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■ A right angle is an angle that measures exactly 90 degrees.. ■ An obtuse angle is an angle that measures more than 90 degrees, but less than 180 degrees.. C OMPLEMENTARY A NGLES Two an

■ A right angle is an angle that measures exactly 90 degrees A right angle is sumbolized by a square at the

vertex

■ An obtuse angle is an angle that measures more than 90 degrees, but less than 180 degrees.

■ A straight angle is an angle that measures 180 degrees Thus, both its sides form a line.

C OMPLEMENTARY A NGLES

Two angles are complimentary if the sum of their measures is equal to 90 degrees.

1 2

Complementary Angles

Straight Angle

180°

Obtuse Angle

Right Angle

Symbol

S UPPLEMENTARY A NGLES

Two angles are supplementary if the sum of their measures is equal to 180 degrees.

A DJACENT A NGLES

Adjacent angles have the same vertex, share a side, and do not overlap.

The sum of all possible adjacent angles around the same vertex is equal to 360 degrees

A NGLES OF I NTERSECTING L INES

When two lines intersect, vertical angles are formed Vertical angles have equal measures and are supple-mentary to adjacent angles

1

2

3

4

∠1 + m∠2 + m∠3 + m∠4 = 360°

m

1 2

∠1 and ∠2 are adjacent

Adjacent Angles

1 2

∠1 + m∠2 = 180°

Supplementary Angles

m

■ m1 = m3 and m2 = m4

■ m1 + m2 = 180°and m2 + m3 = 180°

■ m3 + m4 = 180°and m1 + m4 = 180°

B ISECTING A NGLES AND L INE S EGMENTS

Both angles and lines are said to be bisected when divided into two parts with equal measures

Example:

Therefore, line segment AB is bisected at point C.

35°

35°

A

C

bisector

1

2 3 4

A NGLES F ORMED BY P ARALLEL L INES

When two parallel lines are intersected by a third line, or transversal, vertical angles are formed

■ Of these vertical angles, four will be equal and acute, and four will be equal and obtuse The exception

to this is if the transversal is perpendicular to the parallel lines In this case, each of the angles formed measures 90 degrees

■ Any combination of an acute and obtuse angle will be supplementary

In the above figure:

■ b, c, f, and g are all acute and equal.

■ a, d, e, and h are all obtuse and equal.

Some examples:

mb + md = 180º

mc + me = 180º

mf + mh = 180º

mg + ma = 180º

Example:

In the following figure, if m  n and a  b, what is the value of x?

x °

(x + 10)°

b m

n a

e f

h g

Because x is acute, you know that it can be added to x + 10 to equal 180 Thus, the equation is

x + x + 10 = 180.

Solve for x: 2x + 10 = 180

2x = 170

2

2x = 17 2 0



x = 85

Therefore, mx = 85 and the obtuse angle is equal to 180 – 85 = 95

A NGLES OF A T RIANGLE

The measures of the three angles in a triangle always equal 180 degrees

E XTERIOR A NGLES

An exterior angle can be formed by extending a side from any of the three vertices of a triangle Here are some

rules for working with exterior angles:

B

1

∠4 + m∠3 = 180° and m∠4 = m∠2 + m∠1

2

m

B

A C

3

2

1

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

Equilateral

3

5 1 2