Download MATH 161, Extra Exercises 1. Let A, B, and C be three points such

MATH 161, Extra Exercises
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1. Let A, B, and C be three points such that B lies on AC. Then A − B − C if and only if
AB < AC.
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2. Suppose distinct points A and C are on line l and points B and D are not on l. If AB and
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CD intersect, then B and D are on the same side of l.
3. Given A − B − C, B − C − D, and AB = CD, prove AC = BD.
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4. Given triangle 4ABC, B − C − D, and A − F − B, then there exists point E on line DF
such that A − E − C and D − E − F .
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5. Let rays V A and V B be such that ∠AV B is a right angle, and let C be a point such that
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∠AV C is obtuse. Prove that A and C are on opposite sides of line V B.
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6. Prove that if B − A − C, rays AD and AE lie on opposite sides of the line AB, and if
m(∠BAD) = m(∠CAE), then D − A − E, so (∠BAD, ∠CAE) form a vertical pair of angles.
Hint: if not D − A − E, make a ray such that F − A − D; arrive at a contradiction.
7. Given triangle 4ABC, if B − C − D, A − E − C, and B − E − F , show that F is in the
interior of ∠ACD.
8. If points A, B, and C are non-collinear, then every point of the geometry lies on a line that
intersects the triangle 4ABC in at least two points.
9. Given V A ⊥ V B, prove that if A and C are on opposite sides of V B then ∠AV C is obtuse
or A − V − C.
10. Let l be a line and let H be one of the half-planes determined by l. Prove that the union of
H and l is convex.
11. Suppose that 4ABC is a triangle and line l is not incident on any of A, B, C. Show that l
cannot intersect all three sides of 4ABC. Is it possible for a line to intersect all three sides
of a triangle?
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12. Let AB be a line segment. Then there is one and only one line l such that l is perpendicular
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to line AB and meets line AB at a midpoint of AB.
13. State what it means for a function f with domain a line l and range the real numbers to be
a coordinate function on l.
Show that if f is a coordinate function on l and r is a fixed real number, then the function
g(x) = f (x) + r is a coordinate function on l.
14. If A − B − C and D − E − F and AC ∼
= DF and AB ∼
= DE, then BC ∼
= EF .
15. Prove that if P is a point not on line l, and m is a line incident on P and perpendicular
to l, then m is the only line incident on P and perpendicular to l. (uniqueness of dropped
perpendiculars). You are not being asked to construct a perpendicular.
Start by supposing that there is another one.
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16. If C − A − B then AB ∪ AC = CB.
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17. If AB = CB then A = C.
18. Let A = (x1 , y1 ), B = (x2 , y2 ), and C = (x, y) be three collinear points in R2 with x1 < x2 .
Prove that A − B − C if and only if x1 < x < x2 .
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19. ∠QM N and ∠QM P are a linear pair of angles, with m(∠QM N ) = 40. Assume that M X
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and M Y bisect ∠QM N and ∠QM P respectively. Find m(∠QM X) and m(∠QM Y ).
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20. Prove that for A 6= B, AB ⊆ AB ⊆ AB.
21. Show that the sum of the measures of all adjacent angles sharing the same vertex is equal to
360.
22. Given 4ABC with A − D − B, A − E − C, ∠ABE ∼
= ∠ACD, ∠BDC ∼
= ∠BEC, and
BE ∼
CD,
prove
that
4ABC
is
isosceles.
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23. Given 4ABC with AB ∼
= BC, A − D − E − C, and ∠ABD ∼
= ∠CBE, prove that DB ∼
= EB.
24. Is there a triangle with sides 2 in, 4 in, and 7 in?
25. Let 4ABC be an isosceles triangle with AC ∼
= BC. Prove that the triangle with base AB
and sides given by the two angle bisectors of the base angles of 4ABC is isosceles.
26. Consider the figure below. Show that δ(4ABC) = 0 if and only if δ(4ADC) = δ(4DBC) =
0.
27. Use the Hinge Theorem to prove SSS.
28. Let D ∈ Int(4ABC).
(a) Prove that at least two of the angles ∠ADB, ∠BDC, and ∠CDA are obtuse.
(b) Give an example of a triangle in the Euclidean plane such that exactly two of the angles
in (a) are obtuse, and another example when all three are obtuse.
29. Show that equilateral triangles exist.
30. Prove that in a right triangle, the hypothenuse is the longest side.
31. Let ABCD be such that ∠A and ∠B are right. Show that m(∠C) ≥ m(∠D) if and only if
AD ≥ BC.
32. We know that Saccheri quadrilaterals exist, and that they have exactly two right angles or
four right angles. Do quadrilaterals with exactly three right angles exist? Discuss.
33. A rhombus is a quadrilateral with four congruent sides (recall that squares do not exist in
the hyperbolic plane).
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(a) Prove that if AB and CD are two segments that share a common midpoint and AB ⊥ CD,
then ABCD is a rhombus.
(b) Prove that rhombi exist by constructing one.
