Prepared by: David Crockett Math Department

If two lines cross, we say they intersect and the point where they cross is called the point of intersection.

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Two lines which are in the same plane must either intersect or not intersect. When two lines in the same plane do not intersect they are called parallel lines.

Intersecting lines Parallel lines

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Angles (Continued)If two lines make square corners at the point of intersection, they are called perpendicular lines and the angles formed are called right angles.

We can draw a little square at the point of intersection to indicate that all four angles formed are right angles.

4 right angles

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Angles (Continued)Two right angles form a straight angle.

Straight angle

An angle smaller than a right angle is called an acute angle.

acute acute

An angle greater than a right angle is called an obtuse angle.

obtuse

obtuse2 acute angles and 2 obtuse angles

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Angles (Continued)A right angle has a measure of 90 degrees.

A straight angle has a measure of 180 degrees.

90° in a right angle

180° in a straight angle

A circle has a measure of 360 degrees.

90°

90°90°

90°90°

90°90°

360° in a circle

A polygon is a special type of geometric figure. The word polygon is formed from the Greek roots poly, which means “more than one” or “many,” and gonon, which means “angle.” Therefore, the word polygon literally means “more than one angle.”

The modern understanding of a polygon is a simple, closed, flat geometric figure whose sides are line segments. The following are examples of figures that are not polygons.

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a) not a polygon b) not a polygon c) not a polygon d) not a polygon

lines crossnot simple

lines crossnot simple

not aclosedfigure has a

curvedside

Polygons (continued)

Triangle3 sides3 vertices

Quadrilateral4 sides4 vertices

Pentagon5 sides5 vertices

Hexagon6 sides6 vertices

Heptagon7 sides7 vertices

Octagon8 sides8 vertices

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Polygons (continued)

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Each segment of a polygon is called a side. Each endpoint of a side and also the points where two sides meet is called a vertex. The plural of vertex is vertices.

For each polygon, the number of sides is always equal to the number of vertices.

Polygons are named according to the number of sides they have:

• The polygon with the fewest number of sides (3) is the triangle.• A polygon with 4 sides is called a quadrilateral. • A polygon with 5 sides is called a pentagon.• A polygon with 6 sides is called a hexagon.• A polygon with 7 sides is called a heptagon.• A polygon with 8 sides is called a octagon.• A polygon with 9 sides is called a nonagon.• A polygon with 10 sides is called a decagon.• A polygon with 11 sides is called a undecagon.• A polygon with 12 sides is called a dodecagon.

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Concave and ConvexPolygons

If a polygon has an indentation (or cave), the polygon is called a concave polygon. Any polygon that does not have an indentation is called a convex polygon.

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

Concave polygon Convex polygon

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Regular PolygonsIf all the sides of a polygon have the same length, the polygon is called an equilateral polygon.

Equilateral polygon Equiangular polygon

If all the angles of a polygon have the same measure, the polygon is called an equiangular polygon.

Polygons in which all sides are the same length and all angles have the same measure are called regular polygons.

Regular polygon

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Regular PolygonsThe following are examples of regular polygons.

Regular Triangle(Equilateral Triangle)

Regular Pentagon Regular Hexagon

Regular Quadrilateral(Square)

60° 60°

60°

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TrianglesRemember that the polygon with the fewest number of sides is the triangle. A triangle has three sides and three angles.

EquiangularTriangle

RightTriangle

60° 60°

60°

The sum of the measures of the three angles in any triangle is 180°.

If the triangle has a right angle (90°), the triangle is a right triangle.

AcuteTriangle

If all angles have a measure less than 90°, the triangle is an acute triangle.

If one angle has a measure greater than 90°, the triangle is an obtuse triangle.

ObtuseTriangle

If all three angles have the same measure, the triangle is an equiangular triangle.

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TrianglesTriangles are also classifed according to the relative lengths of their sides.

IsocelesTriangle

An isosceles triangle is a triangle that has at least two sides of equal length.

An equilateral triangle is a triangle in which the lengths of all sides are equal.

EquilateralTriangle

If all the sides of a triangle have different lengths, the triangle is called a scalene triangle.

ScaleneTriangle

The lengths of the sides of a triangle and the measures of the angles opposite these sides are related. In any triangle, the angles opposite sides of equal lengths have equal measures. Also, the sides opposite angles of equal measures have equal lengths.

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Quadrilaterals

Trapezoid

A trapezoid is a quadrilateral that has exactly two parallel sides (1 pair).A parallelogram is a quadrilateral that has at two pairs parallel sides (2 pair).A rectangle is a parallelogram with four right angles.

Remember that the polygon with four (4) sides is the quadrilateral. We will discuss five different types of quadrilaterals.

A rhombus is an equilateral parallelogram.

Parallelogram Rectangle

Rhombus Square

A square is an rhombus with four right angles.

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Find x.

30°

130°

x°

Remember the sum of the measures of the three angles in any triangle is 180°. Therefore,

130 30 180x CLT 160 180x

S(160) 20x

20°

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Find x and y.

Remember the sides opposite angles of equal measure are equal. Therefore,

50 180x y Sub 50 50 180y CLT

50x

50°

y°

x°

and

100 180y S(100) 80y

50°

80°

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Find x and y.

Remember the sides opposite angles of equal measure are equal. Therefore,

110 180x x Sub 2 110 180x S(110)

x y

and

2 70x D(2) 35x

110° y°

x°

8

835°

35°

Prepared by: David Crockett Math Department