Geometry & Measurement Maureen Steddin
table of contents To the Student......................... v Part 1: Introduction.................................... 1 General Approach to Math Questions................... 1 Content-Specific Strategies for Measurement and Geometry Problems.................. 1 Part 2: Reference Information........................ 7 Test-Taking Glossary................................. 7 Content-Specific Definitions and Formulas............... 8 Part 3: Measurement Conversion................... 13 Set 1 (Standard and Metric Conversions)............... 13 Set 2 (Scale Drawings and Models).................... 15 Part 4: Polygons...................................... 17 Set 3 (Properties of Polygons)......................... 17 Set 4 (Perimeter and Area)........................... 19 Set 5 (Perimeter and Area)........................... 22 Set 6 (Perimeter and Area)........................... 24 Part 5: Circles......................................... 27 Set 7 (Circumference and Area)....................... 27 Part 6: Lines and Angles............................. 29 Set 8 (Angle Relationships and Measures)............... 29 Set 9 (Angle Relationships and Measures)............... 31 Set 10 (Angle Relationships and Measures).............. 33 Set 11 (Angle Relationships and Measures).............. 35 iii
Part 7: Congruence and Similarity.................. 37 Set 12 (Congruent and Similar Figures)................ 37 Part 8: Finding Volume............................... 39 Set 13 (Volume).................................... 39 Set 14 (Volume).................................... 41 Part 9: Transformations.............................. 43 Set 15 (Transformations)............................. 43 Answer Key................................... 47 iv
to the student Test Your Best! We all have to take tests. Often, our abilities are measured by how well we test. Each year, more and more tests are added to our lives. District, state, and national assessments reflect student progress, teacher abilities, administrative skills, and curriculum standards. In other words, a lot is riding on these tests. It is important for you to take them seriously, just as your superintendent, your principal, and your teachers do. The books in the Test Time! series were designed to help you practice your test-taking skills. They also provide you with successful strategies and tips to follow at test time. As you well know, practice makes perfect. The more you practice, the higher you score. When you do well, not only are you successful, but your teachers, your administrators, and your state legislators are, too. This means that they took the testing seriously and wanted to help you be successful. It s a team effort. With all that in mind, be confident that you can succeed. You have the power; now just practice the skills. Good luck! v
PART 1 introduction General Approach to Math Questions Read each question carefully. Think about what math facts or formulas you will need. If you know what to do right away, you can start solving. On a multiple-choice question, it might help to look at the answer choices first. If some are unreasonable, they can be eliminated. The choices might also push you in the correct direction. If all the choices are given in minutes, you should figure out your answer in minutes, too. For some questions, you might need to draw a diagram or mark up one you are given with information from the question. Remember to answer the question that is asked. If you are working on a problem with lots of steps, you could lose track of what you are looking for. For example, a question might ask you to find each person s share of an equally divided restaurant bill. To do so, you will need to figure out the total amount of the bill first. If you are not careful, you might stop at this total and choose it as your answer. You may get partial credit on open-ended questions. That means that you should always do as much as you can on these. Even if you do not make it all the way to the final answer, you may score some points for the work you do show. Content-Specific Strategies for Measurement and Geometry Problems STRATEGY: Use the conversion value to switch between measurements. Multiply by the conversion value to convert from a larger unit to a smaller unit. Divide by the conversion value to convert from a smaller unit to a larger unit. Because 12 inches = 1 foot, 12 is the conversion value. Convert 4 feet into inches. This is going from a larger unit to a 2004 Walch Publishing 1 Test Time! Geometry & Measurement, 5 6
