G73 Measuring and Drawing Angles and Triangles Pages


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1 G73 Measuring and Drawing Angles and Triangles Pages Curriculum Expectations Ontario: 5m51, 5m52, 5m54, 6m48, 6m49, 7m3, 7m4, 7m46 WNCP: 6SS1, review, [T, R, V] Vocabulary angle vertex arms acute obtuse Goals Students will measure and draw angles. PRIOR KNOWLEDGE REQUIRED Knows what an angle is Can name an angle and identify a named angle Materials dice geoboards and elastics grid paper Dynamic geometry software (optional) Review the concept of an angle s size. Draw two angles: ASK: Which angle is smaller? Which corner is sharper? The diagram on the left is larger, but its corner is sharper, and mathematicians say that this angle is smaller. The distance between the ends of the arms in both diagrams is the same, but this does not matter; angles are made of rays and these can be extended. What matters is the sharpness. The sharper the corner on the outside of the angle, the narrower the space between the angle s arms. Explain that the size of an angle is the amount of rotation between the angle s arms. The smallest angle is closed; both arms are together. Draw the following picture to illustrate what you mean by smaller and larger angles. Smaller Larger You can show how much an angle s arm rotates with a piece of chalk. Draw a line on the board then rest the chalk along the line s length. Fix the chalk to one of the line s endpoints and rotate the free end around the endpoint to any desired position. You might also illustrate what the size of an angle means by opening a book to different angles. Draw some angles and ask your students to order them from smallest to largest. A B C D E E6 Teacher s Guide for Workbook 7.1
2 Define acute and obtuse angles in relation to right angles. Obtuse angles are larger than a right angle; acute angles are smaller than a right angle. ExTRa PRaCTICE: 1. Copy the shapes onto grid paper and mark any right, acute, and obtuse angles. Which shape has one internal right angle? What did you use to check? 2. Which figures at left have a) all acute angles? b) all obtuse angles? c) some acute and some obtuse angles? ExaMPLE: Introduce protractors. On the board, draw two angles that are close to each other say, 50 and 55 without writing the measurements and in a way that makes it impossible to compare the angles visually. ask: How can you tell which angle is larger? Invite volunteers to try different strategies they suggest (such as copying one of the angles onto tracing paper and comparing the tracing to the other angle, or creating a copy of the angle by folding paper). Lead students to the idea of using a measurement tool. Explain that to measure an angle, people use a protractor. A protractor has 180 subdivisions around its curved edge. These subdivisions are called degrees ( ). Degrees are a unit of measurement, so just as we write cm or m when writing a measurement for length, it is important to write the symbol for angles. origin base line Using protractors. Show your students how to use a protractor on the board or on the overhead projector. Identify the origin (the point at which all the degree lines meet) and the base line (the line that goes through the origin and is parallel to the straight edge). When using a protractor, students must place the vertex of the angle at the origin; position the base line along one of the arms of the angle. You could draw pictures (see below) to illustrate incorrect protractor use, or demonstrate it using an overhead projector and a transparent protractor. Geometry 73 E7
3 Introduce the degree measures for right angles (90 ), acute angles (less than 90 ), and obtuse angles (between 90 and 180 ). Point out that there are two scales on a protractor because the amount of rotation can be measured clockwise or counterclockwise. Students should practise choosing the correct scale by deciding whether the angle is acute or obtuse, then saying whether the measurement should be more or less than 90. (You may want to do some examples as a class first.) Then have students practise measuring angles with protractors. Include some cases where the arms of the angles have to be extended first. Introduce angles in polygons, then have students measure the angles in several polygons. Students can draw polygons (with both obtuse and acute angles) and have partners measure the angles in the polygons. Drawing angles. Model drawing angles step by step (see Workbook p. 104 or BLM Measuring and Drawing Angles and Triangles, p E42). Emphasize the correct position of the protractor. Have students practise drawing angles. You could also use Activity 2 for that purpose. Have students practise drawing lines that intersect at a given angle. Another way to practise drawing angles is to construct triangles with given angle measures. ACTIVITIES Students can use geoboards and elastics to make right, acute, and obtuse angles. When students are comfortable doing that, they can create figures with different numbers of given angles. EXAMPLES: Sample ANSWERS: d) e) a) a triangle with 3 acute angles b) a quadrilateral with 0, 2, or 4 right angles c) a quadrilateral with 1 right angle d) a shape with 3 right angles e) a quadrilateral with 3 acute angles process Expectation Selecting tools and strategies, Technology 2. Students will need a die, a protractor, and a sheet of paper. Draw a starting line on the paper. Roll the die and draw an angle of the measure given by the die; use the starting line as your base line and draw the angle counterclockwise. Label your angle with its degree measure. For each next roll, draw an angle in the counterclockwise direction so that the base line of your angle is the arm drawn at the previous roll. The measure of the new angle is the sum of the result of the die and the measure of the angle in the previous roll. Stop when there is no room to draw an angle of the size given by the roll. For example, if the first three rolls are 4, 5, and 3, the picture will be as shown. 3. Teach students to draw and measure angles using Geometer s Sketchpad. Then ask them to try moving different points (on the arms of the angle, or its vertex) so that the size of the angle becomes, say, 50. Is it easy or hard to do? When you move the line segments, does E8 Teacher s Guide for Workbook 7.1
4 the angle change? (yes) When you move the vertex or other point on the arms, does the angle change? (yes) Show students how to draw an angle of fixed measure (using menu options). Will moving the endpoints change the size of the angle now? (no) Show students how to draw angles equal to a given angle. process Expectation Technology, Reflecting on the reasonableness of the answer 4. Have students draw polygons in Geometer s Sketchpad and measure the size of the angles and the length of the sides of these polygons. Have students check that the angle measures they obtain make sense. For example, clicking on three vertices of a quadrilateral and then using menu options to measure the angle might produce different angles, depending on the order in which the vertices were selected. Also, the software sometimes measures angles in the wrong direction, producing an answer more than 180. Extensions 1. Angles on an analogue clock. What is the angle between the hands at 12:24? 13:36? 15:48? (Draw the hands first!) To guide students to the answers, draw an analogue clock that shows 3:00 on the board. Ask your students what angle the hands create. What is the measure of that angle? If the time is 1:00, what is the measure of the angle between the hands? Do you need a protractor to tell? Have students write the angle measures for each hour from 1:00 to 6:00. Which number do they skip count by? An hour is 60 minutes and a whole circle is 360. What angle does the minute hand cover every minute? (6 ) How long does it take the hour hand to cover that many degrees? How do you know? (12 minutes, because the hour hand covers only one twelfth of the full circle in an hour, moving 12 times slower than the minute hand) If the time is 12:12, where do the hour hand and the minute hand point? What angle does each hand make with a vertical line? What is the angle between the hands? (ANSWER: The hour hand points at one minute and the angle that it makes with the vertical line is 6. The minute hand points at 12 minutes and the angle that it makes with the vertical line is 12 6 = 72. The angle between the hands is 72 6 = 64.) 2. Some scientists think that moths travel at a 30 angle to the sun when they leave home at sunrise. Note that the sun is far away, so all the rays it sends to us seem parallel. evening sun s rays moth home flower 30 morning W N S E Geometry 73 E9
5 a) What angle do the moths need to travel at to find their way back at sunset? Hint: Where is the sun in the evening? b) A moth sees the light from the candle flame and thinks it s the sun. The candle is very near to us, and the rays it sends to us go out in all directions. Where does the moth end up? Draw the moth s path E10 Teacher s Guide for Workbook 7.1
6 G74 Perpendicular Lines Pages Curriculum Expectations Ontario: 7m4, 7m5, 7m46 WNCP: 6SS1, 7SS3, [CN, V, T] Vocabulary angle perpendicular slant line right angle arms Goals Students will identify and draw perpendicular lines. PRIOR KNOWLEDGE REQUIRED Knows what a right angle is Can name an angle and identify a named angle Materials Dynamic geometry software (optional) protractors set squares Introduce perpendicular lines (lines that meet at 90 ) and show how to mark perpendicular lines with a square corner. Draw several pairs of intersecting lines on the board and have students identify the perpendicular lines. Include pairs of lines that are not horizontal and line segments that intersect in different places and at different angles (see examples below). Invite volunteers to check whether the lines are perpendicular using a corner of a page, a protractor, and/or a set square. Ask students where they see perpendicular lines or line segments also called perpendiculars in the environment (sides of windows and desks, intersections of streets, etc.). Extra practice: Which lines look like they are perpendicular? B C F K P a) b) c) J d) O G L A D H I M N R Q E A perpendicular through a point. Explain that sometimes we are interested in a line that is perpendicular to a given line, but we need an additional condition the perpendicular should pass through a given point. In each diagram below, have students identify first the lines that are perpendicular to the segment AB, then the lines that pass through point P, and finally the one line that satisfies both conditions. D K G a) b) E G B P C P F A B D C F H E H Constructing a perpendicular through a point. Model using a set square (and then a protractor) to construct a perpendicular through a point that is not on the line. Emphasize the correct position of the set square (one A Geometry 74 E11
7 side coinciding with the given line, the other touching the given point) and the protractor (the given line should pass through the origin and through the 90 mark). Have students practise the construction. Circulate among the students to ensure that they are using the tools correctly. Then invite volunteers to model the construction of a perpendicular through a point that is on the line. (Emphasize the difference in the position of the set square: the square corner is now at the point, though one arm still coincides with the given line.) Then have students practise this construction as well. Extra practice: Draw a pair of perpendicular slant lines (i.e., lines that are neither vertical nor horizontal) and a point not on the lines. Draw perpendiculars to the slant lines through the point. What quadrilateral have you constructed? (rectangle) Bonus Draw a slant line and a point not on the line. Using a protractor and a ruler or set square, draw a square that has one side on the slant line and one of the vertices at the point you drew. (ANSWER: Draw a perpendicular through the point to the given line. Measure the distance from the point to the line along the perpendicular. Then mark a point on the given line that is the same distance from the intersection as the given point. Now draw a perpendicular to the given line through this point as well. Finally, draw a perpendicular to the last line through the given point.) connection Science Why perpendiculars are important. Discuss with students why perpendiculars are important and where are they used in real life. For example, you can explain that it is easy to determine a vertical line (using gravity just hang a stone on a rope and trace the rope), but you need a right angle to make sure that the floor of a room is horizontal. ACTIVITIES 1 2 process Expectation Technology 1. Have students use Geometer s Sketchpad to: a) Draw a line. Label it m. b) Mark a point A on the line m. c) Draw another line through point A. d) Measure the angle between the two lines. e) Try to make the angle a right angle by moving the points around. Is it easy or hard to do? f) Check whether the lines stay perpendicular when you move any of the points in the picture. g) Repeat parts a) through f) with a point not on the line. Note that when point A is not on the line, the second line might be modified so that it does not intersect the line m, and the angle you measure disappears. Explain that you need a method to draw perpendicular lines that will keep them perpendicular even if the points are moved around. E12 Teacher s Guide for Workbook 7.1
8 Teach students to draw perpendicular lines using the perpendicular line command from the menu. Do these lines stay perpendicular to the given line even if points are moved around? (yes) process Expectation Technology process Expectation Selecting tools and strategies 2. Have students draw a triangle using the polygon tool in Geometer s Sketchpad. Ask them to move the points around to make it look like a right triangle. Then ask them to measure the angles of the triangle and to check whether it is indeed a right triangle. Is it easy to draw a perfect right triangle this way? (no) If you move the points around, does the triangle remain a right triangle? (no) Ask students to think about how they draw a right triangle on paper. What tools do they use and why? (a protractor or a set square) What tools could we use instead of protractors and set square in Geometer s Sketchpad? (perpendicular lines) Then have students draw a right triangle in Geometer s Sketchpad. Ask them to check that the triangle remains a right triangle even if points are moved around. Students can also add the measures of the acute angles in the right triangle they created, and check that the sum remains 90 even when the triangle is modified. Extension process Expectation Visualizing Discuss with your students whether there can be more than one line perpendicular to a given line through a given point, and whether such a perpendicular always exists. You can use the diagrams below to help students visualize the answers. P P A B A B Geometry 74 E13
9 G75 Perpendicular Bisectors Pages Curriculum Expectations Ontario: 7m1, 7m3, 7m4, 7m48 WNCP: 6SS1, 7SS3, [C, R, V, T] Vocabulary line segment midpoint right angle perpendicular bisector Goals Students will identify and draw perpendicular bisectors. PRIOR KNOWLEDGE REQUIRED Can identify and construct a perpendicular using a set square or a protractor Can identify and mark right angles Can name line segments and identify named line segments Can draw and measure with a ruler Materials paper circles or BLM Circles (p E51) Dynamic geometry software (optional) Example: Introduce the notation for equal line segments. Explain that when we want to show that line segments are equal, we add the same number of marks across each line segment. This is particularly useful for sketches, when you are not drawing everything exactly to scale. Introduce the midpoint. Model finding the midpoint of a segment using a ruler, and mark the halves of the line segment as equal, then have students practise this skill. Introduce bisectors. Explain that a bisector of a line segment is a line (or ray or line segment) that divides the line segment into two equal parts. There can be many bisectors of a line segment. Ask students to draw a line segment with several bisectors. ASK: Can a line have a bisector? What about a ray? (no, because lines and rays have no midpoints) Extra practice: Draw a scalene triangle. Choose a side and draw a bisector to that side that passes through the vertex of the triangle that is opposite that side. Repeat with the other sides. What do you notice? (ANSWER: All three bisectors, called medians, pass through the same point.) Introduce perpendicular bisectors. Of all the bisectors of a line segment only one is perpendicular to the line segment, and it is called the perpendicular bisector. The perpendicular bisector of a line segment divides the line segment into two equal parts AND intersects the line segment at right angles (90 ). Point out that there are two parts in the definition, and both must be true. ASK: How can we draw a perpendicular bisector? How is that problem E14 Teacher s Guide for Workbook 7.1
10 process Expectation Changing into a known problem similar to constructing a bisector? Constructing a perpendicular? Lead students to the idea that they should first find the midpoint of the line segment, then construct a perpendicular through that point. Have students practise drawing perpendicular bisectors using set squares and protractors. E F G D J C Present several diagrams that combine intersecting line segments, perpendicular lines, and bisectors, such as the one at left, and have students identify equal segments, then find perpendicular lines and bisectors. (Example: Find a bisector of EG. Is it a perpendicular bisector?) Ask other questions about the diagram, such as: H A B J is the midpoint of what segment? (CG) Why not AD? How do you know? (We do not have any information about the lengths of DJ and AJ. They look equal, but might have different lengths.) Name three line segments GJ is perpendicular to. Finally, have students identify all the perpendicular bisectors and the line segments they bisect. (ANSWERS: AD, JA, and JD bisect line segment CG; CJ, JG, and CG bisect line segment FH) process assessment Workbook Question 8f) [C,], 7m3 process Expectation Technology, Selecting tools and strategies Ask students to find examples of equal segments, midpoints, and perpendicular bisectors in the classroom or elsewhere, such as in letters of the alphabet, in pictures or photographs, and so on. ACTIVITIES Show students how to construct the midpoint of a line segment in Geometer s Sketchpad. Explain that there is no command that will construct a perpendicular bisector of a line segment. Challenge students to find a way to construct a perpendicular bisector of a line segment using the tools they are familiar with in Geometer s Sketchpad. (Find the midpoint of the line segment, then construct a line through the midpoint and perpendicular to the given line segment) PROMPT: Think of the way you construct a perpendicular line segment on paper. Which tools can you use? (protractor or set square and ruler) Which tool replaces a protractor or a set square in Geometer s Sketchpad? ( Perpendicular Line in the Construct menu) B A C 2. Paper folding and line segments Draw a line segment AB dark enough that you can see through the paper. Fold the paper so that A meets B. What line has your crease made? (ANSWER: a perpendicular bisector) Use a ruler and protractor to check your answer. 3. Paper folding and circles Give each student a circle (you can use BLM Circles) and ask students to draw and label a triangle on their circle. a) Fold the circle in half so that A meets B. Geometry 75 E15
11 Look at the line that the crease in your fold makes. Is it a bisector of angle C? Is it a perpendicular bisector of line segment AB? b) Fold the circle in half again, this time making A meet C. What two properties will the crease fold have? c) Repeat, making B meet C. At what point in the circle will all three perpendicular bisectors meet? Extensions 1. Given a triangle ABC, how can you use perpendicular bisectors to help you draw the circle going through the points A, B, and C? Draw an acute scalene triangle. Draw the perpendicular bisectors by hand do not cut and fold the triangle. A C B 2. Start with a paper circle. Choose a point C on the circle and draw a right angle so that its arms intersect the circle. Label the points where the arms intersect the circle A and B and draw the line segment AB. Repeat with several circles to produce different right triangles. (You can use BLM Circles for this Extension.) Fold the circle in two across the side AC so that A falls on C (creating a perpendicular bisector of AC). Mark the point where B falls on the circle. Repeat with the side BC, marking the point where A falls on the circle. What do you notice? (The image of A is the same as the image of B.) What type of special quadrilateral have you created? (a rectangle) Repeat the exercise starting with an obtuse or an acute angle C. Do the images of A and B coincide? E16 Teacher s Guide for Workbook 7.1
12 G76 Parallel Lines Pages Curriculum Expectations Ontario: 7m1, 7m2, 7m46 WNCP: 5SS5, 7SS3, [C, R, PS] Vocabulary line segment parallel right angle perpendicular Goals Students will identify and draw parallel lines. PRIOR KNOWLEDGE REQUIRED Can identify and construct a perpendicular using a set square or a protractor Can identify and mark right angles Can name a line segment and identify a named line segment Can draw and measure with a ruler Materials BLM Distance Between Parallel Lines (p E46) BLM Drawing Parallel Lines (p E44) Introduce parallel lines straight lines that never intersect, no matter how much they are extended. Show how to mark parallel lines with the same number of arrows. Have students identify parallel lines in several diagrams. Then have students identify and mark parallel sides of polygons. Include polygons that have pairs of parallel sides that are neither horizontal nor vertical. Introduce the symbol for parallel lines, label the vertices of the polygons used above, and have students state which sides are parallel using the new notation (EXAMPLE: AB CD). m process assessment Workbook Question 8 [R, C], 7m2? p n Ask students to think about where they see parallel lines. Some examples of parallel lines in the real world are a double centerline on a highway and the edges of construction beams. Determining if two lines are parallel. Have students draw a triangle on grid paper. Then ask them to draw line segments that are parallel to the sides of the triangle. For each pair of parallel lines (say, m and n) ask students to draw a perpendicular (say, p) to one of the lines (m) so that it intersects both lines. Ask students to predict what the angle between n and p is. Ask students to explain their prediction. Then have them measure the angle between the lines. Was the prediction correct? Have students check the prediction using the other two pairs of parallel lines they drew. Explain that this property If one line in a pair of parallel lines interests a third line at a right angle, the other parallel line also makes a right angle with the same line allows us to check whether two lines are parallel and to construct parallel lines. Drawing a line segment parallel to a given line segment. Have students problemsolve how to construct parallel lines using what they ve just learned about a perpendicular to parallel lines. As a prompt, you could use the Geometry 76 E17
13 process assessment [PS], 7m1 rectangle and square problems from G74 (Extra practice and Bonus, p E12 students are required to draw a rectangle or a square using a point and a pair of parallel lines). ASK: What do you know about the sides of a rectangle? How does constructing a rectangle mean that you constructed a pair of parallel lines? Model drawing the line parallel to a given line through a point using a protractor (see p 111 in the Workbook or BLM Drawing Parallel Lines), and then model doing the same thing using a set square. Have students practise drawing parallel lines using both tools. ACTIVITIES Paper folding. Draw a line dark enough so you can see it through the page. Fold the paper so that you can find a line perpendicular to AB that does not bisect AB. How can you use this crease to find a line parallel to AB? How can you use a ruler or any right angle to find a line parallel to AB? ANSWER: Fold the paper so that the perpendicular to AB falls onto itself. The crease is perpendicular to the perpendicular, so it is parallel to AB. 2. Students can investigate distances between parallel lines with BLM Distance Between Parallel Lines. Extensions 1. A plane is a flat surface. It has length and width, but no thickness. It extends forever along its length and width. Parallel lines in a plane will never meet, no matter how far they are extended in either direction. Can you find a pair of lines not in a plane that never meet and do not intersect? process assessment 7m1, [PS] 2. Draw: a) a hexagon with three parallel sides b) an octagon with four parallel sides c) a heptagon with three pairs of parallel sides d) a heptagon with two sets of three parallel sides e) a polygon with three sets of four parallel sides f) a polygon with four sets of three parallel sides Sample ANSWERS: a) b) c) E18 Teacher s Guide for Workbook 7.1
14 d) e) f) 3. Lines are parallel if they point in the same direction that s why we use arrows to show parallel lines! We can regard direction as an angle with a horizontal line. For example, if two lines are both vertical, they both make a right angle with a horizontal line, and they are parallel. The choice of a horizontal line as a benchmark is arbitrary it is just a convention; any line could be used for that purpose. Indeed, any two lines perpendicular to a third line are parallel. Students can complete the Investigation on BLM Properties of Parallel Lines (p E45) to learn what happens when parallel lines meet a third line at different angles. 4. Ask students if they can draw a parallelogram that s not a rectangle using only a ruler and a set square. SOLUTION: Step 1: Draw one side of the parallelogram. Step 2: Draw two of the parallel sides using the triangle. Step 3: Use the ruler to complete the figure. Geometry 76 E19
15 G77 Angle Relationships Pages Curriculum Expectations Ontario: 7m3, 7m7, 7m46, 7m47 WNCP: 6SS2, 7SS1, [C, R, V] Goals Students will find angles in triangles, discover the sum of the angles in a triangle, and use this sum to solve problems. PRIOR KNOWLEDGE REQUIRED Can use a protractor to measure angles Materials Vocabulary straight angle acute, obtuse, right angle adjacent angles intersecting lines linear pair BLM Sum of the Angles in a Triangle (pp E47 E48) A straight angle is formed when the arms of the angle point in exactly opposite directions and form a straight line through the vertex of the angle. Adjacent angles share an arm and a vertex. Two adjacent angles are a linear pair if, together, they form a straight angle. a b a b c a d b f Have students name all the pairs of adjacent angles in the picture at left. ASK: Which angles make a linear pair? Which two combinations of angles make a straight angle? + = 1 3 cup 1 2 cup 5 6 cup Angle measures in adjacent angles add to the measure of the large angle. Draw a line segment divided into two smaller segments. Mark the lengths of the smaller line segments. ASK: What is the length of the whole line segment? How do you know? What do you do with the lengths of the smaller line segments to obtain the length of the whole line segment? 4 cm cm 3 = 6 cm Point out that the same procedure applies to capacities, volumes, and areas. What happens with angles? Their measures are also added: the measure of the large angle at left is = 75. Sum of the angles around a point. Ask students to draw a pair of intersecting lines, measure the angles, and write the measures on the picture. Then ask them to add up the measures. What is the sum of the angles around the point? Did they all get the same answer? Show the E20 Teacher s Guide for Workbook 7.1
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