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Computer Number ___ Names _____________________ Date _____________________ Activity 1a 1. Geometer’s Sketchpad has a set of six tools at the left edge. Choose the Straightedge Tool: Click in two places on the screen and Geometer’s Sketchpad will produce a line segment connecting two points where you clicked. The segment will be highlighted. 2. Click on one of the endpoints of the segment and somewhere else not on the segment. Geometer’s Sketchpad will produce another line segment connecting the two points where you clicked. The new segment will be highlighted and the old one will no longer be highlighted. 3. Choose the Text Tool. Click on one of the two endpoints that do not connect the two segments. Geometer’s Sketchpad will mark that point A. Click on the endpoint that connects the two segments. Geometer’s Sketchpad will mark that point B. Click on the remaining endpoint. Geometer’s Sketchpad will mark that point C. 4. Choose the Selection Arrow Tool: Click on the segment that is not highlighted. Now both segments will be highlighted. This shows that both are selected. Click somewhere not on the segments. Now nothing will be highlighted showing that nothing is selected. Click on the three points A B and C (in that order). They will all be highlighted. Computer Number ___ Names _____________________ Date _____________________ 5. At the top of the Geometer’s Sketchpad window are nine menus. Click on the Construct menu and choose Angle Bisector. Click and drag point A. Observe what happens. Question 1a-1 What does the Angle Bisector construction do? ___________________________________ ___________________________________ ___________________________________ 6. The Angle Bisector construction only works if three points are selected and the second one is the vertex of the angle. In the Construct menu the constructions that are printed in grey require other objects to be selected. Select one segment. Observe which constructions listed below are possible. Try them to see what they do. Select two points. Observe which constructions listed below are possible. Try them to see what they do. Select a point and a segment. Observe which constructions listed below are possible. Try them to see what they do. For each construction, record what objects must be selected and what it does: Construction Midpoint Parallel Line Perpendicular Line Circle by Center and Point Circle by Center and Radius What objects must be selected What it does Computer Number ___ Names _____________________ Date _____________________ Activity 1b 1. Choose New Sketch from the File Menu. Notice that the Straightedge Tool has a small triangle in one corner: . This means you can click and hold it to reveal tools for drawing rays and lines. Choose the Line Tool (with two arrows). Click in two places and Geometer’s Sketchpad will draw a line through those two points. 2. Use the Point Tool to create a point that is not on the line. Select the line and the point and use the Parallel Line construction to construct a new line parallel to the first line and passing through the point. Use the Line Tool again to create a new line that intersects the two parallel lines. 3. Choose the Point Tool and create points in the positions indicated in the diagram above. (Notice that Geometer’s Sketchpad indicates when a point will be created on another object by changing the colour of the object). Choose the Text Tool and click on the points in the order shown above, to label them A, B, C, etc. 4. Select Points A, B and C (in that order). Click on the Measure menu and choose Angle. Geometer’s Sketchpad will display the measure of angle ABC. Measure the other angles and fill in the first row of the table below. Click and drag a point and observe how the angle measures change. Fill in the second row. Angle ABC ABD CBE DBE BEF BEH FEG HEG Measure before dragging Measure after dragging Computer Number ___ Names _____________________ Date _____________________ Questions 1b-1 Which angles have the same measure as angle ABC?_____________________ ______________________________________________________________________ 1b-2 The relationship between angles ABD and BEH is sometimes described by saying it makes a letter F. What other pairs of angles make an F? (You may need to imagine an F upside down or in a mirror) ______________________________________________________________________ 1b-3 The relationship between angles DBE and BEF is sometimes described by saying it makes a letter Z. What other pairs of angles make a Z? (You may need to imagine a Z upside down or in a mirror) ______________________________________________________________________ 1b-4 The relationship between angles ABC and DBE is sometimes described by saying it makes a letter X. What other pairs of angles make an X? ______________________________________________________________________ 1b-5 Which angles’ measures add up to 180°? ______________________________ ______________________________________________________________________ 1b-6 How would you describe the relationship between the angles whose measures add up to 180°? ______________________________________________________________________ ______________________________________________________________________ Computer Number ___ Names _____________________ Date _____________________ Activity 2a 1. Open the file Six_parameters.gsp. You will see six measurements, three lengths (5 cm, 6 cm and 7 cm) and three angle measure (50°, 60° and 70°). These measurements are called parameters. Double click one of the parameters. A window will open in which you can change the value of the parameter by typing a new number and clicking OK. If the parameters are not 5 cm, 6 cm and 7 cm and 50°, 60° and 70°, change the ones that are wrong. 2. The Custom Tool includes new tools that have been added to Geometer’s Sketchpad. This document includes two new tools: Angle from Parameter and Segment from Parameter. 