IMPORTANT! Make liberal use of the Geometry Student Solutions Guide, not so much
to just see the answer but to study and understand the method(s) to reach a
solution.
PROOFS are special kinds of problems and require special attention. The
expected outcome in completing proofs is an understanding and appreciation
of logical thought (and not to memorize a bunch of postulates and theorems).
It is important to realize that there are many right ways to structure a
proof. The answer in the solutions guide is just one correct method. In general, proofs do not appear on standardized tests. While it is important to attempt proofs, it is more important to understand the sequential thinking that goes into a solution. Please utilize technical support as needed.
In grading a student's homework look for a reasonable and logical flow of
proof. While working with proofs, encourage the student to have handy a
list of definitions, postulates, and theorems for reference and verify that
all of the items in the REASONS column are definitions, postulates, theorems
, or a close variation of one.
Don't hesitate to consult the Student Solutions Guide and compare the
method in the solutions guide with the method used by the student. If a
student continually gets stuck with proofs, stop for a day or so and try
this: Go back one or two sections and consider 4-5 proofs that were
attempted before. Read over each proof problem and look carefully at the
flow of the proof in the solutions guide. The idea is to understand the
logic flow, not to memorize the proof. Then close the solutions guide and
try the problems from scratch.
One strategy that may help you in determining the steps involved is to think
backwards through the problem. That is, think about what information you
need in order to conclude the last step. Then think about the step that
needs to precede that, and so on.
If the formal statements and reasons format is intimidating, have the
student write the "logic path" for the proof in a kind of narrative form
before using the two-column format. Be flexible in grading. Allow for
creativity. Learning proofs is a process. Valuable assets in the learning
process include patience, think backwards, realize there are several correct
ways to complete a proof, give yourself some latitude with informality in
the REASONS column, make liberal use of the solutions guide when you get
stuck, positive attitude, and keep your sense of humor.
It is neither advised nor necessary for a student to memorize theorems,
postulates and definitions. The student should make a list of each and add
to the lists as the course progresses and refer to the lists frequently when
completing a proof. Complete lists can be found at the end of the textbook.
Objectives in learning proofs include:
1. Understanding and improving the skill of logical thinking.
2. Developing appreciation and skill in forming a well conceived
"argument".
3. Improving problem solving skills, both personal and mathematical.
Allowing lists to be used advances the additional benefit of teaching the
student the value and use of reference tools in study and problem solving.
Corrections and Clarifications
Geomtery 6th edition
Section 1.4, problem 4b. In the solutions guide, the answer should be "adjacent."
Chapter 1 Test, problem 19. In the solutions guide, the construction is not correct. A correct illustration has points at A and B, the endpoints of the given segment, and the crossing arcs above and below the given segment with a line drawn that contains the points of intersection of the crossing arcs. The line is perpendicular to the segment.
Section 1.6, problem 23. The proof in the solutions guide does not correspond with the problem. The correct proof is below.
Section 1.6, problem 25. The proof in the solutions guide does not correspond with the problem. The correct proof is below.
Chapter 3 Review, #3. The proof in the solutions guide does not correspond with the problem. The correct proof is below.
Geomtery 5th edition
Section 1.7, problem 22. In the solutions guide, the measure of angle 2 should be 70 degrees rather than 110 degrees.
Chapter 2 Review, problems 4 and 5. The solutions require that segments AB and CD are parallel and that information is not given. Please omit the problem.
Chapter 2 Test, problem 6. The construction in the solutions guide does not correspond with the problem in the textbook. The correct construction has the following steps.
1. With the point of the compass at point A and using a fixed opening greater than the distance to the line, draw an arc that intersects the line at two points. Label the points B and C. [Only the intersection points need to be shown.]
2. With the point of the compass at B and using an opening greater than half the distance between B and C, draw an arc below the line and approximately below point A.
3. Using the same opening, place the point of the compass C and cross the previous arc. Label the crossing point D.
4. Draw line AD. [Point may be labeled differently or not at all. They are used for clarity in this description.]
Section 2.6, problem 4. In the solutions guide, the upper case letter "I" as printed in the textbook has point symmetry.
