Geometry – Study Guide
Shapes
Triangles
Right triangles have one angle that equals . The symbol represents . Here are some examples:
Isosceles triangles have two sides of the same length and two congruent angles. Here are some examples:
Equilateral triangles have 3 sides of equal length and three congruent angles.
This is true because all angles of the triangle add up to
. With this given information,
angles equal to
Angles:
Right angles are angles that measure exactly . When forming perpendicular lines, right angles are formed. Here are some examples:
Acute angles are angles that measure
less than . Here are some examples:
Obtuse angles are angles that measure
greater than and
less than . Here are some examples:
Complementary angles are angles that add up to
. Here are some examples
Supplementary angles are angles that add up to
. Here are some examples:
Equations to Solve Angles and Sides of Triangles
Law of Sines and Cosines
The Law of Siens and Law of Cosines can be used on any triang,e regardless of whether it is a right triangle to find a missing angle or side length.
Law of Sines

Law of Cosines

Pythagorean Theorem:
The Pythagorean Theorem can be applied to right triangles. Given any 2 sides of a right triangle, we can use the Pythagorean Theorem to determine the third side.
It states:
Where a and b are the legs of the triangle, and c is the hypotenuse (the longest side).
Example:
Solve for x.
Step 1 Identify that
(Pythagorean Theorem) for right triangles
Step 2 Substitute so that
becomes
Step 3 Simplify to
Step 4 Add to get
Step 5 Take the square root of both sides to solve for x. x=5
How would this problem change if you were asked to solve for x in the triangle below?
Example:
Step 1 Identify that
(Pythagorean Theorem) for right triangles
Step 2 Substitute so that
becomes
 Note that the difference between this equation and the one in the previous example, because 10 is the hypotenuse.
Step 3 Simplify to
Step 4 Subtract to get
Step 5 Square root both sides to solve for x. x=8
Special Right Triangles
These common types of right triangles have certain side length ratios, which can be useful in solving geometry problems.
Example:
Find x.
We know that this is a
triangle because the legs are of equal length. The hypotenuse of a
triangle is equal to
times the length of a leg. We can use this fact to calculate
.
Example:
Find x.
Because this triangle has a right angle and an angle of
, the other angle must be
. (The angles must sum to
. Since the long leg of a
right triangle is equal to
times the short leg, we can calculate x to equal
.
Area of Basic Shapes
Area of Triangles and Rectangles
Two of the most fundamental shapes in geometry are triangles and rectangles. We can often find the area of more complex shapes by breaking them down into smaller rectangles and triangles. So it is very important that we can find the area of these basic shapes.
The area of any rectangle is the length multiplied by the width. The area of any triangle is one half of the base multiplied by the height.
Examples:
Circles
Another basic shape in geometry is the circle. To find the area of a circle we need , which is the ratio of the circumference of a circle to its diameter. The area of a circle is multiplied by the radius squared.
Example:
Find the area of the circle.
Area=
=78.54 units
Area of Parallelograms, Trapazoids, Rhombuses, and Kites
There are lots of foursided shapes besides rectangles and squares.
Hierarchy of FourSided Shapes
 Quadrilaterals –all foursided polygons
 Trapezoids — one pair of parallel sides
 Parallelograms — two pairs of parallel sides
 Rectangles — four right angles, four sides.
 Rhombus — four equal length sides, with perpendicular diagonals
 Kites — two adjacent pairs of equal length sides
 Square — foursided polygon having equallength sides meeting at right angles, with perpendicular diagonals
Note: A square is both a rectangle and a rhombus.
 To find the area of a parallelogram we need to multiply the base by the height.
Parallelogram:
 The area of a trapezoid is one half of the height multiplied by the sum of the upper and lower bases.
Trapezoid:
Example: Find the area of the parallelogram.
To solve this problem we will need to find the height of the parallelogram. The Pythagorean Theorem can be used with the right triangle formed by the vertical line to find h.
Next, use the formula
to calculate the area.
units
 The area of a Rhombus is equal to one half of the product of the two diagonal lengths. The area of a kite can be found in exactly the same way.
Examples: