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Name: VERTICALLY OPPOSITE, ALTERNATE AND CORRESPONDING ANGLES After completion of this workbook you should be able to: know that when two lines intersect, four angles are formed and that the

intersected by a third line, the alternate angles (Z-angles) are equal; use the properties of vertically opposite angles, corresponding angles and alternate angles to calculate unknown angles; know

1 Name: VERTICALLY OPPOSITE, ALTERNATE AND CORRESPONDING ANGLES After completion of this workbook you should be able to: know that when two lines intersect, four angles are formed and that the vertically opposite angles (X-angles) are equal; know that when two parallel lines are intersected by a third line, the corresponding angles (F-angles)are equal; know that when two parallel lines are intersected by a third line, the alternate angles (Z-angles) are equal; use the properties of vertically opposite angles, corresponding angles and alternate angles to calculate unknown angles; know that the sum of the internal angles of a triangle is 180 o and correctly use this to calculate unknown angles; calculate the sum of the internal angles of a polygon. Institute for Mathematics and Science Teaching University of Stellenbosch Private Bag X1, Matieland 7602 Tel: , Fax: Copyright IMSTUS/IWWOUS 2011 under the Creative Commons License. You need the following for this workbook: ruler, pencil/pen, eraser; calculator, 1cm grid paper.

2 Vertically opposite angles 1 a Below are four diagrams of lines cutting each other. Use your protractor to measure the sizes of all the angles in each sketch. Clearly mark them on each diagram. (Remember to write the degree sign ) b What do you observe if you look at the size of each angle? c Look at the diagrams below. The angles that are marked the same are equal. These angles are directly opposite each other and are called vertically opposite angles or X-angles (to help you remember how they look). Explain why vertically opposite angles are equal. d Which angles would you call adjacent angles? e The sum of all the angles around a point is called a revolution. What is this sum? 2 Copyright IMSTUS / Kopiereg IWWOUS 2011 under the Creative Commons License.

3 Parallel lines and calculating angles You have worked with parallel lines and angles. You have learnt how to use a protractor to measure angles and do constructions. You also learnt that parallel lines never intersect. The (perpendicular) distance between parallel lines is the same everywhere. (Remember: distance is the shortest space between two points). You also know that a parallelogram is a quadrilateral in which the opposite sides are parallel. In this workbook we will be dealing with parallel lines and angles. When parallel lines are cut by another line (transversal), we can indicate angles that are the same. 2 a Look around in your classroom and record where you see parallel lines. b Give everyday examples of situations where we would find parallel lines. 3 a Draw two parallel lines such that the distance between the lines is 3cm. b Describe how you drew the two parallel lines 3 cm apart. A parallelogram is a quadrilateral in which the opposite sides are parallel. 4 a Draw a parallelogram in which an internal angle is 50 o and where one side is 4 cm long and another 5 cm long. Measure the distance(s) between the opposite sides. b Draw a parallelogram in which the distance between one pair of sides is 3cm and the distance between the other pair of opposite sides is 5 cm. (There are various solutions.) Workbook 16 VERTICALLY OPPOSITE-ALTERNATE- AND CORRESPONDING ANGLES 3

4 Corresponding angles and Alternate angles 5 Look at this drawing of part of a tiled floor. Each tile has the shape of a parallelogram. Indicate with arrows which lines are parallel on your sketch. What do you do if there are different pairs of parallel lines? Remember: Lines are parallel when the (perpendicular) distance between them is the same everywhere. a Angles marked the same are equal. You know that vertically opposite angles are equal. Indicate, using different markers, all angles that are equal. Use the same mark to denote equal angles. You may use your protractor to check. b Below we have the same tiled floor but line segments are drawn on the floor to form the letter F. Some of the F s look a bit funny. In each F-diagram, three angles are marked. What can you note about these angles in the F? 4 Copyright IMSTUS / Kopiereg IWWOUS 2011 under the Creative Commons License.

5 c Below we have the same tiled floor, but line segments are drawn on the floor to form the letter Z. Some of the Z s look a bit funny. In each Z-diagram, two angles are marked. What do you notice about these angles in the Z? First indicate with arrows which lines are parallel on your sketch. Remember: Lines are parallel when the (perpendicular) distance between them is the same everywhere. 6 In the diagram below we also drew a F-figure and Z-figure. a Are there any parallel lines in the figures above? b Are there angles that are equal in the F-figure and Z-figure? c What conclusion can you draw about the angles in the F-figure and Z-figure? Workbook 16 VERTICALLY OPPOSITE-ALTERNATE- AND CORRESPONDING ANGLES 5

6 An F-figure and a Z-figure consist of three line segments. The three line segments include angles. (Indicate which lines are parallel to each other). When two of the three line segments are parallel, we will find angles that are equal. In the F-figure, the corresponding angles are equal and in the Z- figure, the alternate angles are equal. 7 Calculate the size of the angles marked with a question mark. Write you answers in the sketch. (You may not measure them.) Use the properties of vertically opposite angles, corresponding angles and alternate angles for parallel lines to assist you. Also use the fact that the sum of the angles on a straight line = 180º. Lines that are parallel are indicated with arrows. 6 Copyright IMSTUS / Kopiereg IWWOUS 2011 under the Creative Commons License.

7 Extra exercises Example: B D The lines AB, CD and EF are parallel. x = 67º [Alternate angles; AB CD] y = 67º [Corresponding angles; CD EF] z = 67º [vertical opposite angles] A 67º x y z F C 8 a Determine the sizes of the angles marked with small letters. Give reasons for all your answers. B E t G F c d Q b A D s 147º U 57º a A P S L 60º q p C P r V Workbook 16 VERTICALLY OPPOSITE-ALTERNATE- AND CORRESPONDING ANGLES 7

8 b Determine the sizes of the angles marked with small letters. Give reasons for all your answers. A A B h g j C u m 105º n S D f k e 77º v x 75º y B S L W K P c Complete: g + j = k + e = y + n = We call these pairs of angles co-interior angles. d In the following sketches the sizes of angles are given. Indicate which lines are parallel and give reasons for your answers: B L G 58º Z 98º 122º Q 82º F 122º H A W P 8 Copyright IMSTUS / Kopiereg IWWOUS 2011 under the Creative Commons License.

9 9 Internal angles of a triangle On a clean piece of paper draw a large triangle and cut it out. Use the letters A, B and C and mark off the angles on the triangle.. Neatly tear the angles from the triangle, as shown in the first sketch. (NB: Do not cut the angles, because you might forget which angle is the angle of the triangle. Paste (as shown in third sketch) the three torn angles of the triangle next to each other. Write down what you observe about angles A, B and C. 10 Look at the drawing of part of a tiled floor. Each tile has the shape of a triangle. a In one tile, all the angles have a specific mark. Now mark all angles such that all angles that are equal have the same mark. (Hint: parallel lines...) b At each vertex (point/corner) on the tiled floor, we have six triangular tiles meeting each other. How many angles, of different sizes, can you observe at each vertex? c What do you notice about the angles of one triangle? Is that the case for all the triangles? Workbook 16 VERTICALLY OPPOSITE-ALTERNATE- AND CORRESPONDING ANGLES 9

10 11 In the triangle drawn below, a line PQ is drawn through (point) vertex C such that it is parallel to AB. In this figure we have two Z-figures. P Q a Mark all equal angles with the same mark. b Write down all that you notice. You have now discovered, via three methods, that the sum of the internal angles of a triangle = 180º. 12 Calculate the size of the angles marked with a question mark. (You may not measure them.) Write your answers in the sketch. Use the properties of vertically opposite angles, corresponding angles and alternate angles for parallel lines to assist you. Also use the fact that the sum of the angles of a triangle = 180º. 10 Copyright IMSTUS / Kopiereg IWWOUS 2011 under the Creative Commons License.

11 13 Now concentrate. Peter writes down the following: The sum of the angles of triangle ADC = 180 o The sum of the angles of triangle DBC = 180 o Therefore the sum of the angles of triangle ABC = 360 o What is wrong with his reasoning? Why? 14 The South African flag is drawn below. NOT TO SCALE. Using the correct colours, colour in the flag. What is the meaning of each colour? The sizes of two angles are indicated on the sketch. Calculate all the other angles. You are not allowed to measure the angles! Write your answers in the drawing. Hint: The flag is a special parallelogram namely a rectangle. The triangles in the sketch are isoceles. 146 º 67 º Workbook 16 VERTICALLY OPPOSITE-ALTERNATE- AND CORRESPONDING ANGLES 11

12 Internal angles of a polygon 15 For each of the polygons below, accurately measure each angle in each sketch. For each polygon, calculate the sum of the internal angles. What can you deduce about the sum of the internal angles of each polygon? fig 1 fig 2 fig 3 fig 4 16 In the adjacent sketch, a quadrilateral, a pentagon and a hexagon are drawn on the lefthand side. On the right-hand side, the same polygons are drawn but diagonals are drawn in from one vertex. 12 Copyright IMSTUS / Kopiereg IWWOUS 2011 under the Creative Commons License.

13 a The diagonals divide the polygons into a number of triangles. Complete the table: number of sides of polygon number of triangles in polygon Sum of internal angles of polygon b On separate pages, neatly draw the following: A triangle with three equal sides and three equal angles. A quadrilateral with four equal sides and four equal angles. A pentagon with five equal sides and five equal angles. A hexagon with six equal sides and six equal angles. You have to show your planning; indicate the sizes of the angles and write down your conclusions. 17 The sum of the internal angles of a polygon is 1800º. How many vertices does this polygon have? Extra cheese 18 On the same piece of paper, draw the adjacent triangle twice. Cut out each of the triangles. You can form a quadrilateral by placing these two triangles next to each other. a How many different quadrilaterals can you form? Draw these quadrilaterals. b Which of these quadrilaterals are parallelograms? c What are the sizes of each of the internal angles of this parallelogram? 19 Calculate (not measure!) the size of the nine unknown angles in the adjacent parallelogram. 20 a One of the angles of a parallelogram is 30 o. What are the sizes of the other angles? b Draw a parallelogram in which the angle between the diagonals is 90 o. There are many possibilities. What property is common to all these parallelograms? 21 State if the following are true or false: a There exists a parallelogram in which all the angles are equal. b There exists a parallelogram in which all the angles are acute. c There exists a parallelogram with angles of 30 o and angles of 120 o. Workbook 16 VERTICALLY OPPOSITE-ALTERNATE- AND CORRESPONDING ANGLES 13

22 Use the system of axes below and do the following: a Plot the points: A(-1 ; -1), B(3 ; -1) and C (2 ; 1 ). b Draw parallelogram ABCD. c What are the coordinates of point D?

14 22 Use the system of axes below and do the following: a Plot the points: A(-1 ; -1), B(3 ; -1) and C (2 ; 1 ). b Draw parallelogram ABCD. c What are the coordinates of point D? 23 Use the system of axes below and do the following: a Plot the points: A(-4 ; -2), G (3 ; 0 ) and R (0 ; 2 ). b Draw parallelograms such that A, G and R form 3 of the 4 vertices. c What is noticeable about the total picture on your system of axes? 14 Copyright IMSTUS / Kopiereg IWWOUS 2011 under the Creative Commons License.

15 24 A number of triangles have been drawn below. Calculate (without using a protractor) the sizes of the angles indicated with a question mark (?). Give reasons for your answers. Workbook 16 VERTICALLY OPPOSITE-ALTERNATE- AND CORRESPONDING ANGLES 15

Boardworks Ltd KS3 Mathematics. S1 Lines and Angles

Boardworks Ltd KS3 Mathematics. S1 Lines and Angles 1 KS3 Mathematics S1 Lines and Angles 2 Contents S1 Lines and angles S1.1 Labelling lines and angles S1.2 Parallel and perpendicular lines S1.3 Calculating angles S1.4 Angles in polygons 3 Lines In Mathematics,