Section 2.6, problem 8. In the solutions guide, item c, the isosceles does not have point symmetry. Generally, when a figure has point symmetry, if the figure is rotated 180 degrees about the point of symmetry, it will appear identical to the original figure.
Section 3.1, problem 34. In the solutions guide, the answer should be yes. The proof is:
Section 3.2, problem 4. In the solutions guide, in Statement 2 the first part should say angle M is congruent to angle R (rather than angle Q).
Section 3.2, problem 12. In the solutions guide, Reason #2, the second "if" should be replaces with "it".
Section 3.3, problem 10. In the solutions guide, the answer should be, the two sets are equivalent.
Chapter 3 Review, #3. The proof in the solutions guide does not correspond with the problem. The correct proof is below.
Section 5.1, problem 2(c). In the solutions guide, the answer should be 5/8 rather than 8/5.
Section 5.5, problem 2. In the solutions guide, both answers should be "a" rather than 1.
Chapter 6, Mid-Chapter Review, problem 23(f). There is a misprint in the solutions guide. In the first step of 3 steps, the sign between the arc measures should be plus rather than minus.
Section 6.3, problem 9. In the solutions guide, when all the terms of the quadratic equation are on one side of the equal sign, the constant should be +48 rather than -48.
Section 6.4, problem 23. In the solutions guide, the answers in both parts are incorrect. By the way, those answers are correct in the back of the textbook.
Section 7.2, problem 4. In the solutions guide, the correct answer is orthocenter.
Section 9.1, problem 6. In the solutions guide, the unit used to measure its lateral area is square inches and the unit used to measure its volume is cubic inches.
Section 9.3, problem 6. In the solutions guide, the replacement for the radius in the formula should be 1.5 rather than 1.05. The correct value is used in the calculation so the answer is correct.
Chapter 9 Review, problem 20. In the solutions guide, the answer should be 1/9 rather than 1/2.
Chapter 10 Test, problem 2. In the solutions guide, point D is plotted incorrectly. The correct location for point D is 9 units above the origin (on the y-axis).
Chalk Dust Special 2nd Edition
Geomtery 4th edition - Since October 2007
Section 1.3, problem 17. In the solutions guide, the first step should read 2x + 1 + 3x + 2 = 6x - 4.
Section 1.4, problem 4b. In the solutions guide, the answer should be "adjacent."
Section 1.4, problem 26. In the solutions guide the expression for Part C is actually the simplification of the expression in Part B. Part C should be
90 - (2x + 5y)
90 - 2x - 5y
Section 1.5, problem 29. In the solutions guide, step 5 should be "Addition" and step 6 can be "Division Property."
Section 1.6, problem 12.
Section 1.6, problem 26.
Clarification of statements in the proof
The steps starting with statement 4...
4. 0 < x < 90
5. x + m angle 2 = 90
6. m angle 2 = 90 - x
7. -x < 0 < 90 - x
This is from statement 4,
0 < x < 90
Subtract x in all three parts of the compound inequality to get statement 7.
Following statement 4, there are two things that must be addressed. One is the manipulation in steps 5 and 6 that allows for a substitution later in the proof and the other is a continuation with the compound inequality. It looks like a couple of skipped steps, but if steps 5 and 6 are not in place there, then they will have to be added later in the proof and, again, it would look like a skip in the flow.
8. 90 - x < 90 < 180 - x
This is the result of adding 90 in all parts of statement 7.
9. 0 < 90 - x < 90
Here is the real dilemma. It looks like statement 9 must come from statement 8 and if you assume that then statement 9 does not make sense because to clear -x in the outside parts of the inequality in statement 8, x has to be added to all parts and that would make the middle part 90 + x rather than 90 - x.
However, the author of the solutions guide did not come up with statement 9 by manipulating statement 8. Rather, two previous steps were put together (transitive property) to make statement 9. Specifically, part of statement 7 is 0 < 90 - x and part of statement 8 is 90 - x < 90. Put them together to get statement 9.
Further, the reason given in the solutions guide is "Transitive Property of Inequality" but statement 9 does not represent the normal usage of the transitive property because normally the left side is made less than the right side and the middle part is left out. However, the validity of the statement should be clear in spite of the unusual use of the transitive property.
I hope this helps clear things up.
