SAT Mc Graw Hill 2011

CHAPTER 10 / ESSENTIAL GEOMETRY SKILLS 365

Angles in Polygons

Remembering what you learned about parallel lines
in the last lesson, consider this diagram:

We drew line so that it is parallel to the opposite side
of the triangle. Do you see the two Z’s? The angles
marked aare equal, and so are the angles marked c.
We also know that angles that make up a straight line
have a sum of 180°, so a
b
c180. The angles in-
side the triangle are also a, b,and c.

Therefore, the sum of angles in a triangle is

always 180°.

Every polygon with nsides can be divided into n− 2
triangles that share their vertices (corners) with the
polygon:

Therefore, the sum of the angles in any polygon

with n sides is 180(n2)°.

Angle-Side Relationships in Triangles

A triangle is like an alligator mouth with a stick in it:

The wider the mouth, the bigger the stick, right?

Therefore, the largest angle of a triangle is al-

ways across from the longest side, and vice

versa. Likewise, the smallest angle is always

across from the shortest side.

Example:
In the figure below, 72 70, so ab.

70 °^72 °

a b

a°

a°

b°

c°

c°

An isosceles triangle is a triangle with two

equal sides. If two sides in a triangle are equal,

then the angles across from those sides are

equal, too, and vice versa.

The Triangle Inequality

Look closely at the figure below. The shortest path

from point Ato point Bis the line segment connect-

ing them. Therefore, unless point Cis “on the way”

from Ato B,that is, unless it’s on AB

––

, the distance

from Ato Bthrough Cmust be longer than the direct

route. In other words:

The sum of any two sides of a triangle is always

greater than the third side. This means that the

length of any side of a triangle must be be-

tween the sum and the difference of the other

two sides.

The External Angle Theorem

The extended side of a triangle forms an exter-

nal angle with the adjacent side. The external

angle of a triangle is equal to the sum of the

two “remote interior” angles. Notice that this

follows from our angle theorems:

a + b + x= 180 and c + x= 180;

therefore, a + b = c

a°

b°

x° c°

A

C

B

12

10

Lesson 2: Triangles

12 − 10 <AB< 12 + 10

5 sides, 3 triangles^2 <AB<^22

= 3(180°) = 540°

7 sides, 5 triangles

= 5(180°) = 900°