Lesson 1: Polygons, Triangles, Transversals and Proportional Segments

Definitions often change. The definition of a polygon is a good example. The word is formed from the Greek roots poly, which means “more than one” or “many”, and gonon, which means “angle”. Thus, polygon literally means “more than one angle.”

Some figures have and indentation that we can think of as a cave, and these polygons are called concave. Any polygon that does not have a cave is a convex polygon.

Any two points in the interior of a convex polygon can be connected with a line segment that does not cut a side of the polygon.

Notice that in each figure the number of vertices (corners) is the same as the number of sides.

Some polygons of more than 12 sides have special names, but these names are not used often. Instead, we use the word polygon and tell the number of sides or use the number of sides with the suffix –gon. Thus, if a polygon has 143 sides, we would call it “a polygon with 143 sides” or “143-gon.” The endpoints of one side of a polygon are called consecutive vertices, and two adjacent sides are called consecutive sides. A diagonal of a polygon is a line segment that connects two nonconsecutive vertices.

The sum of the measures of the three angles of any triangle is 180° The sum of the measures of the three angles of any triangle is 180°. The greatest angle is opposite the longest side, and the smallest angle is opposite the shortest side.

Example: What is the longest side of the triangle and why?

Answer: Side c

If two sides of a triangle have equal lengths, the angles opposite these sides have equal measures. If two angles of a triangle have equal measures, the sides opposite these angles have equal lengths.

When the three sides of a triangle have equal lengths, all three angles are 60° angles. If the three angles of a triangle are equal, they must be 60° angles and the three sides must have equal lengths.

Example: Find x and y. X Y

Answer: X = 70° Y = 70°

A transversal is a line that cuts or intersects two or more other lines. If a transversal intersects two or more lines that are parallel and if the transversal is perpendicular to one of the parallel lines, it is perpendicular to all the parallel lines.

Example: Find the measures of angles 2, 7 and 8.

Answer: <2 = 127° <7 = 127° <8 = 53°

When three or more parallel lines are cut by two transversals, the lengths of the corresponding segments of the transversals are proportional. This means that the lengths of the segments of one transversal are related to the lengths of the corresponding segments of the other transversal by a number called the scale factor.

Example: The arrowheads indicate that the lines are parallel. Find x.

Answer: x = 9

HW: Lesson 1 #1-30