Section 1.7, problem 20. In the solutions guide, the measure of angle 2 should be 70 degrees rather than 110 degrees.
Section 2.1, problem 26. In the solutions guide, omit the "||" symbol in the third reason.
Section 2.1, problem 27. The labeling of the illustration is incorrect in the solutions guide. Please swap the numbers 1 and 2.
Section 2.2, problem 13. In the solutions guide, the notation used in the figure is not consistent with the notation used in the textbook. Please omit the problem.
Section 2.2, problem 14. The solution in the solutions guide does not match the problem in the textbook. Please omit this problem.
Section 2.3, problem 18. In the solutions guide, the last statement and reason should be labeled "6" rather than "3".
Section 2.3, problem 21. The proof in the solutions guide does not correspond with the problem. The correct proof follows:
| Statements |
Reasons |
| 1. Ray DE bisects angle CDA |
1. Given |
| 2. Angle 2 is congruent to Angle 3 |
2. Definition of angle bisector |
| 3. Angle 3 is congruent to Angle 1 |
3. Given |
| 4. Angle 2 is congruent to Angle 1 |
4. Transitive Property (no substitution) |
| 5. Segment ED parallel to Segment AB |
5. Theorem 2.3.2 |
Section 2.4, problem 41.
Section 2.4, problem 44.
Section 2.5, problem 27.
Section 2.5, problem 40.
In the solutions guide, the segment to be drawn should be BD rather than BC.
Section 2.6, problem 4.
The letter "I" also has point symmetry.
Chapter 2 Review, problem 14. In the solutions guide, some of the steps in the solution are incorrect. Here is one way to work the problem.
From the given information, two equations emerge.
Since a || b,
Equation 1: 2x - y = 3x + 2y
In the left-side triangle, the sum of the measures of the angles is 180. So,
Equation 2: (2x - y) + 100 + x = 180
There are many ways to solve the system of equations. Begin by simplifying both equations.
(1)
2x - y = 3x + 2y
2x - 3x = 2y + y
-x = 3y
x = -3y
(2)
2x - y + 100 + x = 180
3x - y = 80
Use substitution and replace x with -3y in the second equation.
(2)
3(-3y) - y = 80
-9y - y = 80
-10y = 80
y = -8
Use either equation to find x.
(1)
x = -3y
x = -3(-8)
x = 24
Chapter 2 Review, problem 17.
In the solutions guide, step 2, the right side of the equation should be 111 rather than 222.
Chapter 2 Review, problem 18.
In the solutions guide, the third equation is supposed to be the simplification of the first so the first term should be 8x rather than 3x.
Chapter 2 Test, problem 6. The construction in the solutions guide does not correspond with the problem in the textbook. The correct construction has the following steps:
1. With the point of the compass at point A and using a fixed opening greater than the distance to the line, draw an arc that intersects the line at two points. Label the points B and C. [Only the intersection points need to be shown.]
2. With the point of the compass at B and using an opening greater than half the distance between B and C, draw an arc below the line and approximately below point A.
3. Using the same opening, place the point of the compass at C and cross the previous arc. Label the crossing point D.
4. Draw line AD.
[Point may be labeled differently or not at all. They are used for clarity in this description.]
Section 3.1, problem 39. In the solutions guide, the angles listed in step 7 are not correct. The angles should be angle ABC and angle DEC.
Section 3.2, problem 29. In the solutions guide, the angle in the first statement should be SRU rather than RSU.
Section 3.3, problem 10. In the solutions guide, the solution is incorrect. The sets are equivalent.
Section 3.3, problem 11. In the solutions guide, replace "S" with "O".
Section 3.4, problem 30.
Chapter 3 Review, problem 3.
Section 4.1, problem 25.
Section 4.1, problem 36. In the solutions guide the solution is not consistent with the problem in the textbook. The answer is, "The bisectors should bisect each other."
Section 4.2, problem 27. In the solutions guide, reason #11, the symbol for angles should be replaced with the word "segments."
Section 4.2, problem 35.
In the solutions guide, statement 13, segment ND should be segment BC.
Section 4.3, problem 22.
Section 4.3, problem 26. In the solutions guide, the answer should be (b).
Section 4.3, problem 35. In the solutions guide, statement 4 should be "segment AE is congruent to segment CE."
Section 4.3, problem 36. In the solutions guide, there are two mistakes in the proof. Statement 7 should be, segment BE is congruent to segment BE. Reason 8 should be SAS.
Section 4.4, problem 30. In the solutions guide, in statements 5, 7, and 8, the parallel symbol should be replaced with the congruence symbol. Also, the reason for statement 9 is incorrect. It should be, "If the legs of a trapezoid are congruent, the trapezoid is an isosceles trapezoid."
Section 5.1, problem 2. In the solutions guide, the answers for parts (c) and (d) should be swapped.
Section 5.2, problem 24. In the solutions guide, the first numerator is the length of side AD so the numerator should be 4 rather than x. The mistake is corrected in the next step.
Section 5.3, problem 22. In the solutions guide, the last item is incorrect. It should read R6. CSSTP
Section 5.4, problem 43. The proof in the solutions guide contains a typographical error. Rather than "angle CDB is congruent to angle ACD", it should be "angle CDB is congurent to angle ADC."
Section 5.5, problem 2. In the solutions guide, the answers should be:
a. AC = a b. BC = a
Section 5.5, problem 36b. In the solutions guide, the length of CF should be 2x rather than 12.
Section 5.6, problem 30. In the solutions guide, statement 2, the denominator on the right side of the equation should be SY.
Section 6.1, problem 19. In the solutions guide, there are several errors. Please omit this problem.
Section 6.3, problem 9. In the solutions guide, the constant term in the quadratic equation should be +48 rather than -48.
Section 6.3, problem 30. In the solutions guide, the ratio of RT / RS is calculated incorrectly.
If TO = x, then RO = x and RS = 2x. Since RT = x sqrt 2,
RT / RS = x sqrt 2 / 2x,
RT / RS = sqrt 2 / 2
Section 6.4, problem 20.The problem is not valid. It appears the author used inscribed angles when central angles were inrended. Please omit the problem.
Section 6.4, problem 23.In the solutions guide the inequality in part "a" should be "greater than".
Section 6.4, problem 33. In the solutions guide, the tangent requested should be drawn.
Section 6.6, problem 4. In the solutions guide, the answer should be Orthocenter rather than Circumcenter.
Chapter 6 Review, problem 10. In the solutions guide, the measure of arc AD should be 180 degrees rather than 80 degrees.
Chapter 6 Review, problem 33.
The solutions guide does not provide enough information for one to know where the listed equation comes from and it is not intuitively obvious.
Make a sketch of the problem — a right triangle with an inscribed circle. In order for my description to match your diagram, place the right angle of the triangle at the lower left with acute angles above and to the right of the right angle. The hypotenuse should be slanting from upper left to lower right.
From the center of the circle, draw segments to the three sides of the triangle at the point of tangency. Notice a right angle is formed by the segments drawn to the horizontal and vertical legs of the triangle and a square is formed with one vertex at the right angle of the right triangle. Label 6 on all sides of the square.
Draw segments from the center of the circle to the vertices of the acute angles of the triangle. Notice a little pair of congruent right triangles formed at each acute angle.
Move around the outer edge of the original right triangle and label every little segment. Starting at the right angle and moving up and then clockwise, the first segment is the 6 labeled earlier.
Label x the segment from the point of tangency on the vertical side to the upper left vertex.
Label x the segment along the hypotenuse from the upper left vertex to the point of tangency on the hypotenuse.
Since the hypotenuse is 29 cm, the segment from the point of tangency on the hypotenuse to the vertex at the lower right is (29 - x).
Also label the segment from the vertex at the lower right of the figure to the point of tangency on the horizontal leg (29 - x).
The segment from the point of tangency on the hoizontal leg of the triangle to the vertex of the right angle is already labeled 6.
Now take a look at the original big right triangle and consider the lengths of the sides.
The vertical left side is (x + 6).
The hypotenuse is 29.
The horizontal bottom side is 6 + 29 - x or (35 - x).
Using those side lengths, apply the Pythagorean Theorem and you have the equation in the solutions guide.
Section 7.1, problem 34. In the solutions guide, the solution for this problem is missing; they duplicated the solution for problem 38 instead.
There are 3 similar right triangles in the figure. Consider the small one (with hypotenuse "a") and the large one (with hypotenuse "c"). Similar trianlges have the characteristic that corresponding sides are proportional. That is, ratios of corresponding sides can be set equal to one another.
One such equation can be formed using the pattern,
(long leg / hypotenuse) = (long leg / hypotenuse)
[small triangle] [large triangle]
h / a = b / c Multiply both sides by "a" to get...
h = ab / c
Section 7.1, problem 49.
Section 7.3, problem 12. In the solutions guide, since the area and perimeter are equal, they cancel in the last step given in the solutions guide. Therefore, a = 2.
Section 7.5, problem 18. In the solutions guide, while calculating the area of the sector, the radius is incorrect. The radius should be 10 / sqrt 3 rather than just 10. The change results in an answer of (100pi - 75sqrt3)/9.
Section 7.5, problem 23. In the solutions guide, the diagram does not correspond with the problem in the textbook. However, the calculation of the area of the circle is correct.
To find the radius of the circle, draw a vertical segment from the top vertices down to the bottom side. Since the trapezoid is isosceles, the base of the right triangles formed (on both sides) is 5. Calculate the length of the vertical leg using the Pythagorean Theorem.
Section 8.1, problem 32. In the solutions guide, the factor 2/3 should be 1/3 (because 4 inches in 1/3 foot). The correct answer is 13 1/3 cubic yards.
Section 8.2, problem 35.
In the solutions guide, the answers given are correct but part (c) is not included, the method of solution for parts (a) and (b) are not clear and the figure does not seem to match the situation.
Since the regular tetrahedron in the text is composed of 4 congruent equilateral triangles with sides of length e, begin with the sketch of an equilateral triangle. Draw the triangle with a horizontal base at the bottom of the figure. Draw a segment from the top vertex vertically to the base.
The segment drawn is a height of the equilateral triangle and the point of intersection with the bottom side bisects the bottom side. Therefore, the length of each half is e/2.
Two right triangles were formed by the construction of the height of the equilateral triangle. Each right triangle has a leg of length e/2 and hypotenuse e. Find the length of the vertical leg using the Pythagorean Theorem.
a^2 + b^2 = c^2
(e/2)^2 + b^2 = e^2
e^2 / 4 + b^2 = e^2
b^2 = e^2 - e^2 / 4
Create a common denominator on the right by multiplying e^2 by 4/4...
b^2 = 4e^2 / 4 - e^2 / 4
b^2 = 3e^2 / 4
b = sqrt (3e^2 / 4)
b = e sqrt 3 / 2
(a) Since b is the height of the little right triangles, the area of each of them is...
A = (1/2)bh
A = (1/2)(e/2)(e sqrt 3 / 2)
A = (e^2 sqrt 3) / 8
Since there are two such right triangles forming the equilateral triangle, the area of the equilateral triangle is
2 (e^2 sqrt 3) / 8
(e^2 sqrt 3) / 4
(b) Since there are 4 congruent triangles forming the overall figure, the total area is 4 times the area of one triangle...
Total area = 4 (e^2 sqrt 3) / 4
Total area = e^2 sqrt 3
(c) When e = 4
Total Area = 4^2 sqrt 3
Total Area = 16 sqrt 3
Section 8.2, problem 36. In the solutions guide there is a typo in the third step. The fraction 1/2 should be 1/12.
Chapter 8 Review, problem 20. In the solutions guide, the answer to the first part should be 1/9 rather than 1/2.
Section 9.3, problem 24. In the solutions guide, the equation in the third to last step is 2st = -3st. It should be 2st = -2st.
Chapter 9 Test, problem 2. In the solutions guide, point D should be located 9 units above the origin on the y-axis.
Section 10.4, problem 39. The solution in the solutions guide does not match the problem. Draw the segment Q, perpendicular to side MN, intersecting MN at point R. In the right triangle MQR,
sin y = QR/a So
QR = a sin y
The area of the parallelogram is found by multiplying the base times the height.
Area = base x height
Since the base of the paeallelogram is b and the height is QR,
A = b(QR)
Replace QR with a sin y to get
A = b(a sin y) or
A = ab sin y
Chapter 10 Test, problem 3(b). In the solutions guide, replace "tan" with "sin".
