SAMPLE. Classification of angles: Answer: Answer: Answer: Answer: Answer:

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1 SMPL

2 The ancient Romans are famous for their achievements in architecture and engineering. One of ancient Rome s greatest accomplishments was the Roman aqueduct system. The aqueducts brought clean water into Rome for drinking and washing. Q1 nswer: 35 nswer: 110 nswer: 220 nswer: The design of the aqueducts had to be just right to allow the water to run into the city without becoming stagnate in the channels or flowing too rapidly into the city. Therefore, the engineers had to be very skilled with geometry and couldn t afford not to understand angles! lassify each angle according to its size. lassification of angles: right angle is an angle of 90 n acute angle is an angle less than 90 n obtuse angle is an angle above 90 but below 180 straight angle is an angle of 180 reflex angle is an angle of more than 180 but less than 360 id you know? To lift heavy stones, the Romans used cranes powered by slaves or animals turning a treadmill! SMPL 180 nswer:

3 Q2! ircle the different angles in different colours. cute = red, obtuse = blue, right angle = purple, reflex = orange and straight = green. Q3 a) c) Using your knowledge of angles, determine the value of. Show all working. b) d) SMPL 125

4 e) g) I) k) f) h) j) SMPL id you know? The Roman aqueducts were over 55 feet high. This allowed control of the water flow but also made it more difficult for someone to steal or poison the water.

5 Roman roads were engineered so well that some can still be seen and used today. Their roads were mostly straight and thus formed by two parallel lines. Have a look at the example below, and then try the questions. Note that the diagrams are not drawn to scale. XMPL: The straight road is crossed by a transversal line. If the line makes an angle of 50 with the side of the road, what is the value of missing angle x? Q What is the value of missing angle x? What is the value of missing angle x? Q2 x What is the value of missing angle x? x hoose the correct answer. Write the answer in the box provided.? How? When a line (a transversal) crosses two parallel lines many angles are formed. ach angle formed has a pair and each pair has a special name. 1. orresponding angles: are equal (a=b), draw an shape (in red) to recognise. 2. lternate angles: are equal (a=b), draw a Z shape (in red) to recognise. 3. o-interior angles: add up to 180 (a +b = 180 ), raw a shape (in red) To recognise. Therefore the angles in the example were co-interior angles: 50 + x = 180, x = 130 Q3 Q What is the value of missing angle x? x a b hoose the correct answer. a a b Write the answer in the box provided. SMPL x x

6 Q5 a) and? b) and? c) and? d) and? e) The Roman in charge of supervising the construction of roads has to know more details about angles. Use the hint box to label the given angles as either alternate, complementary, corresponding, co-interior, supplementary or vertically opposite. f) Hint: omplementary angles are angles that add up to 90 Supplementary angles are angles that add up to 180 Vertically opposite angles are angles that are opposite, share an apex and are equal The symbol means angle SMPL and? and?

7 Q6! a)? b)? c)? d) 35 Using your knowledge of the value of pairs of angles, solve the missing angle. Show all working e)? f)? g)? h) 91 G SMPL??

8 an you calculate the interior angles of a triangle? Have a look at the example below, and then try the questions. Note that the diagrams are not drawn to scale. XMPL: wooden ladder makes an angle of 65 with a vertical stone wall. What is the value of missing angle x? Q assius noticed the blade of a sword outside his home. What is the value of missing angle x? Q2 hoose the correct answer Roman social structure can be presented as a triangle with the mperor at the apex, the plebeians and slaves at the base of the triangle and the patricians in the middle. What is the value of missing angle x (the emperor s position)? Write the answer in the box provided.? ny given triangle has an interior angle sum of 180. In other words, all three angles within any triangle add up to a total of 180. In this example, two angles within the triangle are already given, 70 and 90 (the right angle). Therefore, in order to find the value of x, the answer is simply: x = = 20 Hence, the value of x is 20. Q3 The Romans used columns in their architecture extensively. What is the value of missing angle x? Q4 How? hoose the correct answer The Romans traded goods throughout their vast empire. They used ships to transport the goods to and from each country. What is the value of missing angle x? Write the answer in the box provided. SMPL

9 Q5 Help the Roman soldiers work out the value of the missing angle for each triangle. a) b) c) SMPL id you know? The Roman army provided most of the labour for building roads, forts and walls!

10 Q6! a) The angle sum of a quadrilateral is 360. Work out the value of the missing angles and show all working b) c) d) Hint: Parallelogram: Opposite angles are equal in size Rectangle and square: ll angles equal 90 Trapezium: ase angles are equal Kite: One pair of opposite angles are equal (the angles between the sides of unequal length) SMPL

11 e) f) g) Y Y Z h) i) j) 88 SMPL

12 SMPL

13 irstly, thank you for your support of Mighty Minds and our resources. We endeavour to create highquality resources that are both educational and engaging, and results have shown that this approach works. To assist you in using this resource, we have compiled some brief tips and reminders below. bout this resource This Mighty Minds undamentals Lesson focusses on one subtopic from the NPLN Tests and presents this skill through a theme from the ustralian urriculum (History, Science or Geography). This lesson is also targeted at a certain skill level, to ensure that your students are completing work that is suited to them. How to use this resource Our undamentals Lessons are split into two main sections, each of which contain different types of resources. The student workbook contains The main title page; and The blank student worksheets for students to complete. The teacher resources section contains This set of instructions; The Teacher s Guide, which offers information that may be needed to teach the lesson; The Item escription, which gives a brief overview of the lesson and its aims, as well as extension ideas; The student answer sheets, which show model responses on the student worksheets to ensure that answers to the questions are clear; The teacher s answer sheets, which provide a more detailed explanation of the model responses or answers; and inally, the end of lesson marker. We suggest that you print the student workbook (the first set of pages) for the students. If students are completing this lesson for homework, you may also like to provide them with the student answer pages. eedback and contacting us SMPL We love feedback. Our policy is that if you us with suggested changes to any lesson, we will complete those changes and send you the revised lesson free of charge. Just send your feedback to resources@mightyminds.com.au and we ll get back to you as soon as we can.

14 Introduction When two lines meet, they form an angle. ngles are measured in degrees, and the name of an angle depends on its size. The types of angle students will likely come by will be: cute (less than 90 ) Right (exactly 90 ) Obtuse (greater than 90 but less than 180 ) Measuring ngles ngle sizes are measured with a protractor, which looks like this: Reflex (Greater than 180 but less than 360 ) SMPL When measuring an angle, line the point where the two lines intersect up with the middle section (yellow) so that the angle s bottom arm points to 0. Whichever number the top arm aligns with is the size of the angle. This teaching guide is continued on the next page...

15 ...This teaching guide is continued from the previous page. or example: This acute angle is roughly 44. This acute angle is exactly 90, so is a right angle. This obtuse angle is roughly 149. Triangles The sum of angles in a triangle is 180 and there are also 180 in a straight line. Often students are asked to calculate the value of a missing angle in a triangle. Missing angles are usually represented as a letter from either the alphabet (a-z) or a Greek symbol, usually theta, which looks like this:. e.g., SMPL 45 Missing angle = (right angle) = 45 This teaching guide is continued on the next page...

16 ...This teaching guide is continued from the previous page. Quadrilaterals The angle sum of a quadrilateral is 360. ny four-sided shape is a quadrilateral. Often students are given three angles of a quadrilateral and asked to calculate the missing value. Like triangles, missing angles are usually represented as a letter from either the alphabet (a-z) or a Greek symbol, usually theta, which looks like this:. e.g., Missing angle = = 95 It is important to note that different quadrilaterals have different properties regarding their angles. Squares and rectangles (all angles equal 90 ) Parallelogram (opposite angles are equal) a b b Trapezium a Y Z W 85 SMPL Kite (the angles between each pair of uneven sides are equal) a b b c This teaching guide is continued on the next page...

