MATH 90 CHAPTER 4 PART I MSJC ~ San Jacinto Campus Math Center Workshop Series Janice Levasseur

Polya’s 4 Steps to Problem Solving 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 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. Check: x = 36 degrees  other angle = 2(36) – 18 = 54 degree  36 + 54 = 90 degrees! Yes, Yes!

Ex: Solve for x 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 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): 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 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 b c e d f g 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 c e d f 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 t b g 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 t b c e d f g 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 < c = < g < d = < h

Ex: Given l1 || l2 find the measure of angles a and b 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 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 b c < 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 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. 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? 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 x 20+x We need to find the measures of the three angles

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 20+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 + 20 = 180 4x = 160  x = 40  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 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: Don’t say, “I hate word problems!” Take a deep breath and tackle the word problem using Poyla’s 4 steps PRACTICE, PRACTICE, PRACTICE Get help (Instructor, Math Center Bldg 300, study buddy/group, SI) Good Luck . . . You can do it!