

(AB) is in (Q) and perp. to the inters. of (P) and (Q)

Then, (AB): perp. (P) ===> perp. (D)

So, (AB) perp. to (D) and (D')

===> [AB] is common perp. to (D) and (D')

## Axis of Circle or other

## Definition

## Property

## Proof

1- Prove line (AO) perp. to circle (or other)

2- Prove AB=AC=AD (HA=HB=HC)

## Bisector of Dihedral

## Definition

[IM] is also the bisector of the dihedral

## Property

M is a point on the bisector, therefore, MN=MN'

- Any point on the bisector of the dihedral is equidistant from the planes