Question 1 DAВFIGURE 3.22: µ(LBAD) µ(LBAC).Section 3.5The Crossbar Theorem and the Linear Pair Theorem55If D lies on AĆ, then µ(LBAD) = µ(LBAC). Otherwise, C is in the interior of ZBAD(Lemma 3.4.4). Therefore, µ(LBAD) µ(LBAC) (by the first half of the theorem). OThe Betweenness Theorem for Rays is used in the proof of the existence of anglebisectors just as the Betweenness Theorem of Points was used in the proof of the existenceof midpoints.Definition 3.4.6. Let A, B, and C be three noncollinear points. A ray AD is an anglebisector of LBAC if D is in the interior of LBAC and µ(LBAD) = µ(LDAC).AВFIGURE 3.23: The angle bisectorTheorem 3.4.7 (Existence and Uniqueness of Angle Bisectors). If A, B, and C are threenoncollinear points, then there exists a unique angle bisector for LBAC.Proof. Exercise 1.KERCISES 3.41. Prove existence and uniqueness of angle bisectors (Theorem 3.4.7).`AB, and let2. Let A and B be two points, let H be one of the half-planes of l ={LBAE| E E H}. Define ƒ : A –→ (0, 180) by f(LBAE) = µ(LBAE).(a) Prove that f is a one-to-one correspondence.(b) Prove that AF is between AB and AÉ if and only if f(LBAF) is between 0 andf(LBAE).A5 THE CROSSBAR THEOREM AND THE LINEAR PAIR THEOREMBefore proceeding to the final postulate, we pause to prove two fundamental theoremsof plane geometry, the Crossbar Theorem and the Linear Pair Theorem. Both are oftentaken as axioms. Readers who are anxious to move more quickly past the foundationscan also do that: it is logically acceptable to take the statements as additional axioms andto move on. The proofs can be revisited later when the reader is in a better position toappreciate the power and importance of the results. The main goal we wish to accomplishin this chapter is to lay out explicitly all the basic facts that Euclid took for granted in hisproofs; the goal of making those assumptions explicit is much more important that thesecondary goal of trying to find a minimal set of necessary assumptions.The proof of the first theorem relies heavily on the following preliminary result,which is known as the "Z-Theorem" because of the shape of the diagram that accompaniesit. The Z-Theorem is an easy consequence of Theorem 3.3.9.Theorem 3.5.1 (The Z-Theorem). Let l be a line and let A and D be distinct points on l.If B and E are points on opposite sides of l, then AB N DÉØ.

Question

Question 1 DAВFIGURE 3.22: µ(LBAD) < µ(LBAC) if and only if D is in the interior of LBACWe will prove the contrapositive of the second half of the theorem. Suppose AD isnot between rays AB and AĆ (hypothesis). We must prove that µ(LBAD) > µ(LBAC).Section 3.5The Crossbar Theorem and the Linear Pair Theorem55If D lies on AĆ, then µ(LBAD) = µ(LBAC). Otherwise, C is in the interior of ZBAD(Lemma 3.4.4). Therefore, µ(LBAD) > µ(LBAC) (by the first half of the theorem). OThe Betweenness Theorem for Rays is used in the proof of the existence of anglebisectors just as the Betweenness Theorem of Points was used in the proof of the existenceof midpoints.Definition 3.4.6. Let A, B, and C be three noncollinear points. A ray AD is an anglebisector of LBAC if D is in the interior of LBAC and µ(LBAD) = µ(LDAC).AВFIGURE 3.23: The angle bisectorTheorem 3.4.7 (Existence and Uniqueness of Angle Bisectors). If A, B, and C are threenoncollinear points, then there exists a unique angle bisector for LBAC.Proof. Exercise 1.KERCISES 3.41. Prove existence and uniqueness of angle bisectors (Theorem 3.4.7).`AB, and let2. Let A and B be two points, let H be one of the half-planes of l ={LBAE| E E H}. Define ƒ : A –→ (0, 180) by f(LBAE) = µ(LBAE).(a) Prove that f is a one-to-one correspondence.(b) Prove that AF is between AB and AÉ if and only if f(LBAF) is between 0 andf(LBAE).A5 THE CROSSBAR THEOREM AND THE LINEAR PAIR THEOREMBefore proceeding to the final postulate, we pause to prove two fundamental theoremsof plane geometry, the Crossbar Theorem and the Linear Pair Theorem. Both are oftentaken as axioms. Readers who are anxious to move more quickly past the foundationscan also do that: it is logically acceptable to take the statements as additional axioms andto move on. The proofs can be revisited later when the reader is in a better position toappreciate the power and importance of the results. The main goal we wish to accomplishin this chapter is to lay out explicitly all the basic facts that Euclid took for granted in hisproofs; the goal of making those assumptions explicit is much more important that thesecondary goal of trying to find a minimal set of necessary assumptions.The proof of the first theorem relies heavily on the following preliminary result,which is known as the &#34;Z-Theorem&#34; because of the shape of the diagram that accompaniesit. The Z-Theorem is an easy consequence of Theorem 3.3.9.Theorem 3.5.1 (The Z-Theorem). Let l be a line and let A and D be distinct points on l.If B and E are points on opposite sides of l, then AB N DÉØ.

Accepted Answer

To prove the existence and uniqueness of angle bisectors: If A, B, and C are three noncollinear…