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C LASSIFYING T RIANGLES It is possible to classify triangles into three categories based on the number of equal sides: Scalene Triangle: no equal sides Isosceles Triangle: at least two e

m1 = m3 + m5

m4 = m2 + m5

m6 = m3 + m2

■ The sum of the exterior angles of a triangle equal to 360 degrees

Triangles

More geometry questions on the GRE pertain to triangles than to any other topic The following topics cover the information you will need to apply when solving triangle problems

C LASSIFYING T RIANGLES

It is possible to classify triangles into three categories based on the number of equal sides:

Scalene Triangle: no equal sides

Isosceles Triangle: at least two equal sides

Equilateral Triangle: all sides equal

Scalene

Isosceles

3

5 1 2

It is also possible to classify triangles into three categories based on the measure of the greatest angle:

Acute Triangle: greatest angle is acute

Right Triangle: greatest angle is 90 degrees

Obtuse Triangle: greatest angle is obtuse

A NGLE -S IDE R ELATIONSHIPS

Knowing the angle-side relationships in isosceles, equilateral, and right triangles will be useful when you take the GRE

■ In isosceles triangles, equal angles are opposite equal sides

A B

C

m∠A = m∠B

70°

150°

Obtuse Right

Acute

■ In equilateral triangles, all sides are equal and all angles are equal.

■ In a right triangle, the side opposite the right angle is called the hypotenuse The hypotenuse is the

longest side of the triangle

P YTHAGOREAN T HEOREM

The Pythagorean theorem is an important tool for working with right triangles.

It states: a2+ b2= c2, where a and b represent the length of the legs and c reprecents the length of the hypotenuse

This theorem allows you to find the length of any side as long as you know the measure of the other two

a2+ b2= c2

2

1

√¯¯¯5

Hypoten

use

Right

Equilateral

60

60 60

5

45-45-90 R IGHT T RIANGLES

A right triangle with two angles each measuring 45 degrees is called an isosceles right triangle In an isosceles

right triangle:

■ The length of the hypotenuse is 2 multiplied by the length of one of the legs of the triangle

■ The length of each leg is multiplied by the length of the hypotenuse

x = 1

1

0

30-60-90 R IGHT T RIANGLES

In a right triangle with the other angles measuring 30 and 60 degrees:

■ The leg opposite the 30-degree angle is half the length of the hypotenuse (And, therefore, the

hypotenuse is two times the length of the leg opposite the 30-degree angle.)

■ The leg opposite the 60-degree angle is 3 times the length of the other leg

102

 2

2



2

10

x

x

2

 2

45 °

45 °

x  2 7  14 and y  3

Circles

A circle is a closed figure in which each point of the circle is the same distance from a fixed point called the

center of the circle

A NGLES AND A RCS OF A C IRCLE

Minor Arc

Major Arc

Central Angle

60 °

30 °

x

y

7

60

30

2s

s

s3

■ A central angle of a circle is an angle that has its vertex at the center and that has sides that are

radii

■ Central angles have the same degree measure as the arc it forms

L ENGTH OF A RC

To find the length of an arc, multiply the circumference of the circle, 2πr, where r  the radius of the circle,

by the fraction 36x0, where x is the degree measure of the arc or central angle of the arc.

Example:

Find the length of the arc if x  36 and r  70.

L = 33660 × 2(π)70

L = 110× 140π

L = 14π

A REA OF A S ECTOR

A sector of a circle is a region contained within the interior of a central angle and arc.

A

B C

shaded region = sector

r o