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 \(\angle X = \angle Z\) .
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
+9466 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. |