3. Choose Segment from Parameter. Click on a blank area to create a new point. This point will be one endpoint of the segment. Click on a parameter that gives a length. Geometer’s Sketchpad will create a segment with that length. Drag the endpoints of the segment. Notice that the length always stays the same. Choose the Selection Arrow Tool and double click the parameter you used for the length of the segment. Change the value of the parameter. Observe what happens to the segment. Change the value of the parameter back to its original value. 4. Choose the Custom Tool Segment from Parameter again. Click on one endpoint of the segment you created. Then click on a different length parameter than you used the first time. Geometer’s Sketchpad will create a new segment connected to the first one. Create a third segment connected to the free endpoint of one of the segments. Questions 2a-1 Can you drag the three segments so they form a triangle? _________________ 2a-2 If so, what are the angle measures of the triangle? _______________________ 2a-3 If not, why can’t you form a triangle? _________________________________ ______________________________________________________________________ Computer Number ___ Names _____________________ Date _____________________ 5. If you could not create a triangle with the three given length parameters, change the parameters so you can form a triangle. Record your new parameters here: ____________ Drag the three segments and try to form two different triangles with sides of the same three lengths. Questions 2a-4 Can you drag the three segments so they form two different triangles? _______ 2a-5 If so, what are the angle measures of the two triangles? ____________________ 2a-6 If not, why can’t you form two different triangles? ________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 2a-7 Can you always form at least one triangle with sides measuring three given lengths? ________________________________________________________________________ 2a-8 If so, can you explain why this is possible? _______________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 2a-9 If not, can you explain when this is impossible? ___________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Computer Number ___ Names _____________________ Date _____________________ Activity 2b 1. If you have a file open, choose Save As… from the File menu. Save your file, calling it “Activity 2a”. Open the file Six_parameters.gsp again. You will see the six parameters. If the angle measure parameters are not 50°, 60° and 70°, change the ones that are wrong. 2. Choose the Custom Tool Angle from Parameter. Click on a blank area to create a new point. This point will be the vertex of the angle. Click somewhere else to create another point. This point will be on one arm of the angle. Click on a parameter that gives an angle measure. Geometer’s Sketchpad will create a angle with that angle measure. Drag the points of the angle. Notice that the angle measure always stays the same. Change the value of the parameter you used for the measure of the angle. Observe what happens to the angle. Change the value of the parameter back to its original value. 4. Choose the Custom Tool Angle from Parameter again. Click on the newest point of the angle you created. Then click on the vertex. The click on a different angle measure parameter than you used the first time. Geometer’s Sketchpad will create a new angle connected to the first one. Create a third angle connected to the new angle in the same way. Questions 2b-1 Can you drag the three angles so they form a triangle? ____________________ 2b-2 If so, what are the lengths of the sides of the triangle? _____________________ 2b-3 If not, why can’t you form a triangle? ___________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Computer Number ___ Names _____________________ Date _____________________ 5. If you could not create a triangle with the three given angle measure parameters, change the parameters so you can form a triangle. Record your new parameters here: ___________________________________________________________________ Drag the three angles and try to form two different triangles with angle of the same three measures. Questions 2b-4 Can you drag the three angles so they form two different triangles? _________ 2b-5 If so, what are the side lengths of the two triangles? _______________________ 2b-6 If not, why can’t you form two different triangles? ________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 2b-7 Can you always form at least one triangle with angles of three given measures? ________________________________________________________________________ 2b-8 If so, can you explain why this is possible? _______________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 2b-9 If not, can you explain when this is impossible? ___________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Computer Number ___ Names _____________________ Date _____________________ Activity 2c 1. Choose Save As… from the File menu. Save your file, calling it “Activity 2b”. Open the file Six_parameters.gsp again. You will see the six parameters. If the angle measure parameters are not 50°, 60° and 70°, change the ones that are wrong. 