Polygons/triangles/transversals/proportional segments

Lesson 1:

Polygons

Polygon: simple, closed, flat geometric figures whose sides are straight lines. Names of some polygons; triangle,

quadrilateral, pentagon, hexagon, heptagon, octagon.

Polygons have the same number of vertices as they do sides

Concave polygon: a polygon with an indentation

Convex polygon: a polygon with no indentation.

Polygons

Regular polygon: all angels have equal measures and all sides have equal lengths.

Irregular polygon: any polygon that is not a regular polygon.

The name of a polygon tells the number of sides the polygon has

3 sides triangle 4 sides quadrilateral 5 sides pentagon 6 sides hexagon 7 sides heptagon 8 sides octagon 9 sides nonagon 10 sides decagon 11 sides undecagon 12 sides dodecagon N sides n-gon

Consecutive vertices: endpoints of one side of a polygon

Consecutive sides: two adjacent sides of a polygon

Diagonal: a line segment that connects two nonconsecutive vertices.

Triangles

Right triangle: a triangle with an angle of 90 degrees

Acute triangle: all angles in the triangle have a measure less than 90 degrees

Obtuse triangle: all angle measures are more than 90 degrees

Equilateral triangle: measure of all angles are equal. The lengths of all sides are equal

Isosceles triangle: a triangle with at least two sides of equal length.

Scalene triangle: a triangle with no two sides the same length

Angles and sides connection The sum of the measure of the three angles in

a triangle is 180 degrees The greatest angle is opposite the longest side,

and the smallest angle is opposite the shortest side.

The angles opposite sides of equal length also have equal measures.

The sides opposite of equal measures also have equal lengths.

If all the sides of a triangle have different measures, then all measures of the angles will be different.

If all three sides of a triangle have equal length, what does that tell us about the angles?

What is the measurement of those three angles?

Find x and y

50

y

x

In any triangle the angles opposite equal sides are equal. Thus x is 50 degrees. The sum of three angles in a triangle is 180 degrees, so y must be 80 degrees.

Find x and y

x

40

y

150

The 150 degree angle is supplementary to angle x. So angle x=180-150=30 degrees. Since angle B is 40 degrees, y=180-30-40=110 degrees. You can double check by adding all the angles together. 40+30+110=180 degrees.

B

Triangle ABD is anEquilateral triangle.Find the measurementOf angle D.

D

Since Triangle ABD is an equilateral, all angles inside ABD are 60 degrees. Angle D is supplementary to a 60 degree angle. So 180-60=120. angle D is 120 degrees

CB

A

Transversals

Transversal: a line that cuts or intersects two or more 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.

parallel not parallel

transversal

F

F N so T is not perpendicular to either F or N.

T

N

J P M so T is not only perpendicular to J, but also to P and M

T

J

P

M

If the transversal is not perpendicular to the lines, two groups of equal angles are formed.

Half the angles are “large angles” that are equal angles and are greater than 90 degrees. Half the angles are “small angles” and are less than 90 degrees.

There are 2 sets of vertical angles and 2 sets of supplementary anglesat each intersection

Proportional segments

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 length of the segments of one transversal are related to the lengths of the corresponding segments of the other transversals by a number called the scale factor.

In this figure p, m and n are parallel. The left to right scalar factor for this figure is 3/2. The arrowhead tells us that the scale factor is from left to right. This means that 3/2 times the length of any segment on the left equals the length of the corresponding segment on the right

2

N

P

M

9/2

3

3

Segment Length

Times Scale Factor

Corresponding segment length

2 2x3/2 =3

3 3x3/2 =9/2

Find x using the scale factor

2

N

P M

x

8/3

5

The segments whose lengths are 2 and 8/3 are corresponding segments. Thus, 2 times the left-to-right scale factor equals 8/3

To find the scale factor we set up this equation. 2(SF)=8/3 and solve

½(2)(SF)=8/3(1/2)

SF=4/3

To solve for x we set up this equation. 5(SF)=x

plug in for SF 5(4/3)=x

20/3=x

Solve for x

2

N

P

Mx

1

8

X=4

Questions?!

Homework worksheet due tomorrow