smaller unit, so multiply by 12: 12(4) = 48 inches. Convert 63 inches into feet. This is going from a smaller unit to a larger unit, so divide by 12: 63 3 12 = 5 ----- = 5.25 feet. 12 STRATEGY: Move the decimal point in metric conversions. The metric system is based on the power of 10. This means that you can multiply or divide by a power of 10 to make metric conversions. To multiply by a power of 10, move the decimal point right as many places as that power. To divide by a power of 10, move the decimal point left as many places as that power. Example: Convert 347 centimeters (cm) to meters (m). There are 100 centimeters in a meter, so the conversion value is 100 or 10 2. You are converting from a smaller unit to a larger unit, so you divide. The decimal point in 347 is understood to be after the 7. Moving it two places to the left gives you 3.47. Therefore, 347 cm = 3.47 m. STRATEGY: Use a proportion to solve problems with scale drawings. To solve a question with a scale drawing or model, you need to know its scale ratio. Use it to set up a proportion. The other ratio in the proportion should compare a scale measurement to an actual measurement. Cross multiply to solve for the unknown. STRATEGY: Plug the right values into the right formulas. Always make sure to use the right formula and plug in the right values. It is easy to mix up area and perimeter formulas or to plug in the wrong measurement from a figure. Say that the perimeter of a square is 36 feet. You are asked to find its side length. Many people see the 36 in a question about a square and say that the side is 6. This is the result of using the area formula (s 2 = 36, s = 36, s = 6) instead of the perimeter formula: 4s = P, 4s = 36, s = 9. STRATEGY: Check your work. Try to make time to check your work on math problems. This is especially important for questions that do not give you answers from which to choose. Do not let a careless error lose points for you. 2004 Walch Publishing 2 Test Time! Geometry & Measurement, 5 6
STRATEGY: Check the reasonableness of answers. It is good to check the reasonableness of answers. For multiple-choice questions, this can help you eliminate some unreasonable choices. Say that you are given the amount of a bill at a restaurant and are asked how much must be left to include a 15% tip. You know that the correct answer must be greater than the amount of the bill alone. Eliminate any choices that are less than this, and you will have a better chance of choosing the correct answer. It is likely that the amount of the tip alone will be a choice. Doing this reasonable check will help make sure that you do not jump at this choice before adding it to the bill for the correct total amount. STRATEGY: Plug in a value to make the problem more concrete. Sometimes it can help to use real numbers to figure out the results of a general situation. For example, you might be asked what happens to the area of a square if you double its side length. Choose a number for its side length and figure out the original area. Then double this side length and figure out the new area. If the original side length is 5, doubling it gives you 10. Area of a square is s 2, where s is a side of the square. That makes the original area 5 2 = 25 and the new area 10 2 = 100. This is 4 times the original area. Now you can see that if you double the side of a square, you multiply its area by 4. STRATEGY: Organize the information you are given, and take it one step at a time. When there is a lot of information in a question, it is important to take the time to organize it correctly. If not, you might understand the math in a question but make a careless mistake and get it wrong. For example, you may be asked to solve for the measure of an angle identified as 3a. If you have enough information to solve for the value of a, do so. Do not choose that number as the answer, though. Remember that you are asked for 3a, so multiply the number you find by 3. 2004 Walch Publishing 3 Test Time! Geometry & Measurement, 5 6
STRATEGY: Know what happens when parallel lines are crossed by a transversal. When parallel lines are crossed by a transversal, eight angles are formed. If the transversal is perpendicular to the parallel lines, all eight angles are right angles containing 90. If the transversal is not perpendicular, four of the angles formed are acute and four are obtuse. All four acute angles are equal to one another. All four obtuse angles are equal to one another. Every acute angle is supplementary to every obtuse angle. STRATEGY: Know the relationship among the three angles of a triangle. The three interior angles of a triangle add up to 180. If you know the values of two of the three angles, you can solve for the