2. In this activity you are going to explore whether it is possible to construct many triangles, a unique triangle or no triangles at all given two angle measures and one side length. There are two cases: the given side can be between the two angles, or it can be one of the two sides not between the two angles. The abbreviations ASA and AAS stand for these two possibilities. ASA AAS 3. Choose the Custom Tool Segment from Parameter. Create a segment using one of the three length parameters. Choose the Custom Tool Angle from Parameter. Create an angle by clicking on the two endpoints of the segment and one of the angle measure parameters (Recall that the first point you click on with the Angle from Parameter tool is the vertex of the angle). Create another angle by clicking on the two endpoints of the segment in the opposite order and a different angle measure parameter. You will notice that the angle goes the wrong way to create a triangle. Instead it makes a Z shape. This is because the Angle from Parameter tool always makes the angle going counter clockwise from the segment. To get the angle to go the direction you want, change the parameter to its negative. Before: 60° angle After: -60° angle 4. Drag the points and try to make a different triangle with the same two angle measures and side length. Computer Number ___ Names _____________________ Date _____________________ Question 2c-1 Based on your investigations do you think it is possible to construct many triangles, a unique triangle or no triangles at all, given two angle measures and one side length, when the given side is between the two angles (ASA)? ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 5. Choose the Custom Tool Segment from Parameter. Create a new segment using one of the three length parameters. Choose the Custom Tool Angle from Parameter. Create an angle by clicking on the two endpoints of the segment and one of the angle measure parameters. Create another angle by clicking on the two endpoints of the new segment created by the Angle from Parameter tool. Click on the free endpoint first so that the angle has that point as its vertex. 6. Drag the points and try to make a different triangle with the same two angle measures and side length. Question 2c-2 Based on your investigations do you think it is possible to construct many triangles, a unique triangle or no triangles at all, given two angle measures and one side length, when the given side is not between the two angles (AAS)? ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Computer Number ___ Names _____________________ Date _____________________ Activity 2d 1. Choose Save As… from the File menu. Save your file, calling it “Activity 2c”. Open the file Six_parameters.gsp again. You will see the six parameters. If the angle measure parameters are not 50°, 60° and 70°, change the ones that are wrong. 2. In this activity you are going to explore whether it is possible to construct many triangles, a unique triangle or no triangles at all given two side lengths and one angle measure. There are two cases: the given angle can be between the two sides, or it can be one of the two angles not between the two sides. The abbreviations SAS and SSA stand for these two possibilities. Given side 1 Given ang le Given side 1 Given ang le Given side 2 Given side 2 SAS SSA 3. Choose the Custom Tool Segment from Parameter. Create a segment using one of the three length parameters. Choose the Custom Tool Angle from Parameter. Create an angle by clicking on the two endpoints of the segment and one of the angle measure parameters. You now have an angle with one arm of a fixed length and the other one of variable length. To create the SAS configuration it is necessary to replace the variable arm with one of fixed length. To do this, select the vertex of the angle and a length parameter. Choose the Circle by Center+Radius construction from the construct menu. Computer Number ___ Names _____________________ Date _____________________ Geometer’s Sketchpad will create a circle with the given radius. Create a point where the circle crosses the variable length arm of the angle (If the variable length arm is not long enough to intersect with the circle, drag it to make it longer). Create a segment joining the intersection point with the vertex of the angle. You won’t be able to see it because it overlaps with the variable length arm. Select the circle and the variable length arm of the angle. Choose Hide Path Objects from the Display menu. Geometer’s Sketchpad will hide the circle and the variable length arm of the angle, leaving an angle with two fixed length arms. 4. Drag the points and try to make a different triangle with the same two side lengths and angle measure. Question 2d-1 Based on your investigations do you think it is possible to construct many triangles, a unique triangle or no triangles at all, given two side lengths and one angle measure, when the given angle is between the two sides (SAS)? ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Computer Number ___ Names _____________________ Date _____________________ 5. Summarise your findings so far in this table: Case No triangle A unique triangle possible possible SSS (Three side lengths given) AAA (Three angle measures given) ASA AAS SAS Many triangles possible Question 2d-2 Based on your findings so far, what would you predict for the SSA case (two side lengths and one angle measure given, with the given angle not between the two sides): many triangles, a unique triangle or no triangles at all? Why? ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 6. Use the Custom Tool Segment from Parameter to create a segment 5 cm long. Use Angle from Parameter to create a 50° angle at one end of the segment. Use Segment from Parameter to create a segment 4 cm long at the other end of the segment. Drag the points and try to make different triangles with the same two side lengths and angle measure. Question 2d-3 Based on your investigations do you think it is possible to construct many triangles, a unique triangle or no triangles at all, given two side lengths and one angle measure, when the given angle is not between the two sides (SSA)? ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Computer Number ___ Names _____________________ Date _____________________ Activity 3a For this activity you will need to recall these definitions: A quadrilateral is a polygon with four sides. A square is a quadrilateral with four sides of equal length and four 90° angles. A rectangle is a quadrilateral with four 90° angles. A rhombus a quadrilateral with four sides of equal length. A parallelogram is a quadrilateral with two pairs of parallel sides. 