17 ...This teaching guide is continued from the previous page. orresponding (ngles are equal) orresponding angles can be identified by drawing an shape. In the first example, these are corresponding angles: and, and G, and, and H o-interior (ngles add up to 180 ) The angles formed can be classified into the following pairs: G o-interior angles can be identified by drawing a shape H lternate (ngles are equal) lternate angles can be identified by drawing a Z shape in the first example, these are the alternate angles: and, and Vertical (Opposite are equal) SMPL Vertical angles are directly opposite each other. In the first example, these are co-interior angles: and, and in the first example, these are the opposite angles: G and, H and, and, and

18 Item escription Please note: any activity that is not completed during class time may be set for homework or undertaken at a later date. queduct ngles, Roman Roads and Roman Triangles ctivity escription: The first activity requires the students to identify the basic classification of each angle and then apply that knowledge to solve for missing values. The second activity introduces the students to angles formed by parallel lines crossed by a transversal. Once the students are able to classify the angle they are required to solve the missing values. The third activity teaches students about the angle sum of a triangle and quadrilateral and asks the students to solve for the value of the missing angle. Purpose of ctivity: The purpose of the activity is to increase the students understanding of classifying angles and how they can use each classification to solve for a missing values. The activity also encourages the use of multiple steps of classification and problem solving of different angles. KLs: Mathematics, History s: Recognising letters, words and other symbols (α1) Interpreting the meaning of words or other symbols (α4) Interpreting the meaning of pictures/ illustrations (α5) Translating from one form to another (α7) Identifying shapes in two and three dimensions (α51) lassifying (β30) alculating with or without calculators (Ф16) Substituting in formulae (Ф19) pplying a progression of steps to achieve the required answer (Ф37) SMPL Suggested Time llocation: This lesson is designed to take approximately one hour to complete 20 minutes per activity. This Item escription is continued on the next page...

19 Item escription continued This Item escription is continued from the previous page. queduct ngles, Roman Roads and Roman Triangles Teaching Notes: Students should complete each activity as individuals. The teacher may choose to run through the examples as a class and encourage students input. To increase the difficulty of each lesson, ask the students to justify each step of their problem solving. The students will need coloured pencils and a calculator to complete the lesson. ollow Up/ lass iscussion Questions: How can students use triangles to build quadrilaterals? an students work out the angle sum of a quadrilateral from the angle sum of triangles? In what professions would the knowledge of angles be important? SMPL

20 The ancient Romans are famous for their achievements in architecture and engineering. One of ancient Rome s greatest accomplishments was the Roman aqueduct system. The aqueducts brought clean water into Rome for drinking and washing. Q1 nswer: Right angle 35 nswer: cute angle 110 nswer: Obtuse angle 220 The design of the aqueducts had to be just right to allow the water to run into the city without becoming stagnate in the channels or flowing too rapidly into the city. Therefore, the engineers had to be very skilled with geometry and couldn t afford not to understand angles! lassify each angle according to its size. nswer: Reflex angle lassification of angles: right angle is an angle of 90 n acute angle is an angle less than 90 n obtuse angle is an angle above 90 but below 180 straight angle is an angle of 180 reflex angle is an angle of more than 180 but less than 360 id you know? To lift heavy stones, the Romans used cranes powered by slaves or animals turning a treadmill! SMPL 180 nswer: Straight angle

21 Q2! ircle the different angles in different colours. cute = red, obtuse = blue, right angle = purple, reflex = orange and straight = green. R Q3 Using your knowledge of angles, determine the value of. Show all working. a) b) = = 25 c) 335 R S R S R S R R O d) O 165 = = 53 R 275 R SMPL O 125 = = 102 = = 95

22 e) 65 = = 295 g) 284 = = 76 I) = 180 ( ) = = 58 k) = 360 ( ) = = 58 f) h) j) = = 257 = = = 180 ( ) = = 106 SMPL id you know? The Roman aqueducts were over 55 feet high. This allowed control of the water flow but also made it more difficult for someone to steal or poison the water.

23 queduct ngles Question One: Students were given examples of angles and asked to classify them as either a right, acute, obtuse, reflex or straight angle. elow are the correct answers. a) b) c) d) e) nswer: Right angle (90 ) nswer: cute angle (<90 ) nswer: Obtuse angle (>90, <180) SMPL nswer: Reflex angle (>180, <360 ) 180 nswer: Straight angle (180 ) This answer guide is continued on the next page...

24 ...This answer guide is continued from the previous page. Question Two: Students were given examples of different angles and asked to circle each type in a different colour. cute angles = red, obtuse angles = blue, right angle = purple, reflex angle = orange and straight angles = green. The correct answers are shown below. Question Three: Students were given angles and asked to determine the value of using the given value and what they have already learnt about angles. The students were also required to show all working. The correct answers are shown below. a) S 65 R R 335 R S 250 R S R R O O 165 R 275 R SMPL O nswer: = = 25 This answer guide is continued on the next page...