third. If you have information about the relative sizes of the three angles, you can solve for their values. STRATEGY: Use a proportion to solve questions with similar figures. If two figures have the same shape but are different sizes, they are similar. The corresponding angles of similar figures are equal. The corresponding sides of similar figures are proportional. Use a proportion to solve for an unknown length in a question about similar figures. STRATEGY: Keep your transformations straight. Remember that a reflection is a flip, a translation is a slide, and a rotation is a turn. STRATEGY: Find an estimate. An estimate will not give you the exact answer to a question. However, if you are running out of time, an estimate can help you choose the correct choice. Sometimes an estimate is all you need to solve. You know you can use an estimate when a question includes words such as about, approximately, or closest to. Another clue is when all the answer choices are round numbers. To find an estimate, round the numbers in the question. Then do the math. If a question involves π, you can estimate it as 3.14 or just a little more than 3. 2004 Walch Publishing 4 Test Time! Geometry & Measurement, 5 6
STRATEGY: Try a backdoor approach. For some questions, you can plug the answer choices back into the question to see which is correct. This strategy works when the answer choices are numbers. This can take a lot of time. However, you often do not need to try every choice to see which is correct. And if you do not know how to solve the question, it is worth it to work backward. When you are plugging answer choices back into a question, always start with the middle value. If you plug in the middle value and it is too big, you can eliminate it and all bigger choices. If it is too small, you can eliminate it and all smaller choices. If time were running out, you would have a better chance of guessing correctly after eliminating one or more choices. 2004 Walch Publishing 5 Test Time! Geometry & Measurement, 5 6
PART 6 lines and angles Set 8 1. In ABC, what is the measure of ABC? B 65 A C A. 25 B. 45 C. 90 D. 115 2. What is the best estimate for the measure of QRS? Q 3. Which angle is NOT obtuse? A. 60 B. 100 C. 120 D. 150 E. 175 4. A triangle has angles with measures 63, 85, and x. Which of the following is the value of x? A. 22 B. 32 C. 95 D. 148 5. If the measure of WXZ is 59, what is the measure of ZXY? Z R A. 15 B. 45 C. 75 D. 100 E. 150 S W X Y A. 31 B. 59 C. 121 D. 239 2004 Walch Publishing 29 Test Time! Geometry & Measurement, 5 6
Set 8 (continued) 6. Angles 1 and 2 are vertical angles. If the measure of angle 1 is 37, what is the measure of angle 2? 9. An isosceles triangle has equal base angles of 75. Draw this triangle, identifying all of its angle measures. A. 37 B. 53 C. 74 D. 143 7. Which of the following lines appears to be parallel to AB? C E G A I B 10. If two angles are supplementary, A. the sum of their measures is 90. A. CD B. EF C. GH D. IJ H F D J B. the sum of their measures is 180. C. the sum of their measures is greater than 180. D. their measures are equal. 8. In triangle XYZ, X is twice as big as Y. If X = 50, what is the measure of Z? A. 25 B. 30 C. 75 D. 100 E. 105 STOP 2004 Walch Publishing 30 Test Time! Geometry & Measurement, 5 6
Set 9 1. Two angles of a triangle have measures 25 and 45. What is the measure of the third angle of this triangle? A. 45 B. 50 C. 70 D. 90 E. 110 2. Which angle is NOT acute? A. A 20 B. 3. In the figure below, WY bisects XWZ. If the measure of YWZ is 32, what is the measure of XWZ? X Y W Z A. 16 B. 32 C. 58 D. 64 Questions 4 and 5 refer to the figure below. M C. 55 B L 140 N O 4. What is the measure of MNL? A. 40 C 90 B. 50 C. 90 D. D. 140 75 D 5. If the measure of LMN is twice the measure of MNL, what is the measure of MLN? A. 20 B. 60 C. 80 D. 120 2004 Walch Publishing 31 Test Time! Geometry & Measurement, 5 6
Set 9 (continued) 6. Draw a line perpendicular to AB at point X. Identify the measure of any angles that are formed. 9. If line 1 is parallel to line 2, what is the value of the unknown angle? 110 l 1? 7. If the measure of WAY is 77, what is the measure of ZAX? W A X B Y A. 13 B. 38.5 C. 77 D. 103 E. 283 A Z X A. 20 B. 55 C. 70 D. 110 10. In a certain triangle, A = B. If C = 80, what is the value of B? A. 10 B. 20 C. 40 D. 50 l 2 8. Angles 1 and 2 are complementary angles. Which of the following could be the measures of angles 1 and 2? A. angle 1 = 30 and angle 2 = 30 B. angle 1 = 30 and angle 2 = 150 C. angle 1 = 30 and angle 2 = 45 D. angle 1 = 45 and angle 2 = 45 STOP 2004 Walch Publishing 32 Test Time! Geometry & Measurement, 5 6