Mathematics Spring 2015 Dr. Alexandra Shlapentokh Guide #3

Mathematics 3301-001 Spring 2015

Dr. Alexandra Shlapentokh Guide #3

The problems in bold are the problems for Test #3. As before, you are allowed to use statements above and all postulates in the proofs of statements below and partial credit will be awarded for every problem attempted.

(1) Show that each line contains infinitely many points.

(2) Prove the following: if `,m are two distinct lines, then they can have at most one point in common.

(3) Prove the following: if `,m are two distinct lines intersecting at a point O, a point P 6= O is such that P ∈ `, a point Q 6= O is such that Q ∈ m, then the points P,Q,O are not collinear.

(4) Prove the following: if `,m are two distinct lines intersecting at a point O, then there is a unique plane containing both lines.

(5) Suppose A, B are points and under some placement of a ruler along line ^←→ AB we have that x A = 1 and x B = 2, where x A , x B are the real numbers assigned to A and B . Is there a ruler placement such that x A = 1 and x B = 3? Justify your answer.

(6) Let A, P, B be collinear points such that under some ruler placement they were assigned coordinates x _A , x _P , x _B respectively,with x _A ≤ x P ≤ x B . Prove that under any other ruler placement assigning coordinates y _A , y _P , y _B to A, P, B respectively we must have that ei- ther y _A ≤ y P ≤ y B or y _B ≤ y P ≤ y A .

(7) How do we use Ruler Postulate to define a “segment”?

(8) Prove that a “segment” is well defined, i.e. it does not depend on a placement of a ruler.

(9) How do we use Ruler Postulate to define a “ray”?

(10) Prove that a “ray” is well defined, i.e. it does not depend on a placement of a ruler.

(11) Let A, B,C be distinct collinear points such that C 6∈ ~ AB . Show that ~ AC ∩ ~ AB = {A} and AC ∪ ~ ~ AB = AB . ^↔

(12) Prove the following: if `,m are two distinct lines intersecting at a point O, a point A ∈ ` is such that A 6= O, a point B ∈ ` is such that B 6∈ ~ O A, then A and B are in different half- planes with respect to m.

(13) Let ` be a line and let P ↔ x P be a ruler placement. Let C be a real number. Prove that P ↔ y P = x P +C is a ruler placement by showing

(a) P ↔ y P is a bijection, and

(b) the distance requirement is satisfied.

(14) Let ` be a line and let P ↔ x P be a ruler placement. Prove that P ↔ y P = −x P is a ruler placement by showing

(a) P ↔ y P is a bijection, and

(b) the distance requirement is satisfied.

(15) Show that Ruler Postulate implies Ruler Placement Postulate.

(16) Prove the segment construction theorem: given a point P , a line ` such that P ∈ ` and a real number d > 0, there are two points on ` exactly d units away from P.

(17) Given three points A, B,C , define ∠ ABC and its interior. Don’t forget the collinear case.

(18) Define a convex set.

(19) Show that the intersection of convex sets is convex.

(20) Show that the interior of an angle is convex.

(21) Show that a union of two distinct lines is not convex.

(22) Prove that a line is convex.

(23) Prove that a ray is convex.

(24) Prove that if ` is a line, a point A ∈ `, a point B 6∈ `, then all points of AB \{A} are in H `,B . (25) Prove that a half-plane with its boundary is convex.

(26) Rewrite (A ∪ B) ∩ (C ∪ D) using distributive law. 5 points

(27) Let ` be a line, a point B ∈ `, a point A 6∈ `. Show H `,A ∩ B A= ^−→ B A \{B }. ^−→

(28) Show that any ∠ ABC , where A, B,C are not collinear, is convex by showing that ∠ ABC = (H

AB ,C ∪ ^←→ AB ) ∩ (H

C B ,A ∪ C B ). ^←→

(29) Let B ∈ AC . Show ∠ ABC is convex.

(30) Show that AB ∩ ^−→ B A= AB. ^−→

(31) Show: if P ∈ ^←→ AC \ AC , then P 6∈ ^−→ ∠ ABC . See picture above. Assume points A, B,C are not collinear.

(32) Show that if P ∈ ^←→ AC ∩ ∠ ABC , then P ∈ AC . See picture above. Assume points A,B,C are

(36) Let A, B,C be non-collinear. Show that the interior of the triangle 4ABC is equal to H

AB ,C ∩ H

BC ,A ∩ H

C A,B . (37) Show (H

AB ,C ∪ ^←→ AB ) ∩ (H

BC ,A ∪ BC ) ∩ (H ^←→

AC ,B ∪ ^←→ AC ) = ( ∠ ABC ∩ ^←→ AC ) ∪ ( ∠ ABC ∩ H

AC ,B ).

(38) Show ∠ ABC ∩ H

AC ,B = ( B A ∩H ^−→

AC ,B ) ∪ ( BC ∩H ^−→

AC ,B ) ∪ (H

AB ,C ∩ H

BC ,A ∩ H

C A,B ).