25 ...This answer guide is continued from the previous page. b) c) d) e) f) nswer: = = 53 nswer: = = 102 nswer: = nswer: = 180 ( ) = 95 = = 58 nswer: = g) h) I) = 295 k) j) nswer: = = 76 nswer: = = 116 nswer: = 180 ( ) = = 106 nswer: = 360 ( ) nswer: = = = 257 = SMPL 52

26 Roman roads were engineered so well that some can still be seen and used today. Their roads were mostly straight and thus formed by two parallel lines. Have a look at the example below, and then try the questions. Note that the diagrams are not drawn to scale. XMPL: The straight road is crossed by a transversal line. If the line makes an angle of 50 with the side of the road, what is the value of missing angle x? Q What is the value of missing angle x? What is the value of missing angle x? Q2 x What is the value of missing angle x? x hoose the correct answer. Write the answer in the box provided.? How? When a line (a transversal) crosses two parallel lines many angles are formed. ach angle formed has a pair and each pair has a special name. 1. orresponding angles: are equal (a=b), draw an shape (in red) to recognise. 2. lternate angles: are equal (a=b), draw a Z shape (in red) to recognise. 3. o-interior angles: add up to 180 (a +b = 180 ), raw a shape (in red) To recognise. Therefore the angles in the example were co-interior angles: 50 + x = 180, x = 130 Q3 Q What is the value of missing angle x? x a b hoose the correct answer. a a b Write the answer in the box provided. SMPL x x = = 120 = = 85

27 Q5 a) and? b) and? Vertically opposite angles c) and? o-interior/supplementary angles d) and? e) omplementary angles Supplementary angles The Roman in charge of supervising the construction of roads has to know more details about angles. Use the hint box to label the given angles as either alternate, complementary, corresponding, co-interior, supplementary or vertically opposite. f) Hint: omplementary angles are angles that add up to 90 Supplementary angles are angles that add up to 180 Vertically opposite angles are angles that are opposite, share an apex and are equal The symbol means angle SMPL and? orresponding angles and? lternate angles

28 Q6! a)? b)? c)? = d) 35 = = 55 = 65 = Using your knowledge of the value of pairs of angles, solve the missing angle. Show all working. 87 = e)? = 75 f)? = 98 g)? = = h) = SMPL = G? = ? = 91 = 79

29 Roman Roads Question One: Students were given a drawing of two parallel lines crossed by a transversal as shown below. They were required to work out the value of angle x. The correct answers are below. x The angles shown are alternate angles. Therefore, they 60 are equal, x = 60 (nswer ) Question Two: Students were given a drawing of two parallel lines crossed by a transversal as shown below. They were required to work out the value of angle x. elow is the correct answer. x 60 Question Three: The angles shown are co-interior angles. Therefore, they add up to 180, x = (nswer is 120 ) Students were given a drawing of two parallel lines crossed by a transversal as shown below. They were required to work out the value of angle x. elow is the correct answer. 80 x Question our: The angles shown are corresponding angles. Therefore, they are equal, x = 80 (nswer ) SMPL Students were given a drawing of two parallel lines crossed by a transversal as shown below. They were required to work out the value of angle x. elow is the correct answer. 95 x The angles shown are co-interior angles. Therefore, they add up to 180, x = (nswer is 85 ) This answer guide is continued on the next page...

30 ...This answer guide is continued from the previous page. Question ive: Students were asked to classify each angle given as either alternate, complementary, corresponding, co-interior, supplementary or vertically opposite. The correct answers are shown below. a) and? omplementary angles b) and? Vertically opposite angles c) and? o-interior/supplementary angles d) and? Supplementary angles e) and? orresponding angles SMPL f) and? lternate angles This answer guide is continued on the next page...

31 ...This answer guide is continued from the previous page. Question Six: Students were asked to solve the missing angle. They were also required to show all working. elow are the model answers. or some there may have been alternative ways of solving the missing angle. a)? = = 55 (omplementary angles) b)? = 65 (Vertically opposite angles) c)? = (Vertically opposite angles) = = 93 (o-interior angles) d) ? = = 79 (Supplementary angles) SMPL This answer guide is continued on the next page...

