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A quadrilateral is called a "parallelogram" if both pairs of opposite sides are parallel. Show that if  \(WXYZ\) is a parallelogram, then \(\angle W = \angle Y\) and \(\angle X = \angle Z\) .

 Jun 29, 2019
 #1
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+2

Its SuerBoranJacobs

I'll refer to the diagram below:

We know that WZ || XY and WX || ZY becuase opposite sides are congruent (Def. of a parallelogram).

Draw ZX and WY as shown (Ruler Postulate).

Label Angles 1, 2, 3, 4, 5, 6, 7, 8 as shown (Ruler Postulate).

Angle W = Angle 3 + Angle 4 (By Construction).

Angle Z = Angle 1 + Angle 2 (By Construction).

Angle Y = Angle 7 + Angle 8 (By Construction).

Angle X = Angle 5 + Angle 6 (By Construction).

Angle 1 = Angle 6 (Alternate Interior Angles Are Congruent).

Angle 2 = Angle 5 (Alternate Interior Angles Are Congruent).

Therefore, Angle Z = Angle X (Parts Make Up A Whole).

Using the same reasoning Angle Y = Angle W.

Q.E.D

 Jun 29, 2019
 #2
avatar+8842 
+5

Here's another way...

 

Let's extend

WZ to point  A,

XY  to point  B,

ZY  to point  C,

WX to point  D,  and

YX  to point  E

 

Like this:

 

 


 

m∠XWZ  =  m∠YZA_____because corresponding angles are congruent.
m∠YZA  =  m∠CYB

 

 

because corresponding angles are congruent.
m∠CYB  =  m∠XYZ because vertical angles are congruent.
Therefore

 

 

 
m∠XWZ  =  m∠XYZ by the transitive property of congruence.

 

 

Likewise...

 

 

 

 

 
m∠WZY  =  m∠XYC because corresponding angles are congruent.
m∠XYC  =  m∠EXD

 

 

because corresponding angles are congruent.
m∠EXD  =  m∠WXY because vertical angles are congruent.
Therefore

 

 

 
m∠WZY  =  m∠WXY by the transitive property of congruence.
 Jun 29, 2019
edited by hectictar  Jun 29, 2019

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