(39) Show a triangle is convex. 10 points

(40) Let E be a point in the interior of ∠ ABC . In this case B E \B is in the interior of ^−→ ∠ ^{ABC .} (Assume A, B,C are non-collinear.)

(41) Let A, B,C be non-collinear points, and let D be in the interior of ∠ ABC . Let E ∈ B D, ^←→

E 6∈ B D. Prove that no point of ^−→ B E is in the interior of ^−→ ∠ ^{ABC .}

(42) Let ^←→ AB ∩ C D= {O} (assume A 6= B, C 6= D). Let P 6∈ ^{←→ ←→} AB ∪ C D. Show P must be in the ^←→

interior of exactly one of the following angles: ∠ ^{CO A,} ∠ ^AOD, ∠ ^DOB, ∠ ^{BOC .}

(43) Let B ∈ AC , let D 6∈ ^←→ AB , let P be in the same half-plane with respect to ^←→ AB as D but P 6∈ B D. In this case, either P is in the interior of ^−→ ∠ ^{DB A or} ∠ ^{DBC .}

(44) Let D, B in the same half-plane with respect to ^←→ AC with m ∠ D AC < m ∠ B AC , then B is not in the interior of ∠ ^{D AC .}

(45) If D, B are in the same half-plane with respect to ^←→ AC with m ∠ D AC < m ∠ B AC , then D is in the interior of ∠ ^{B AC .}

(46) Prove existence and uniqueness of an angle bisector for an angle of positive measure.

(47) Define congruence of segments and angles.

(48) Define congruence of triangles.

(49) Show each segment has a unique midpoint.

(50) Show that supplements and complements of congruent angles are congruent.

(51) Define a linear pair of angles.

(52) Show that angles forming linear pair are supplementary. Hint: use the definition of the angle interior.

(53) Define vertical angles.

(54) Show vertical angles are congruent.

(55) Prove Pasch’s Axiom: If a line ` intersects 4PQR at a point S ∈ PQ, then ` intersects P R or PQ.

(56) Let U ,V,W be non-collinear points. Let X be in the interior of ∠ W UV . Show W and V

are in different half-planes with respect to U X and W V ∩ ^←→ U X 6= ;. See picture below: ^←→

(57) Under assumptions of Problem 56 show that V W ∩ int( ∠ V UW ) = V W \ {V,W }.

(58) Under assumptions of Problem 56 show that V W ∩ U X 6= V,W . ^−→

(59) Under assumptions of Problem 56 show W V ∩ U X must be in the interior of ^←→ ∠ ^{W UV .} (60) Prove Crossbar Theorem: Let U ,V,W be non-collinear points. Let X be in the interior

of ∠ ^{W UV . The} U X ∩W V 6= ;. ^−→

(61) Prove the Isosceles Triangle Theorem: if A, B,C are non-collinear and B A ∼ = BC , then

∠ ^{BC A ∼} = ∠ ^{B AC .}

(62) Prove the Perpendicular Bisector Theorem: If A,C are two distinct points, then B is equidistant from A and C if and only if B ∈ `, where ` ⊥ ^←→ AC and ` intersects AC at the midpoint of the segment.

(63) Prove that for any two distinct points B and C there exists a point D on a BC such that ^−→

C ∈ BD.

(64) Prove that for any collinear points A, B,C with B ∈ AC we have d(A,B) + d(B,C ) = d (A,C ).

(65) Given two distinct points A and B , there exists a point C ∈ AB such that B ∈ AC , and B ^−→

is the midpoint of AC .

(66) Let A 6= B be two points. Let x be a real number such that x is less than the distance

from A to B . Show there exists a point C ∈ AB such that d(A,C ) = x.

(69) Prove that given A, B,C non-collinear, E ∈ AC \ {A,C }, F ∈ B E , E ∈ BF , D ∈ ^−→ BC with C ∈ ^−→

B D, we have that F is in the interior of ∠ AC D. See picture below.

(70) Prove the Exterior Angle Theorem: under assumptions of Problem 69, show that m ∠ AC D >

m ∠ ^{B AC .}

(71) Prove Angle-Side-Angle Congruence Condition. Please state it explicitly with two tri- angles before the proof.

(72) Prove Angle-Angle-Side Congruence Condition. Please state it explicitly with two tri- angles before the proof.

(73) Prove the Converse of the Isosceles Triangle Theorem. Please state it explicitly for a triangle before the proof.

(74) Prove the Inverse of the Isosceles Triangle Theorem. Please state it explicitly for a triangle before the proof.

(75) Prove that if two angles of a triangle are not congruent, then the opposite sides are not congruent. Please state it explicitly for a triangle before the proof.

(76) Let A, B,C be non-collinear and let r be a real number such that r < m( ∠ ABC ). Show that in this case there exists X ∈ int( ∠ ABC ) such that m( ∠ X BC ) = r . (Hint: use # 45.) (77) Prove the triangular inequality for non-collinear points. Please state it explicitly be-

fore the proof.

(78) Prove Hinge Theorem. Please state it explicitly for a pair of triangles before the proof.

(79) Prove SSS-congruence Theorem. Please state it explicitly for a pair of triangles before

the proof.