Linear Pairs, Vertical Angles, and Supplementary （线性双垂直角度,和补充）.pdfVIP

Linear Pairs, Vertical Angles, and Supplementary Angles Definition: Two angles pBAD and pDAC are said to form a linear pair if AB and A C are opposite rays. E D C A B Definition: Two angles pBAC and pEDF are said to be supplementary or to be supplements if their measures add to 180. Our next theorem relates these two definitions. First we need a lemma. Lemma: If C*A*B and D is in the interior of pBAE, then E is in the interior of pDAC. Theorem (The Linear Pair Theorem): If angles pBAD and pDAC form a linear pair, then they are supplementary. That is, μ(∠BAD ) +μ(∠DAC ) 180 . β D E β? α C A B D F α ( + ) - 180 α β β C A B As mentioned in the book, and by now to no one’s surprise, this theorem is often taken as an axiom in order to avoid this somewhat messy proof in a high school class. It was taken as an axiom in the book I last used for Math 362, which also avoided the Betweenness Theorem for Rays by making the Protractor Postulate somewhat stronger, as mentioned above. Definition: Two lines l and m are said to be perpendicular , written l zm, if there exists a point A that lies on both l and m and there exis