Geometry Chapter - Polygons and Angles

How many sides does a regular polygon have if each exterior angle measures 45?

The sum of the measures of two exterior angles of a triangle is 269. What is the measure of the third exterior angle?

What is the sum of the interior angle measures of a 28-gon?

The sum of the interior angle measures of a polygon with 's' sides is 2700. Find 's'.

Find the missing values of the variables x = 123, y = 58.

A road sign is in the shape of a regular decagon. What is the measure of each interior angle on the sign? Round to the nearest tenth.

Regular Polygon

A regular polygon with each exterior angle measuring 45 degrees has 8 sides.
Formula for calculating exterior angle: ( \text{Exterior Angle} = \frac{360}{n} ) where ( n ) is the number of sides.

Exterior Angles of a Triangle

In a triangle, the sum of the measures of all three exterior angles is always 360 degrees.
If the sum of two exterior angles is 269 degrees, the third exterior angle measures 91 degrees.

Interior Angles of a Polygon

The formula for the sum of interior angles of a polygon is ( (n - 2) \times 180 ), where ( n ) represents the number of sides.
For a 28-gon, the sum of the interior angle measures is 4680 degrees.

Finding Sides from Interior Angle Sum

To find the number of sides ( s ) in a polygon where the sum of interior angles is 2700 degrees, use the equation ( 2700 = (s - 2) \times 180 ).
Solving gives ( s = 17 ).

Solving for Variables

Variables x and y can be any values; for this card, x = 123 and y = 58 are examples of specific values to find.

Interior Angle of a Regular Decagon

Each interior angle of a regular decagon can be calculated using the formula ( \frac{(n - 2) \times 180}{n} ).
For a decagon (10 sides), each interior angle measures 144 degrees. The value is rounded to the nearest tenth if necessary.

This quiz covers key concepts related to polygons, including exterior and interior angles. It features calculation formulas and examples for regular polygons, triangles, and variables related to polygon side counts. Test your understanding of geometric principles and calculations!