3.5 question 2 Definition 3.5.8. Two lines l and m are perpendicular if there exists a point A that lies onboth l and m and there exist points B E l and C E m such that LBAC is a right angle.Notation: l m.BnDCMmAEFIGURE 3.32: l and m are perpendicular lines; n is the perpendicular bisector of DETheorem 3.5.9. If l is a line and P is a point on l, then there exists exactly one line m suchthat P lies on m and m 1 l.Proof. Exercise 2.Definition 3.5.10. Let D and E be two distinct points. A perpendicular bisector of DE is aline n such that the midpoint of DE lies on n and n I DE.Theorem 3.5.11 (Existence and Uniqueness of Perpendicular Bisectors). If D and E aretwo distinct points, then there exists a unique perpendicular bisector for DE.Proof. Exercise 3.Definition 3.5.12. Angles LBAC and LDAE form a vertical pair (or are vertical angles) ifrays AB and AÉ are opposite and rays AC and AD are opposite or if rays AB and AD areopposite and rays AĆ and AÉ are opposite.EВDFIGURE 3.33: LBAC and LDAE are vertical anglesTheorem 3.5.13 (Vertical Angles Theorem). Vertical angles are congruent.Proof. Exercise 5.Section 3.5The Crossbar Theorem and the Linear Pair Theorem61We conclude this section with an application of the Crossbar Theorem. The finaltheorem in this section will not be used again until we prove circular continuity inChapter 8, so the remainder of the section can be omitted for now without seriousconsequence.The theorem we will prove is called the Continuity Axiom. It asserts that therelationship between angle measure and distance is a continuous one. It is called an axiombecause Birkhoff [4] stated this fact as part of his version of the Protractor Postulate. Itis included here because it adds interest to the statement of the Crossbar Theorem andbecause it relates the Crossbar Theorem to the more familiar concept of continuity.We first need a lemma about functions defined on intervals of real numbers. Thelemma should seem intuitively plausible: If a function from the real numbers to the realnumbers is strictly increasing, then a jump in the graph would result in a gap in the rangeof the function.Lomma 3 5 14Lot La hl and Le dl be closed intervals of real numbers and let £ : La bl → 11:03Gerard A. Venema Foundations of Geometry 2011PDF - 3 MBFIGURE 3.35: The Continuity AXiomВС. Thedefine a function that relates the distance and the angle measure. Let dRuler Placement Postulate (Theorem 3.2.16) gives a one-to-one correspondence from theinterval [0, d] to points on BC such that C corresponds to 0 and B corresponds to d. LetDx be the point that corresponds to the number x; i.e., Dx is the point on BC such thatCDx= x. Define a function f : [0, d] → [0, µ(LCAB)] by f(x) = µ(LCAD,).Theorem 3.5.15 (The Continuity Axiom). The function f described in the previousparagraph is a continuous function, as is the inverse of f.Proof. Let f be the function described above. By Theorems 3.3.10 and 3.4.5, f is a strictlyincreasing function. By the Crossbar Theorem and Theorem 3.4.5, f is onto. Therefore,f is continuous (Lemma 3.5.14). It is obvious that the inverse of f is increasing and onto,so the inverse is also continuous.SES 3.51. If l I m, then l and m contain rays that make four different right angles.2. Prove existence and uniqueness of a perpendicular to a line at a point on the line(Theorem 3.5.9).3. Prove existence and uniqueness of perpendicular bisectors (Theorem 3.5.11).4. Prove that supplements of congruent angles are congruent.5. Restate the Vertical Angles Theorem (Theorem 3.5.13) in if-then form. Prove thetheorem.6. Prove the following converse to the Vertical Angles Theorem: If A, B, C, D, and E arepoints such that A * B * C, D and E are on opposite sides of AB, and LDBC = LABE,then D, B, and E are collinear.7. Use the Continuity Axiom and the Intermediate Value Theorem to prove the CrossbarTheorem.HE SIDE-ANGLE-SIDE POSTULATESo far we have formulated one axiom for each of the undefined terms. It would bereasonable to expect this to be enough axioms since we now know the basic properties ofeach of the undefined terms. But there is still something missing: The postulates statedso far do not tell us quite enough about how distance (or length of segments) and anglemeasure interact with each other. In this section we will give an example that illustratesthe need for additional information and then state one final axiom to complete the picture.The simplest objects that combine both segments and angles are triangles. We havedefined what it means for two segments to be congruent and what it means for two anglesto be congruent. We now extend that definition to triangles, where the two types ofcongruence are combined.Section 3.6The Side-Angle-Side Postulate63Definition 3.6.1. Two triangles are congruent if there is a correspondence between thevertices of the first triangle and the vertices of the second triangle such that correspondingangles are congruent and corresponding sides are congruent.The assertion that two triangles are congruent is really the assertion that t’six congruences, three angle congruences and three segment congruences.F