1. To construct a square you can use information from the definition. Begin by constructing a segment. To create 90° angles at each endpoint use the Perpendicular Line construction. Select the segment and one endpoint and choose Perpendicular Line from the Construct menu. Do the same at the other endpoint of the segment. To create sides of length equal to the first segment use the Circle by Center+Radius construction. Select the segment and one endpoint and choose Circle by Center+Radius from the Construct menu. Then create a point where the circle and the perpendicular line intersect. Do the same at the other endpoint of the segment. Hide the circles and lines and connect the points with segments. Drag the square to make sure it remains a square. (This way of checking a construction is called the “Drag Test”.) Computer Number ___ Names _____________________ Date _____________________ 2. Construct the two diagonals of the square by making segments joining the opposite corners. Drag the square and observe the diagonals. Question 3a-1 What have you observed about the diagonals of squares? _______________________ ________________________________________________________________________ ________________________________________________________________________ 3. You have probably observed many things about the diagonals, including that they are both the same length, that they meet at a 90° angle, and that the point where they meet is the midpoint of both diagonals. Dragging the square, or perhaps making some measurements, is probably enough to convince you that these three properties are true for the diagonals of all squares. But knowing that something is true is not the same as knowing why something is true. Read over this explanation of the three properties of diagonals of squares: Diagonal AC is the same length as diagonal BD because they are corresponding parts of congruent triangles. Triangle ABC is congruent to triangle BAD by side-angle-side, because segments AB and BA have the same length (they are the same segment), segments ABC and AD have the same length (all the sides of a square have equal length), and angle ABC has the same measure as angle BAD (all the angles of a square measure 90°). Angle AEB measures 90°. This is because angles BAC and DAC have the same measure (they are corresponding parts of congruent triangles, see above), and together they make up angle BAD which measures 90°; therefore they must each measure 45°. Similarly all the angles made by the diagonals with the sides of the square must measure 45°. So in triangle ABE, angles BAE and ABE measure 45°. As the angles of a triangle add up to 180°, the third angle, AEB, must measure 90°. A B A B D C D C A B A B E D E C D C Computer Number ___ Diagonals AC and BD meet at their midpoints. This is because triangles ABE, BCE, CDE, and DAE are all congruent by angle-side-angle (we know from above that all the angles made by the diagonals with the sides of the square measure 45°, and all the sides of a square have the same length). Because they are corresponding parts of congruent triangles, AE must be the same length as CE, so E is the midpoint of AC, and BE must be the same length as DE, so E is the midpoint of BD. Names _____________________ Date _____________________ A B A E D B E C D C Record here any questions or comments you have about this explanation: ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 4. You now know that: IF a quadrilateral is a square THEN the diagonals are of equal length, meet at a 90° angle and meet at their midpoints. But it is not always possible to turn such a statement around the other way. It is true that: IF a person lives in Antigonish THEN that person lives in Nova Scotia, but it is not true that: IF a person lives in Nova Scotia THEN that person lives in Antigonish. Now you will investigate whether: IF a quadrilateral has diagonals that are of equal length, meet at a 90° angle and meet at their midpoints THEN the quadrilateral is a square. To do this, first construct two segments that are of equal length, meet at a 90° angle and meet at their midpoints. It is easiest to begin with the 90° angle and the midpoint. Create a segment. Construct its midpoint. Use the Perpendicular Line construction to construct a perpendicular line through the midpoint. Computer Number ___ Names _____________________ Date _____________________ Now you need to create a segment on this line that is the same length as the original segment and which has the same midpoint. Construct a circle with the original segment as its diameter using the Circle by Center+Radius construction. Where the circle intersects with the perpendicular line make two points, which are the endpoints of the second diagonal. Hide the circle and the perpendicular line. Join the four endpoints of the two diagonals to make a quadrilateral. Drag your figure and observe the quadrilateral. Does it seem to be a square? Provide an explanation for why IF a quadrilateral has diagonals that are of equal length, meet at a 90° angle and meet at their midpoints THEN the quadrilateral is a square. Use the back of the page if necessary. