WHEN MORE THAN ONE ANGLE HAS THE SAME VERTEX,■ A STRAIGHTANGLE IS A...

3. When more than one angle has the same vertex,

■

A straightangle is an angle that measures 180three letters are used, with the vertex alwaysdegrees. Thus, its sides form a straight line.being the middle letter: –1 can be written as∠BADor as ∠DAB;–2 can be written as ∠DACStraight Angleor as ∠CAD.180°Classifying AnglesAngles can be classified into the following categories:acute, right, obtuse, and straight.

–

M E A S U R E M E N T A N D G E O M E T R Y

–

C

OMPLEMENTARY

A

^NGLES

Angles of Intersecting LinesTwo angles are complementaryif the sum of their meas-When two lines intersect, two sets of nonadjacent anglesures is equal to 90 degrees.called vertical angles are formed. Vertical angles haveequal measures and are supplementary to adjacentangles.Complementary 1

2

Angles2

1

3

∠1 + ∠2 = 90°

4

S

UPPLEMENTARY

A

^NGLES

Two angles are supplementaryif the sum of their meas-

■

m∠1 = m∠3 and m∠2 = m∠4ures is equal to 180 degrees.

■

m∠1 + m∠2 = 180 and m∠2 + m∠3 = 180

■

m∠3 + m∠4 = 180 and m∠1 + m∠4 = 180

Supplementary

Angles

Bisecting Angles and Line

2

1

SegmentsBoth angles and lines are said to be bisected when

∠

1 +

∠

2 = 180

°

divided into two parts with equal measures.

A

DJACENT

A

NGLES

ExampleAdjacentangles have the same vertex, share a side, and donot overlap.SA C BAdjacent ∠

^{1 and}

∠

2 are adjacent.

Line segment ABis bisected at point C.The sum of the measures of all adjacent angles aroundthe same vertex is equal to 360 degrees.C35°

∠

1 +

∠

2 +

∠

3 +

∠

4 = 360

°

AAccording to the figure,∠Ais bisected by ray AC.

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Angles Formed by Parallel LinesSolution:When two parallel lines are intersected by a third line,Because both sets of lines are parallel, you knowvertical angles are formed.that xcan be added to x+ 10 to equal 180. Theequation is thus,x+ x+ 10 = 180.

■

Of these vertical angles, four will be equal andacute, four will be equal and obtuse, or all fourwill be right angles.Solve for x:

■

Any combination of an acute and an obtuse angle2x+ 10 = 180will be supplementary.−10 −10

x

=

17

0

2

a bx = 85c dTherefore,m∠x= 85° and the obtuse angle isequal to 180 −85 = 95°.e fAngles of a Triangleg hThe measures of the three angles in a triangle alwaysequal 180 degrees.

B

In the above figure:

b

■

∠b,∠c,∠f, and ∠gare all acute and equal.

■

∠a,∠d,∠e, and ∠hare all obtuse and equal.

■

Also, any acute angle added to any obtuse angle

c

A

C

a

a

+

b

+

c

= 180°

Examplesm∠b+ m∠d= 180°

E

^XTERIOR

A

^NGLES

m∠c+ m∠e= 180°Bm∠f+ m∠h= 180°bm∠g+ m∠a= 180°In the figure below, ifm|| nand a|| b, what isthe value ofx?ab

x

°

a dcmA Cd+c = 180° andd= b+ anAn exterior anglecan be formed by extending a side from

(x + 10)

°

any of the three vertices of a triangle. Here are some rulesfor working with exterior angles:

■

An exterior angle and interior angle that share thesame vertex are supplementary.

■

An exterior angle is equal to the sum of the

70°

nonadjacent interior angles.

Acute

■

The sum of the exterior angles of a triangleequals 360 degrees.

50°

60°

Tr i a n g l e s

Right

Classifying TrianglesIt is possible to classify triangles into three categoriesbased on the number of equal sides:

Scalene Isosceles

Equilateral

(no equal sides)

(two equal sides)

(all sides equal)

Obtuse

150

°

Scalene

Angle-Side RelationshipsKnowing the angle-side relationships in isosceles, equi-lateral, and right triangles will be useful in taking theGED exam.

■

In isosceles triangles, equal angles are oppositeequal sides.

Isoceles

C

Isosceles

m

∠

a = m

∠

b

A

B

b

a

■

In equilateral triangles, all sides are equal and allangles are equal.

Equilateral

60°5 5EquilateralIt is also possible to classify triangles into three cate-gories based on the measure of the greatest angle:5

Acute

Right Obtuse

greatest angle

greatest angle

greatest angle

is acute

is 90°

is obtuse

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