**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* .

_{A}

_{P}

_{B}

_{A}

_{A}

_{P}

_{B}

_{A}

_{B}

## (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.)**

**(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,** ^{←→}

**(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 .}

**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** ^{←→}

**C D= {O} (assume A 6= B, C 6= D). Let P 6∈**

**C D. Show P must be in the**

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

^{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 .}

**(43) Let B ∈ AC , let D 6∈**

**AB , let P be in the same half-plane with respect to**

**AB as D but**

**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 .}

**(44) Let D, B in the same half-plane with respect to**

**AC with m**

**B AC , then B is**

^{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 .}

**(45) If D, B are in the same half-plane with respect to**

**AC with m**

**B AC , then D**

^{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.**

**` 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**

**(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:** ^{←→}

**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 }.**

**(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 .** ^{−→}

**(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**

**(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= ;.** ^{−→}

^{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**

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

**= BC , then**

## ∠ ^{BC A ∼} = ∠ ^{B AC .}

^{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.**

**(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**

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

**(63) Prove that for any two distinct points B and C there exists a point D on a**

**BC such that**

**C ∈ BD.**

**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 ).**

**(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** ^{−→}

**(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 .**

**AC .**

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

**(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.**

**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 ∈** ^{−→}

**(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.**

**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 >*

**(70) Prove the Exterior Angle Theorem: under assumptions of Problem 69, show that m**

*m* ∠ ^{B AC .}

^{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-**

**(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.)**