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Computer Number ___ Names _____________________ Date _____________________ Activity 3b 1. To describe a square one might want to include everything one knows about squares: They have four equal sides, four 90° angles, the diagonals are the same length, and meet at their midpoints at a 90° angle, etc. But for a mathematical definition, it is best to include as few properties as possible. As you have found out above if you know that a quadrilateral has four equal sides and four 90° angles then the diagonal properties must also be present. There is not need to state them in the definition. In fact, it could be argued that the definition of a square given above “A square is a quadrilateral with four sides of equal length and four 90° angles” includes too much information. As you know that all quadrilaterals have four angles that add up to 360°, all you need to say is that “A square is a quadrilateral with four sides of equal length and four equal angles.” You can then deduce that the angles must measure 90°. In Activity 3a you found that if a quadrilateral has diagonals that are of equal length, meet at a 90° angle and meet at their midpoints then the quadrilateral is a square. It is possible that you don’t need to know all three diagonal properties to establish that a quadrilateral is a square. In the following you will investigate whether a quadrilateral with only two of the diagonal properties is a square. There are three cases to explore: diagonals that are of equal length and meet at their midpoints diagonals that meet at a 90° angle and meet at their midpoints diagonals that are of equal length and meet at a 90° angle 2. Construct two diagonals that are of equal length and meet at their midpoints. To do this, create a segment, construct its midpoint, and construct a circle centered on the midpoint with the segment as a diameter. Any other diameter of the circle will be the same length as the original segment. Hide any construction elements you don’t need anymore. Join the four endpoints of the two diagonals to make a quadrilateral. Drag your figure and observe the quadrilateral. Does it seem to be a square? Record your observations here: ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Computer Number ___ Names _____________________ Date _____________________ 3. Construct two diagonals that meet at a 90° angle and meet at their midpoints. To do this, create a segment, construct its midpoint, and construct a perpendicular line through the midpoint. Create a point on the perpendicular line and a circle with its center at the midpoint and passing through the new point. Create a second point where the circle also meets the perpendicular line. Connect those points with a segment, which is the second diagonal you want. Hide any construction elements you don’t need anymore. Join the four endpoints of the two diagonals to make a quadrilateral. Drag your figure and observe the quadrilateral. Does it seem to be a square? Record your observations here: ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 4. Construct two diagonals that are of equal length and meet at a 90° angle. To do this, create a segment, construct its midpoint, and construct a second point anywhere on the segment. Construct a perpendicular line through the second point. Create a new point on the perpendicular line. Measure the distance from the midpoint to one endpoint. Create a circle with its center at new point on the perpendicular line and using the distance you measured as the radius. Create two points where the circle meets the perpendicular line. Connect those points with a segment, which is the second diagonal you want. Hide any construction elements you don’t need anymore. Join the four endpoints of the two diagonals to make a quadrilateral. Drag your figure and observe the quadrilateral. Does it seem to be a square? Record your observations here: ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Computer Number ___ Names _____________________ Date _____________________ ________________________________________________________________________ 5. Construct a parallelogram, by beginning with two segments joined at one endpoint, and constructing lines parallel to each segment through the free endpoint of the other segment. What properties do the diagonals of a parallelogram have? ________________________________________________________________________ ________________________________________________________________________ Construct two diagonals with those properties. Is the quadrilateral formed by the four endpoints of the diagonals necessarily a parallelogram. Why? ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Computer Number ___ Names _____________________ Date _____________________ Activity 4a Create a triangle. Using the Midpoint and Perpendicular Line constructions, construct the perpendicular bisectors of two sides of the triangle. Create a point at the intersection of the two perpendicular bisectors. Measure the distance from the intersection point to the three points of the triangle. Drag the triangle’s points and observe what happens to the distances. Construct a circle with its center at the intersection point of the perpendicular bisectors and passing through one point of the circle. Drag the triangle’s points and observe what happens to the circle. Record your observations here: ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Explain the properties of the distances and the circle that you observed by referring to congruent triangles (Hint: you may want to connect the intersection point to the points of the triangle with segments.) ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Computer Number ___ Names _____________________ Date _____________________ Activity 4b Create a triangle. Using the Angle Bisector construction, construct the angle bisectors of two angles of the triangle. Create a point at the intersection of the two angle bisectors. Measure the distance from the intersection point to the three sides of the triangle. Drag the triangle’s points and observe what happens to the distances. Construct a circle with its center at the intersection point of the angle bisectors and with its radius equal to one of the distances you measured. Drag the triangle’s points and observe what happens to the circle. Record your observations here: ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ Explain the properties of the distances and the circle that you observed by referring to congruent triangles (Hint: you may want to use segments to connect the intersection point to the points on the sides where the circle touches.) ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________