Vertical angles have the same measure. 

From the illustration above, and are vertical angles and would have the same measurement.

and are another set of vertical angles on this illustration and would have equal measurements.

Let’s set it up and solve it  algebraically, letting x be the missing angle and see what we get:

Let’s set it up and solve it algebraically, letting x be the missing angle and see what we get:

Each line segment is called the side. 

The intersection at the endpoints is called the vertex.

Keep in mind that the number of sides and number of interior angles of a polygon are the same. 

Polygons are named by the number of sides they have. 

Two common polygons are

The sum of the measures of the interior angles of an n-sided polygon is .

For example, if you have a triangle, which has 3 sides, the sum of the measures of the interior angles would be 

The area of a polygon is the measure of the enclosed interior.

Putting 5 into the sum of the measures of the interior angles of an n-sided polygon formula we get:

Next we need to figure out what would be the measure of each interior angle of a regular pentagon. 

Since we are talking specifically about a regular pentagon, that means all interior angles have the same measure.  And since the total of those measures is 540, what do you think the measure of each interior angle is?  If you said 108 degrees give yourself a high five. 

Just divide the total, 540, by the number of angles, in this case 5 and viola .

Some of the more common quadrilaterals are 

Note that a square is a special type of rectangle, one in which all four sides are equal to each other

Note that rectangles are a special type of parallelograms. 

Corresponding sides of congruent figures are equal in length.

Note: Corresponding angles and sides are found by matching up the letters of each figure’s name in the order that they are listed. 

Note how A corresponds with F, B corresponds with E, C corresponds with H and D corresponds with G. 

In other words, side AB is the same as side FE, side BC is the same as side EH, side CD is the same as side HG and side DA is the same as side GF.

Matching up the corresponding points, the proper way of to say this is figure ABCD is congruent to figure FEHG.

Example 4:   Figure ABCDE is congruent to figure FGHIJ. 

If C = 50 degrees and E = 75 degrees, what is the measure of J? 

If AB = 20, BC = 30, and CD = 40, then what is the length of GF?

Since J corresponds with E and the figures are congruent, then J = E = 75 degrees.
 

Since side GF (or FG) corresponds to side AB and the figures are congruent, then side GF (or FG) = side AB = 20.

Corresponding sides of similar figures are in proportion to each other. 

Note: Corresponding angles and sides are found by matching up the letters of each figure’s name in the order that they are listed. 

Note how A corresponds with D, B corresponds with E, C corresponds with F. 

All of the sides are in proportion to each other.  In this example, side AB is twice as large as side DE, side BC is twice as large as side EF, and side CA is twice as large as side FD.
 

Matching up the corresponding points, the proper way of to say this is figure ABC is similar to figure DEF.

Example 5:   Figure ABCDEFG is similar to figure HIJKLMN. 

If C = 25, E = 40, and G = 30, what is the measure of L? 

If AG = 5, HN = 20, and BC = 40, what is the length of IJ?

Since L corresponds to E and the figures are similar, then L = E = 40.
 

Since side IJ corresponds to side BC and the figures are similar to each other, then IJ and BC are in proportion to each other.  Similarly, AG and HN are in proportion to each other. 

When setting up the proportion, make sure that you set it up the same on both sides.

*Cross multiply
 
 
 

alternate interior angles are equal.
 

alternate exterior angles are equal.
 

corresponding angles are equal.
 
 

Example 6:   Given the following parallel lines cut by a transversal

Find 8 if 1 = 120 degrees.

Find 4 if 5 = 120 degrees. 

Find 7 if 3 = 60 degrees.

Find 3 if 5 = 120 degrees.

Find 8 if 1 = 120 degrees.

Since 8 and 1 are alternate exterior angles and the two lines are parallel, then 8  = 1 = 120 degrees.

Find 4 if 5 = 120 degrees.

Since 4 and 5 are alternate interior angles and the two lines are parallel, then 4 = 5 = 120 degrees.

Find 7 if 3 = 60 degrees.

Since 7 and 3 are corresponding angles and the two lines are parallel, then 7 = 3 = 60 degrees.

Find 3 if 5 = 120 degrees.

Since 3 and 5 are not alternate exterior, alternate interior or corresponding angles, they are not guaranteed to be equal. 

However, since 3 and 4 make a straight angle (180 degrees) and 4 and 5 are alternate interior angles (which means they are equal), we can find the measure of 3.

The radius (r on the diagram below) is the distance from the center of the circle to any point on the circle and can be shown as a line segment connecting the center to a point on the circle.

The diameter is a line segment that connects two points on the circle and goes through the center of the circle.  It is always twice as long as the radius.

A chord (line segment PQ on the diagram below) is any line segment whose endpoints are any two points on the circle.

The circumference of a circle is the distance around the circle.

When naming an arc, it is best to use three points - the two endpoints and a point in between - versus just the two endpoints.  The reason is you can go clockwise or counterclockwise, which can make a difference when looking at the length of an arc. 

Arc ABC would start at point A and go counterclockwise through B and end at C.  Since a circle is 360 degrees, then Arc ABC is a 360 - 95 = 265 degree arc.

The tangent line and the radius of the circle that has an endpoint at the point of tangency are perpendicular to each other.

In this situation we can also say that the circle is circumscribed about the polygon.

In this same situation we can say that the circle is inscribed in the polygon.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.