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34. We define two segments to be congruent if and only if they have the same length. We denote
this by AB ∼
= CD.
Assume that A − B − C, AB ∼
= P Q, AC ∼
= P R, and BC ∼
= QR. Prove that P − Q − R.
Hint: Triangle Inequality.
35. Let us agree for this problem that a line can be parallel to itself.
Prove that the following statement is equivalent to Playfair’s Axiom:
If lkm and mkk then lkk.
36. Prove that the following statement is equivalent to Playfair’s Axiom:
If lkm and t is a line such that t ∩ l 6= ∅ then t ∩ m 6= ∅.
37. Prove that the following statement is equivalent to Euclid’s fifth postulate:
If lkm and t is a line such that t ⊥ l then t ⊥ m.
38. Prove that the following statement is equivalent to Proposition 29:
If lkm then there is a number d such that d(P, m) = d, for all P ∈ l.
39. In Euclidean geometry. Prove:
(a) pairs of opposite sides of a parallelogram are congruent.
(b) the diagonals of a parallelogram bisect each other.
40. Prove that the perpendicular bisectors of all three sides of a triangle, if two of them meet,
are concurrent. Prove that the point of concurrency is equidistant to the three vertices of the
triangle.
41. Prove that if l and m are two distinct lines, and P and Q are two distinct points on m so
that d(P, l) = d(Q, l), then either l and m intersect at the midpoint of P Q, or lkm.
42. In 4ABC. Let D be so that B−D−C and AD = CD. Prove that 2m(∠ACB) ≤ m(∠ADB).
43. Let ABCD be a quadrilateral with CD = CB. If AC is the bisector of ∠DCB, prove that
AB = AD.
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44. If C and D are on the same side of AB, and AC = AD and BC = BD, prove that C = D.
45. Consider 4ABC and 4DEF . Assume that ∠B ∼
= ∠E, BC = EF , CA = F D, and that ∠A
and ∠D are either both acute or both obtuse. Prove that 4ABC ∼
= 4DEF .
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The following problems are for you to prepare for Exam 3. The first ten of them (and
only those) can be presented in my office for extra credit (10 points each). If you want
these points then you should schedule a time to present them to me by 05/10.
Fo Exam 3, you should also solve work on problems from the notes and assignments.
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1. Consider 4ABC. Let D ∈ BC be so that AD bisects ∠CAB. Let B 0 ∈ AC be so that
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B 0 DkAB. Let C 0 ∈ AB be such that C 0 DkAC. Prove that AB 0 DC 0 is a rhombus.
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2. Consider 4ABC. Let D ∈ BC be so that AD bisects ∠CAB. Let b = AC, c = AB,
b
m
m = CD, and n = BD. Prove that = .
c
n
3. A trapezoid is a quadrilateral with only two sides parallel. The two parallel sides of a
trapezoid are called bases. The median of a trapezoid is the segment joining the midpoints
of the two non-parallel sides.
Prove that the median is parallel to the bases and it measures exactly
b1 + b2
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4. An isosceles trapezoid is a trapezoid whose non-parallel sides are congruent. Prove.
(a) In an isosceles trapezoid the two base angles are congruent, and so are the two summit
angles.
(b) The diagonals of an isosceles trapezoid are congruent.
5. Let ABCD be the trapezoid in the picture below
Let EG be the median of ABCD and F be the midpoint of AB. Assume that EF = F G.
Prove that ABCD is isosceles.
6. In the figure below. Suppose that X, Y , Z, and W are the midpoints of AB, BC, CD, and
DA, respectively. Prove that W XY Z is a parallelogram.
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7. In the figure below. Suppose that XY kAB and Y ZkBC. Prove that XZkAC.
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←→ ←→
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8. In the figure below. Suppose that P Y kAQ and QXkBP . Prove that XY kAB.
←→ ←→
UY
BX
9. In the figure below. Suppose that U V kBC. Prove that
=
.
YV
XC
10. In the figure below. Suppose that ABCD is a parallelogram and AX = AY . Prove that
ABCD is a rhombus.
11. Prove that the triangle formed by the midlines of a 4ABC is similar to 4ABC. What is the
ratio obtained using corresponding sides of these triangles?
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12. Let ABCD be the isosceles trapezoid in the picture below
Is 4ABE isosceles?
13. In hyperbolic geometry. Assume that l and k have a common perpendicular. Prove that l is
parallel to k but not ultraparallel to k.
14. Assuming that whenever two lines are parallel there exist a common perpendicular to both
lines, prove that Axiom P-E holds.
15. Prove that congruent triangles have the same defect.
16. In Euclidean geometry. Prove that if k ⊥ l, l ⊥ m, and m ⊥ n then k ∩ n 6= ∅.
17. Prove that ∼ is an equivalence relation on the set of all triangles.
18. In Euclidean geometry. Assume that 4ABC ∼ 4XY Z and that AB = XY . Prove that
4ABC ∼
= 4XY Z.
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