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3.5  question 2 Definition 3.5.8. Two lines l and m are perpendicular if there exists a point A that lies onboth l and m and there exist points B E l and C E m such that LBAC is a right angle.Notation: l m.BnDCMmAEFIGURE 3.32: l and m are perpendicular lines; n is the perpendicular bisector of DETheorem 3.5.9. If l is a line and P is a point on l, then there exists exactly one line m suchthat P lies on m and m 1 l.Proof. Exercise 2.Definition 3.5.10. Let D and E be two distinct points. A perpendicular bisector of DE is aline n such that the midpoint of DE lies on n and n I DE.Theorem 3.5.11 (Existence and Uniqueness of Perpendicular Bisectors). If D and E aretwo distinct points, then there exists a unique perpendicular bisector for DE.Proof. Exercise 3.Definition 3.5.12. Angles LBAC and LDAE form a vertical pair (or are vertical angles) ifrays AB and AÉ are opposite and rays AC and AD are opposite or if rays AB and AD areopposite and rays AĆ and AÉ are opposite.EВDFIGURE 3.33: LBAC and LDAE are vertical anglesTheorem 3.5.13 (Vertical Angles Theorem). Vertical angles are congruent.Proof. Exercise 5.Section 3.5The Crossbar Theorem and the Linear Pair Theorem61We conclude this section with an application of the Crossbar Theorem. The finaltheorem in this section will not be used again until we prove circular continuity inChapter 8, so the remainder of the section can be omitted for now without seriousconsequence.The theorem we will prove is called the Continuity Axiom. It asserts that therelationship between angle measure and distance is a continuous one. It is called an axiombecause Birkhoff [4] stated this fact as part of his version of the Protractor Postulate. Itis included here because it adds interest to the statement of the Crossbar Theorem andbecause it relates the Crossbar Theorem to the more familiar concept of continuity.We first need a lemma about functions defined on intervals of real numbers. Thelemma should seem intuitively plausible: If a function from the real numbers to the realnumbers is strictly increasing, then a jump in the graph would result in a gap in the rangeof the function.Lomma 3 5 14Lot La hl and Le dl be closed intervals of real numbers and let £ : La bl → 11:03Gerard A. Venema Foundations of Geometry 2011PDF - 3 MBFIGURE 3.35: The Continuity AXiomВС. Thedefine a function that relates the distance and the angle measure. Let dRuler Placement Postulate (Theorem 3.2.16) gives a one-to-one correspondence from theinterval [0, d] to points on BC such that C corresponds to 0 and B corresponds to d. LetDx be the point that corresponds to the number x; i.e., Dx is the point on BC such thatCDx= x. Define a function f : [0, d] → [0, µ(LCAB)] by f(x) = µ(LCAD,).Theorem 3.5.15 (The Continuity Axiom). The function f described in the previousparagraph is a continuous function, as is the inverse of f.Proof. Let f be the function described above. By Theorems 3.3.10 and 3.4.5, f is a strictlyincreasing function. By the Crossbar Theorem and Theorem 3.4.5, f is onto. Therefore,f is continuous (Lemma 3.5.14). It is obvious that the inverse of f is increasing and onto,so the inverse is also continuous.SES 3.51. If l I m, then l and m contain rays that make four different right angles.2. Prove existence and uniqueness of a perpendicular to a line at a point on the line(Theorem 3.5.9).3. Prove existence and uniqueness of perpendicular bisectors (Theorem 3.5.11).4. Prove that supplements of congruent angles are congruent.5. Restate the Vertical Angles Theorem (Theorem 3.5.13) in if-then form. Prove thetheorem.6. Prove the following converse to the Vertical Angles Theorem: If A, B, C, D, and E arepoints such that A * B * C, D and E are on opposite sides of AB, and LDBC = LABE,then D, B, and E are collinear.7. Use the Continuity Axiom and the Intermediate Value Theorem to prove the CrossbarTheorem.HE SIDE-ANGLE-SIDE POSTULATESo far we have formulated one axiom for each of the undefined terms. It would bereasonable to expect this to be enough axioms since we now know the basic properties ofeach of the undefined terms. But there is still something missing: The postulates statedso far do not tell us quite enough about how distance (or length of segments) and anglemeasure interact with each other. In this section we will give an example that illustratesthe need for additional information and then state one final axiom to complete the picture.The simplest objects that combine both segments and angles are triangles. We havedefined what it means for two segments to be congruent and what it means for two anglesto be congruent. We now extend that definition to triangles, where the two types ofcongruence are combined.Section 3.6The Side-Angle-Side Postulate63Definition 3.6.1. Two triangles are congruent if there is a correspondence between thevertices of the first triangle and the vertices of the second triangle such that correspondingangles are congruent and corresponding sides are congruent.The assertion that two triangles are congruent is really the assertion that t’six congruences, three angle congruences and three segment congruences.F

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Introduction:
Two lines l and m are perpendicular if there exists a point A that lies on both l and…

2. Prove existence and uniqueness of a p-