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page 125

#10

Exercises for Section 5.4 Theorems about rays and lines intersecting triangle interiors [9] Illustrate and justify the steps in the proof of Theorem 30 (about a ray with an endpoint on a line) presented on page 124. [10] Prove Theorem 31 ((Corollary of Theorem 30) about a ray with its endpoint on an angle vertex) presented on page 125 . [11] Prove Theorem 32 ((Corollary of Theorem 30.) about a segment that has an endpoint on a line) presented on page 125. [12] Prove Theorem 33 ((Corollary of Thedrem 32.) Points on a side of a triangle are in the interior of the opposite angle.) presented on page 125 . The preceeding theorem has two corollaries. (The word corollary has more than one usage in mathematics. I use the word here to mean a theorem whose proof is a simple application of some other theorem, with no other tricks.) You will be asked to prove both of them in exercises. Theorem 31 (Corollary of Theorem 30 ) about a ray with its endpoint on an angle vertex If a ray has its endpoint on an angle vertex and passes through a point in the angle interior, then every point of the ray except the endpoint lies in the angle interior. Theorem 32 (Corollary of Theorem 30.) about a segment that has an endpoint on a line If a segment that has an endpoint on a line but does not lie in the line, then all points of the segment except that endpoint are on the same side of the line. Here is a corollary of Theorem 32. You will be asked to prove it in an exercise. Theorem 33 (Corollary of Theorem 32.) Points on a side of a triangle are in the interior of the opposite angle. If a point lies on the side of a triangle and is not one of the endpoints of that side, then the point is in the interior of the opposite angle. The next theorem is not so interesting in its own right, but it gets used occasionally throughout the rest of the book in proofs of other theorems. Because of this, I call it a Lemma. Its name will help you remember what the Lemma says, by reminding you of the picture