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# Help!

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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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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
+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