60-90 TRIANGLES1 + 4 = C2IN A RIGHT TRIANGLE WITH THE OTHER ANGLES...

30-60-90 Triangles1 + 4 = c

²

In a right triangle with the other angles measuring 305 = c

²

and 60 degrees:5= c

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The leg opposite the 30-degree angle is half thelength of the hypotenuse. (And, therefore, thehypotenuse is two times the length of the legopposite the 30-degree angle.)

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The leg opposite the 60-degree angle is 3timesthe length of the other leg.

60°

s

2s

30°

s

√¯¯¯

3

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

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ExampleDA60°7 xB30° yCx= 2 ×7 = 14 and y= 73Step 2: Determine whether this is enoughinformation to prove the triangles areComparing Trianglescongruent.Triangles are said to be congruent (indicated by the sym-Yes, two angles and the side between them arebol ) when they have exactly the same size and shape.equal. Using the ASA rule, you can determineTwo triangles are congruent if their corresponding partsthat triangle ABDis congruent to triangle CBD.(their angles and sides) are congruent. Sometimes, it iseasy to tell if two triangles are congruent by looking.However, in geometry, you must be able to prove that the

P o l y g o n s a n d P a r a l l e l o g r a m s

triangles are congruent.If two triangles are congruent, one of the three crite-A polygonis a closed figure with three or more sides.ria listed below must be satisfied.

B

Side-Side-Side (SSS) The side measures for both

C

triangles are the same.Side-Angle-Side (SAS) The sides and the anglebetween them are the same.

A

D

Angle-Side-Angle (ASA) Two angles and the sideExample:Are triangles ABCand BCDcongruent?

F

E

Given:∠ABDis congruent to ∠CBDand ∠ADBis congruent to ∠CDB. Both triangles share side BD.Terms Related to Polygons

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Verticesare corner points, also called endpoints,of a polygon. The vertices in the above polygonare:A, B, C, D, E,and F.

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A diagonalof a polygon is a line segment betweentwo nonadjacent vertices. The two diagonals inthe polygon above are line segments BFand AE.

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A regularpolygon has sides and angles that are allequal.

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An equiangularpolygon has angles that are allStep 1: Mark the given congruencies on theAngles of a Quadrilateraldrawing.A quadrilateralis a four-sided polygon. Since a quadri-lateral can be divided by a diagonal into two triangles,

3 9 9

the sum of its interior angles will equal 180 + 180 = 360degrees.

4

2

b

120°

10

120

°

5

6

3

a c

60

°

18

9

dThese two polygons are similar because theirangles are equal and the ratios of the correspon-m∠a+ m∠b+ m∠c+ m∠d= 360°ding sides are in proportion.Interior AnglesParallelogramsTo find the sum of the interior angles of any polygon, useA parallelogramis a quadrilateral with two pairs of par-this formula:allel sides.

B

C

S= 180(x−2)°, with xbeing the number ofpolygon sidesFind the sum of the angles in this polygon:

A

D

b

In the figure above, line AB|| CDand BC|| AD.

c

a

A parallelogram has:

■

opposite sides that are equal (AB= CDand BC= AD)

e

d

■

opposite angles that are equal (m∠a= m∠cand m∠b= m∠d)S= (5 −2) ×180°

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consecutive angles that are supplementary (m∠aS= 3 ×180°+ m∠b= 180°, m∠b+ m∠c= 180°, m∠c+ m∠dS= 540°= 180°, m∠d+ m∠a= 180°)Exterior Angles

S

^PECIAL

T

^{YPES OF}

P

ARALLELOGRAMS

Similar to the exterior angles of a triangle, the sum of the

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A rectangleis a parallelogram that has four rightexterior angles ofanypolygon equals 360 degrees.angles.Similar PolygonsIf two polygons are similar, their corresponding anglesare equal and the ratios of the corresponding sides are in

AB = CD

proportion.

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A rhombusis a parallelogram that has four equal

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In a square, diagonals have the same length andsides.intersect at 90-degree angles.

D

C

AB = BC = CD = DA

AC = DB

and

A

B

AC DB

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A squareis a parallelogram in which all angles areequal to 90 degrees and all sides are equal to eachother.AB = BC = CD = DA

S o l i d F i g u r e s , P e r i m e t e r,

m∠A = m∠B = m∠C = m∠D

a n d A r e a

The GED provides you with several geometrical formu-las. These formulas will be listed and explained in thissection. It is important that you be able to recognize the

D

IAGONALS

figures by their names and to understand when to useIn all parallelograms, diagonals cut each other into twowhich formulas. Don’t worry. You do not have to mem-equal halves.orize these formulas. They will be provided for you onthe exam.

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In a rectangle, diagonals are the same length.To begin, it is necessary to explain five kinds ofD Cmeasurement: