MATH 90 CHAPTER 4
PART I
MSJC ~ San Jacinto Campus
Math Center Workshop Series
Janice Levasseur
Polya’s 4 Steps to Problem Solving
1.
2.
3.
4.
•
Understand the problem
Devise a plan to solve the problem
Carry out and monitor your plan
Look back at your work and check your
results
We will keep these steps in mind as we
tackle the application problems from the
infamous Chapter 4!
1. Understand the problem
• Read the problem carefully at least twice.
– In the first reading, get a general overview of the
problem.
– In the second reading, determine (a) exactly what you
are being asked to find and (b) what information the
problem provides.
• Try to make a sketch to illustrate the problem.
Label the information given.
• Make a list of the given facts.
Are they all pertinent to the problem?
• Determine if the information you are given is
sufficient to solve the problem.
2. Devise a Plan
to Solve the Problem
• Have you seen the problem or a similar problem
before?
• Are the procedures you used to solve the similar
problem applicable to the new problem?
• Can you express the problem in terms of an
algebraic equation?
• Look for patterns or relationships in the problem
that may help in solving it.
• Can you express the problem more simply?
• Will listing the information in a table help?
2. continued.
• Can you substitute smaller or simpler
numbers to make the problem more
understandable?
• Can you make an educated guess at the
solution? Sometimes if you know an
approximate solution, you can work
backwards and eventually determine the
correct procedure to solve the problem.
3. Carry Out and Monitor Your
Plan
• Use the plan you devised in step 2 to
solve the problem.
• Check frequently to see whether it is
productive or is going down a dead-end
street. If unproductive, revisit Step 2.
4. Look Back at Your Work and
Check Your Results
• Ask yourself, “Does the answer make sense?”
and “Is the answer reasonable?” If the answer is
not reasonable, recheck your method for solving
the problem and your calculations.
• Can you check the solution using the original
statement?
• Is there an alternative method to arrive at the
same conclusion?
• Can the results of this problem be used to solve
other problems?
Geometry
Problems involving angles formed
by intersecting lines
• A unit used to measure angles is the
degree.
is the symbol for degree
< is the symbol for angle
• One complete revolution is 360 degrees.
Angles
• A right angle has measure 90 degrees
• A straight angle has measure 180 degrees
• An acute angle has measure between 0
and 90 degrees
• An obtuse angle has measure between 90
and 180 degrees
• Complementary Angles are two angles
whose measures sum to 90 degrees
• Supplementary Angles are two angles
whose measures sum to 180 degrees
Ex: Solve for x
x
2x – 18
1. What are we being asked to find?
The value of x
2. Can you express the problem in terms of an
algebraic equation?
The two angles are complementary angles
 x + (2x – 18) = 90
3. Using the plan devised in Step 2,
solve the (algebraic) problem
x + (2x – 18) = 90
Solution: x = 36 degrees
 3x - 18 = 90
 3x = 108
 x = 36
4. Did we answer the question being asked?
Is our answer complete? Check the solutions.
Yes, Yes!
Check: x = 36 degrees
 other angle = 2(36) – 18 = 54
degree  36 + 54 = 90 degrees!
Ex: Solve for x
x+20
5x
2x
1. What are we being asked to find?
The value of x
2. Can you express the problem in terms of an
algebraic equation?
The angles together form a straight angle
 5x + (x + 20) + 2x = 180
3. Using the plan devised in Step 2,
solve the (algebraic) problem
5x + (x + 20) + 2x = 180
 8x + 20 = 180
Solution: x = 20 degrees
 8x = 160
 x = 20
4. Did we answer the question being asked?
Is our answer complete? Check the solutions.
Yes, Yes!
Check: x = 20 degrees
 1st angle = 5(20) = 100, 2nd angle =
20 + 20 = 40, 3rd angle = 2(20) = 40 
100 + 40 + 40 = 180 degrees!
Ex: Solve for x and identify the
measure of each angle
3x
5x
4x
6x
1. What are we being asked to find?
The value of x and then the measures of the
four angles
2. Can you express the problem in terms of an
algebraic equation?
The angles together form a complete revolution
 5x + 3x + 4x + 6x = 360
3. Using the plan devised in Step 2,
solve the (algebraic) problem
5x + 3x + 4x + 6x = 360
 3x = 3(20) = 60
 18x = 360
 4x = 4(20) = 80
 x = 20
 6x = 6(20) = 120
 5x = 5(20) = 100
Solution: x = 20 degrees  angles measure 60,
80, 120, and 100 degrees
4. Did we answer the question being asked?
Is our answer complete? Check the solutions.
Yes, Yes!
Check: 60 + 80 + 120 + 100 = 360
degrees!
Lines
• Parallel Lines never meet (are equidistance
apart), l1 || l2
• Perpendicular Lines are intersecting lines
that form right angles, l1 | l2
Intersecting Lines & Their Angles
• Four angles are formed when two line
intersect:
a
d
b
c
• Vertical Angles are on opposite sides of the
intersection and have the same measure:
< a and < c, < b and < d are vertical
< a = < c and < b = < d
• Adjacent Angles share a common side and
are supplementary (sum to 180 degrees):
a
d
b
c
< a & < b, < b & < c, < c & < d, and < d & < a are
adjacent
< a + < b = < b + < c = < c + < d = < d + < a = 180
degrees
Ex: Find the angle measures
5x
3x+22
1. What are we being asked to find?
The angle measures
(therefore we need to solve for x)
2. Can you express the problem in terms of an
algebraic equation?
The angles are vertical and therefore are
equal in measure
 5x = 3x + 22
3. Using the plan devised in Step 2, solve the
(algebraic) problem
5x = 3x + 22
 5x = 5(11) = 55
 2x = 22
 3x+22 = 3(11)+22 = 55
 x = 11
Solution: angles
measure 55 degrees
4. Did we answer the question being asked?
Is our answer complete? Check the solutions.
Yes, Yes!
• A transversal is a line that intersects two
other lines at different points
a
e
g
c
f
b
t
d
h
• If the lines l1 || l2 and t is not perpendicular
to l1 or l2 then all four acute angles have
the same measure and all four obtuse
angles have the same measure
<a = < d = < e = < h
<b=<c=<f=<g
t
e
c
f
d
• Alternate Interior Angles are two nonadjacent
angles that are opposite sides of the transversal
and between the lines.
• Alternate Interior Angles have the same measure
<d=<e
<c=<f
a
g
b
t
h
• Alternate Exterior Angles are two nonadjacent
angles that are opposite sides of the transversal
and outside the parallel lines.
• Alternate Exterior Angles have the same measure:
<a = < h
<b=<g
a
e
g
c
f
b
t
d
h
• Corresponding angles are two angles that
are on the same side of the transversal and
are both acute or both obtuse.
• Corresponding angles have the same measure:
<a=<e
<b=<f
<d=<h
<c=<g
Ex: Given l1 || l2 find the measure of
angles a and b
t
l1
47
a
b
l2
1. What are we being asked to find?
The measures of angles a and b
2. Can you express the problem in terms of an
algebraic equation?
Alt. Int. Angles have the same measure 
< a = 47
Supp. Angles’ measures sum to 180  < a + < b = 180
3. Using the plan devised in Step 2, solve the
(algebraic) problem
< a = 47 degrees
< a + < b = 180
47 + < b = 180
< b = 133 degrees
4. Did we answer the question being asked?
Is our answer complete? Check the solutions.
Yes, Yes!
Ex: Given l1 || l2 find x
t
3x
x+20
b
l1
l2
1. What are we being asked to find?
The value of x
2. Can you express the problem in terms of an
algebraic equation?
