FITE Notes: Geometry

Geometry starts with the smallest pieces. A point is just a location, no size, drawn as a dot. A line goes straight forever in both directions. A plane is a flat surface that goes forever in every direction, like an endless field.

Segments and rays

• ●A segment is a piece of a line with two endpoints, like the yard line between two markers.
• ●A ray starts at one point and goes forever in one direction, like a flashlight beam.

Angle types

• ●Acute angle: less than 90 degrees (sharp).
• ●Right angle: exactly 90 degrees (a perfect corner).
• ●Obtuse angle: more than 90 but less than 180 degrees (wide open).
• ●Straight angle: exactly 180 degrees (a flat line).

Example: if one angle is 30 degrees, its complement is 60 degrees (30 + 60 = 90) and its supplement is 150 degrees (30 + 150 = 180).

A triangle is a three-sided shape, and it shows up everywhere in geometry. You sort triangles two ways: by their sides and by their angles.

By sides

• ●Equilateral: all three sides equal (and all angles 60 degrees).
• ●Isosceles: two sides equal.
• ●Scalene: no sides equal.

By angles

• ●Acute: all angles less than 90 degrees.
• ●Right: one angle exactly 90 degrees.
• ●Obtuse: one angle more than 90 degrees.

Example: if two angles are 50 degrees and 60 degrees, the third is 180 - 50 - 60 = 70 degrees. Sides of 3, 4, and 5 work because 3 + 4 = 7 is greater than 5.

This one is a star player. It works only on right triangles (a triangle with a 90 degree angle). It links the three sides together.

When to use it

Use it whenever you have a right triangle and you know two sides and need the third. The hypotenuse c is always the side opposite the right angle, and it is always the longest.

1. 1Find the right angle and label the hypotenuse c.
2. 2Plug the two known sides into a^2 + b^2 = c^2.
3. 3Square the numbers, then solve for the missing side.
4. 4Take the square root at the end to get the side length.

Example: legs of 3 and 4. Then 3^2 + 4^2 = 9 + 16 = 25, and c = square root of 25 = 5. So the hypotenuse is 5.

Two figures are congruent when they are exactly the same shape and the same size, just maybe turned or flipped. Two figures are similar when they are the same shape but different sizes, like a photo and a smaller copy of it.

Triangle congruence shortcuts

• ●SSS (Side-Side-Side): all three pairs of sides equal.
• ●SAS (Side-Angle-Side): two sides and the angle between them equal.
• ●ASA (Angle-Side-Angle): two angles and the side between them equal.

Proportions in similar figures

In similar figures, matching sides keep the same ratio. Set up a proportion and cross-multiply to find a missing side.

Example: two similar triangles, the small one has a side of 4 matching a side of 8 on the big one (ratio 1 to 2). If another small side is 5, the matching big side is 10, because 4/8 = 5/10.

A polygon is any closed shape made of straight sides. A quadrilateral is a four-sided polygon. Knowing the properties tells you which one you are looking at.

• ●Square: four equal sides, four right angles.
• ●Rectangle: opposite sides equal, four right angles.
• ●Parallelogram: opposite sides parallel and equal, opposite angles equal.
• ●Trapezoid: exactly one pair of parallel sides.

Example: a quadrilateral has n = 4, so its angles add up to (4 - 2) times 180 = 360 degrees. A pentagon (n = 5) adds up to (5 - 2) times 180 = 540 degrees.

Every quadrilateral, no matter the type, has interior angles that add up to 360 degrees.

A circle is all the points the same distance from a center. That distance is the radius. The diameter goes all the way across through the center, so the diameter is twice the radius.

Central angles and arcs

• ●A central angle has its vertex at the center of the circle.
• ●An arc is a piece of the circle's edge.
• ●The whole circle is 360 degrees, so a central angle tells you what fraction of the circle the arc covers.

Example: a circle with radius 5. Circumference = 2 times 3.14 times 5 = 31.4. Area = 3.14 times 5^2 = 3.14 times 25 = 78.5. A central angle of 90 degrees marks off one quarter of the circle.

Perimeter is the distance around the edge of a flat shape. Area is the space inside it, measured in square units. Keep these formulas in your pocket.

Worked example

A triangle with base 10 and height 6 has area = one half times 10 times 6 = 30 square units. A rectangle 8 by 3 has perimeter = 2 times 8 + 2 times 3 = 22 and area = 8 times 3 = 24 square units.

Tip: perimeter is measured in regular units (feet), area is measured in square units (square feet). Watch your units on the test.

Now we go 3-D. Volume is how much space a solid holds (cubic units). Surface area is the total area of all its outside faces (square units).

Example: a box 4 by 3 by 2 has volume = 4 times 3 times 2 = 24 cubic units. A cylinder with radius 3 and height 10 has volume = 3.14 times 3^2 times 10 = 3.14 times 9 times 10 = 282.6 cubic units.