Geometry/Chapter 7

Section 7.1 - Parallel Lines

In Euclidean Geometry, lines are parallel if they do not intersect.
Remember that lines extend infinitely, if they are not parallel they will eventually intersect.

Exercises

draw line AB, draw a line CD that is parrallel to AB
find a way to prove that AB is parrallel to CD

Section 7.2 - Perpendicular Lines

In Euclidean Geometry, lines are perpendicular if they intersect at a 90 degree angle.

Exercises

draw line AB, then draw point C, make a line which intersects C and is perpendicular to C
prove that line C is perpendicular to AB

Section 7.3 - Circles

Congruent Circles

Congruent circles are circles that have the same size

Concentric Circles

Concentric circles are similar circles that share a common center

Exercises

draw a circle, find a way to draw a congruent circle, prove they are congruent
draw a circle, then draw a concentric one, prove that they are concentric.

Radius, Diameter, Chord

A radius is a line segment connecting a point on the circle and the center of the circle.
Diameter is a line segment connecting 2 points on the circle and intersecting the center.
A chord is a line segment connecting 2 points on the circle, does not have to intersect the center.

Section 7.4 - Quadrilaterals

A quadlateral is defined as any 4 sided polygon. This means squares, rectangles, etc are all called quadlaterals. In the following sections we will take a closer look at each of the important quadlaterals, how they are defined, and some special properties of each of them.

Rectangles

A Rectangle is defined by having 4 conditions. The first is that it is a 4 sided polygon. The second is that it has 2 pairs of parallel sides. The third is that the pairs of parallel sides are of equal length. The forth is that all angles must be equal to 90 degrees or a right angle.

Lets take an example. Below is a Rectangle. First we note that it has 4 sides, and that they form a polygon, this makes it a 4 sided polygon so the first condition is satisified. Next we notice that 2 of the sides are parallel to each other and that the other 2 sides are parallel to each other, thus the second condition has been meet. Now we notice that the pairs of parallel sides are of equal length, thus the third condition has been meet. The final condition was that all angles must be 90 degrees, as shown in the image they are infact 90 degrees. Because all 4 conditions have been meet we now know that the polygon below is infact a Reactangle.

Special Properties

Ok, now that the reactangle has been defined we need to know whats so special about rectangles.

Length of a diagonal:

Below is the same reactangle as above except it has the diagonal drawn from 1 vertex to the other. The length of this diagonal is equal to the square root of one side squared added to the length of the adjecent side squared. eg $\sqrt{(}{3}^2+1^2)=\sqrt{(}10)=3.162$

The proof of this is quite simple. But first you have to know the pythagorean theorem so read over this if you don't already know it.

1) Angle ABC = 90 degrees - given in the definition of a rectangle.
2) Side AB and side BC and the diagonal form a triangle. - a 3 sided polygon
3) The triangle is also a right angled triangle because it has a right angle in it. - angle ABC
4) The diagonal of the rectangle is the hypotenuse of the triangle.
5) the hypotenuse of a right angled triangle is equal to the square root of the sum of the squares of the sides - pythagorean theorem
6) thus the diagonal is equal to $\sqrt{(}{b}^2+c^2)$

Following this property we can note that if we draw a diagonal we create 2 congruent triangles inside the reactangle.

proof:

below is such a Rectangle with the diagonal drawn. We shall name the vertices A, B, C, D. The diagonal is called AC because the 2 endpoints are A and C. The 2 triangles are ABC and ADC.

1) The 2 triangles share the a side - AC.
2) AD and BC have an equal length - given.
3) angles ABC and ADC are of equal measure - definition of a rectangle.
4) Each triangle has an angle of equal measure - statement 3.
5) by the side angle side theorem each triangle is congruent.

Square

A square has the same properties as a rectangle except that all 4 sides must be equal length

Vocabulary

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