Alt. Ext. Angles have the same measure 
3x = < b
Supp. Angles’ measures sum to 180  (x+20) + < b = 180
3. Using the plan devised in Step 2, solve the
(algebraic) problem
3x = < b
(x+20) + < b = 180
(x+20) + 3x = 180
4x + 20 = 180
4x = 160
x = 40
4. Did we answer the question being asked?
Is our answer complete? Check the solutions.
Yes, Yes!
Angles of a Triangle
• If the lines cut by a transversal are not
parallel, the three lines intersect at 3
points and form a triangle.
• The angles within the region enclosed by
the triangle are called interior angles and
the sum of the measures of the interior
angles is 180 degrees.
a
c
b
< a + < b + < c = 180
• An angle adjacent to an interior angle is an
exterior angle
m
a
n
< m and < n are exterior angles
• The sum of the measures of an interior
and an exterior angle is 180 degree
< m + a = 180
< n + a = 180
Ex: A triangle has two angles with
measures 42 degrees and 101 degrees.
Find the measure of the third angle.
1. Try to make a sketch to illustrate the
problem. What are we being asked to find?
101
x
42
Let x = measure of the third angle
We need to find the value of x
2. Can you express the problem in terms of an
algebraic equation?
The sum of the measures of the interior angles
of a triangle is 180
 101 + 42 + x = 180
3. Using the plan devised in Step 2,
solve the (algebraic) problem
101 + 42 + x = 180
143 + x = 180
 x = 37 degrees
4. Did we answer the question being asked? Is our
answer complete? Check the solutions. Yes, Yes!
Check: 101 + 42 + 37 = 180
Ex: Given the picture below, find
the measures of angles a and b.
a
b
45
1. What are we being asked to find?
We need to find the measure of angle
a and the measure of angle b
2. Can you express the problem in terms of an
algebraic equation?
c
a
b
45
< a = 45 degrees since vertical angles have equal measure
Let c be the third angle of the triangle.
The sum of the measures of the interior angles of a
triangle is 180
 < a + 90 + < c = 180
 45 + 90 + < c = 180
3. Using the plan devised in Step 2,
solve the (algebraic) problem
since < c and < b
are supplementary
 45 + 90 + < c = 180
 135 + < c = 180
< b + < c = 180
 < c = 45 degrees
 < b + 45 = 180
 < b = 135 degrees
4. Did we answer the question being asked?
Is our answer complete? Check the solutions.
Yes, Yes!
Check: < a + 90 + < c = 45 + 90 + 45 = 180
and < b + < c = 135 + 45 = 180
Ex: In a triangular gable end of a roof, the
peak angle is twice as large as the back
angle. The measure of the front angle is 20
degrees greater than the back angle. How
large are the angles of the gabled roof?
1. Try to make a sketch to illustrate the
problem. What are we being asked to find?
2x
20+x
x
We need to find the measures of the three angles
2. Can you express the problem in terms of an
algebraic equation?
2x
The sum of the measures of the
interior angles of a triangle is 180
20+x
x
 Back angle + peak angle + front angle = 180
3. Using the plan devised in Step 2,
solve the (algebraic) problem
x + 2x + (x + 20) = 180
4x = 160
 x = 40
 4x + 20 = 180
 back angle = 40 degrees
 peak angle = 2(40) = 80 degrees
 front angle = 20 + 40 = 60 degrees
4. Did we answer the question being asked?
Is our answer complete?
Check the solutions.
Yes, Yes!
Check: 40 + 80 + 60 = 180
To be a successful “word” problem
solver:
1. Don’t say, “I hate word problems!”
2. Take a deep breath and tackle the word
problem using Poyla’s 4 steps
3. PRACTICE, PRACTICE, PRACTICE
4. Get help (Instructor, Math Center Bldg
300, study buddy/group, SI)
5. PRACTICE, PRACTICE, PRACTICE
 Good Luck . . . You can do it!