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 \(\angle X = \angle Z\) .

Guest Jun 29, 2019

#1**+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

Guest Jun 29, 2019

#2**+2 **

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. |

hectictar Jun 29, 2019