Triangles, Quadrilaterals, and Polygons

Triangles

Triangles have three vertices.
The interior angles of a triangle add up to 180 degrees.
Exterior angles are named using three letters, with the middle letter indicating the angle being referred to.
The exterior angles of a triangle add up to 360 degrees.
Theorem: For a triangle with interior angles x, y, and z:
x + y + z = 180
Theorem: For a triangle with exterior angles w, u, and v:
w + u + v = 360
A straight line measures 180 degrees.
Supplementary Angle Theorem: If a straight line is cut by a transversal line into angles x and y, then
x + y = 180
A right angle measures 90 degrees.
Complementary Angle Theorem: If a right angle is divided into angles x and y, then
x + y = 90
The exterior angle at each vertex of a triangle is the sum of the interior angles at the other two vertices.

To find a missing interior angle, subtract the sum of the known interior angles from 180 degrees.
To find a missing exterior angle, subtract the sum of the known exterior angles from 360 degrees.

An isosceles triangle has two sides of the same length and two equal angles.
The angles opposite the equal sides are also equal.
To find the missing angles in an isosceles triangle:
- If one angle is known, subtract it from 180, then divide the result by 2 to find the measure of each of the two equal angles.

A quadrilateral is a shape with four sides.
The interior angles of a quadrilateral add up to 360 degrees.
The exterior angles of a quadrilateral add up to 360 degrees.
To find a missing interior angle, subtract the sum of the known interior angles from 360 degrees.
Parallelograms have two pairs of parallel sides. Parallel sides are indicated by arrows.

Alternate Angle Theorem: When parallel lines are cut by a transversal line, the alternate interior angles are equal (forming a Z shape).
Co-interior Angles Theorem: When parallel lines are cut by a transversal line, the co-interior angles add up to 180 degrees (forming a C shape).
Corresponding Angles Theorem: When parallel lines are cut by a transversal line, the corresponding angles are equal (forming an F shape).
Opposite Angle Theorem: When two lines intersect, the opposite angles are equal (forming an X shape).
Example problems demonstrating how to apply these theorems to find missing angles.

Combination of interior and exterior angle problems, solved using supplementary angle theorem and the property that interior angles of a quadrilateral sum to 360 degrees.
Application of co-interior angles theorem to simplify problem-solving.

A convex polygon has all interior angles measuring less than 180 degrees.
A concave polygon has at least one angle greater than 180 degrees.
Formula: The sum of the interior angles of a convex polygon is given by:
Sum = 180(n - 2)
where n is the number of sides (or angles) of the polygon.
To find the measure of each angle in a regular (equal-angled) polygon, divide the sum of the interior angles by the number of sides.
Formula: To determine how many sides does a polygon have if each of its interior angles measures 140,

180(n-2)= 140n

Working backwards to find the number of sides in a polygon when each interior angle measures 140 degrees. This involves using the distributive property and solving the resulting equation.
180n - 360 = 140n
40n = 360
n = 9
A nine-sided polygon is called a nonagon.