32 ...This answer guide is continued from the previous page. e)? = 75 (orresponding angles) f)? = 98 (Vertically opposite angles) g)? = (Vertically opposite angles) = = = 78 (o-interior angles) h) ? = 91 (lternate angles) 91 G SMPL

33 an you calculate the interior angles of a triangle? Have a look at the example below, and then try the questions. Note that the diagrams are not drawn to scale. XMPL: wooden ladder makes an angle of 65 with a vertical stone wall. What is the value of missing angle x? Q assius noticed the blade of a sword outside his home. What is the value of missing angle x? Q2 hoose the correct answer Roman social structure can be presented as a triangle with the mperor at the apex, the plebeians and slaves at the base of the triangle and the patricians in the middle. What is the value of missing angle x (the emperor s position)? Write the answer in the box provided.? ny given triangle has an interior angle sum of 180. In other words, all three angles within any triangle add up to a total of 180. In this example, two angles within the triangle are already given, 70 and 90 (the right angle). Therefore, in order to find the value of x, the answer is simply: x = = 20 Hence, the value of x is 20. Q3 The Romans used columns in their architecture extensively. What is the value of missing angle x? Q4 How? hoose the correct answer The Romans traded goods throughout their vast empire. They used ships to transport the goods to and from each country. What is the value of missing angle x? Write the answer in the box provided. SMPL

34 Q5 Help the Roman soldiers work out the value of the missing angle for each triangle. a) = = 53 b) = = 9 c) = = SMPL id you know? The Roman army provided most of the labour for building roads, forts and walls!

35 Q6! a) The angle sum of a quadrilateral is 360. Work out the value of the missing angles and show all working = = 105 b) = 90 (ll angles of a square equal 90 ) c) d) 75 = 90 (ll angles of a rectangle equal 90 ) = Hint: Parallelogram: Opposite angles are equal in size Rectangle and square: ll angles equal 90 Trapezium: ase angles are equal Kite: One pair of opposite angles are equal (the angles between the sides of unequal length) SMPL = 110 (ase angles are equal)

36 e) = = = 82 f) g) Y 22 Y = Y = 124 = Y Z = = 27 Y = h) i) j) = = = = = = 43 SMPL = = = 172 Y = 49 Z = Z = 52

37 Roman Triangles Question One: Students were given a triangle with two angles given and one missing. Students were then required to choose the correct answer for the missing value. The correct answer is shown below. Question Two: = = 62 Students were given a triangle with two angles given and one missing. Students were then required to work out the missing value. The correct answer is shown below. Question Three: = = 108 Students were given a triangle with two angles given and one missing. Students were then required to choose the correct answer for the missing value. The correct answer is shown below. Question our: = = 105 SMPL Students were given a triangle with two angles given and one missing. Students were then required to work out the missing value. The correct answer is shown below. 66 = = 24 This answer guide is continued on the next page...

38 ...This answer guide is continued from the previous page. Question ive: Students were asked to find the missing angle,. To do this they had to understand that the angle sum of a triangle is 180. They were asked to show their working. Model answers are shown below. a) b) c) = (right angle) = 53 = = 9 SMPL = (alternate angles are equal) = 73 This answer guide is continued on the next page...

39 ...This answer guide is continued from the previous page. Question Six: Students were required to work out the angle sum of the given quadrilaterals. To complete this they were given the angle properties of different types of quadrilaterals. The students were also asked to show all working. Model answers are shown below. a) b) c) d) = = 105 (ngle sum of a quadrilateral) = 90 ( all angles of a square equals 90 ) = 90 (all angles of a rectangle equal 90 ) SMPL = (ase angles are equal) =110 (ngle sum of a quadrilateral) This answer guide is continued on the next page...

40 ...This answer guide is continued from the previous page. e) f) g) h) Y 22 Y Z SMPL = (angles formed by the two uneven sides are equal) = 82 Y = (angle sum of a triangle) Y = 124 = 124 ( angles formed by the two uneven sides are equal) = (base angles of a trapezium are equal) = 27 Y = (angle sum of a triangle) Y = 49 Z = (angle sum of a triangle) Z = = (angle sum of a quadrilateral) = 24 This answer guide is continued on the next page...

41 ...This answer guide is continued from the previous page. i) j) SMPL = (angle sum of a triangle) = = 43 = (right angle = 90 ) = = 172

42 SMPL