H Angles and Shapes

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					Q1.         In the diagram AB and CD are parallel.

                                                                                                                  Not drawn accurately




      (a)    Write down the value of x.

                                                Answer .......................................................... degrees
                                                                                                                                                      (1)

      (b)    Work out the value of y.

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                                                Answer .......................................................... degrees
                                                                                                                                                      (2)
                                                                                                                                         (Total 3 marks)




Q2.         The diagram shows a regular pentagon and a regular decagon joined at side XY.

                                                                                                      Not drawn accurately




                                                                                                                                                            Page 1 of 25
      Show that the points A, B and C lie on a straight line.

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                                                                                                                                           (Total 5 marks)




Q3.      XYZ is an isosceles triangle in which XZ = XY
      M and N are points on XZ and XY such that angle MYZ = angle NZY.




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Prove that triangles YMZ and ZNY are congruent.

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                                                                                                                                     (Total 4 marks)




                                                                                                                                                       Page 3 of 25
Q4.         Three circles fit inside a rectangle as shown.
      Two of the circles are identical and the third is larger.
      The circles have radii 9 cm, 9 cm and 25 cm.




                                                                                            Not drawn accurately




      Calculate the length, l, of the rectangle.

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                                                                        Answer ............................................ cm
                                                                                                                                           (Total 6 marks)




                                                                                                                                                             Page 4 of 25
Q5.         ABCDEF is a regular hexagon.
      AFGH and AJKB are squares.




                                                                                                        Not drawn accurately




      Show that triangle AHJ is equilateral.

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                                                                                                                                           (Total 4 marks)




                                                                                                                                                             Page 5 of 25
Q6.      Airport runways have a two-digit number painted on them.
      These numbers are used to work out the direction of the runway.
      To work out the three-figure bearing, multiply the runway number by 10.
      Here is a diagram of a runway on a three-figure bearing of 280° and a runway on a three-figure
      bearing of 040°.




      (a)   (i)     Write down the three-figure bearing for a runway pointing due South.

                                                  Answer .............................................. °
                                                                                                             (1)

            (ii)    Write down the runway number for a runway pointing due South.

                                                  Answer .................................................
                                                                                                             (1)


            (iii)   A runway has a three-figure bearing of 060°.

                    Write down the runway number.

                                                  Answer .................................................
                                                                                                             (1)




                                                                                                                   Page 6 of 25
(b)   A runway is being painted.




      By measuring the three-figure bearing, work out the runway number.

      .........................................................................................................................

                                                               Answer ..................................................
                                                                                                                                               (2)


(c)   Runways are used in both directions.
      This means that they have two different runway numbers, one at each end.
      A runway has the number 30 at one end.

      What runway number is at the other end?

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                                                               Answer ..................................................
                                                                                                                                               (3)
                                                                                                                                  (Total 8 marks)




                                                                                                                                                     Page 7 of 25
Q7.        A sign maker designs a letter L.
      All arcs are quarter circles of radius 2 cm.




                                                                                                        Not drawn accurately




      Calculate the area of the L.

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                                                                  Answer ................................................. cm 2
                                                                                                                                           (Total 4 marks)




                                                                                                                                                             Page 8 of 25
Q8.         The diagrams show a trapezium and a parallelogram.




                                                                                                      Not drawn accurately

      (a)    Use the trapezium to explain why                              2x + y = 180

             .........................................................................................................................

             .........................................................................................................................
                                                                                                                                                      (1)


      (b)    The parallelogram can be used to form another equation connecting x and y.

              Tick a box to show the correct equation.


                                                3x + y = 130                                                      3x + y = 230



                                                3x = y – 50                                                       3x + y = 410

                                                                                                                                                      (1)


      (c)    Hence, or otherwise, work out the values of x and y.

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                                                                      Answer x = ................ , y = …...............
                                                                                                                                                      (3)
                                                                                                                                         (Total 5 marks)




                                                                                                                                                            Page 9 of 25
M1.         (a)     41
                                                                B1

      (b)    180 – 67
                         71 + their 41 oe 360 – 41 – 67 – 139
                                                                M1

              113
                                                                A1
                                                                     [3]




                                                                       Page 10 of 25
M2.      360 ÷ 10 or 360 ÷ 5
                      or 36 or 72 or 144 or 108
                      NB Angles may be marked on diagram
                                                           M1

      144 and 108
                    or 36 and 72
                                                           A1




                                                                Page 11 of 25
∠BXC = 360 – (144 + 108)
               or ∠BXD = 36 + 72 or 108
               (X is point where decagon and pentagon meet between B and C)
                                                                              M1

∠XBC = ∠XCB = (180 – 108) ÷ 2
            or ∠XBC = 36
                                                                              M1




                                                                                   Page 12 of 25
∠ABX + ∠XBC = 144 + 36 ( = 180)
             oe
             eg, ∠CBX calculated from ΔBXC equals exterior angle
                 of decagon
                                                                   A1
                                                                        [5]




                                                                          Page 13 of 25
M3.       YZ = ZY
                                                                          B1

      Angle MZY = angle NYZ
      base angles of (Isosceles) ∆ XYZ
                        Note Reason necessary
                        eg you might see
                        If XZ = XY then angle XZY = angle XYZ
                                                                          B1

      Angle MYZ = angle NZY
                                                                          B1

      Triangles congruent, ASA
                       Note Dependent on earning first 3 marks
                       Must give correct reason for congruence (ASA)
                       Only allow AAS if complete argument stating
                       ‘third angles equal’
                                                                       B1 dep
                                                                                [4]




                                                                                  Page 14 of 25
M4.        Side of 34
                                                                B1

      50 – 18 or 25 – 9
                                                                M1

      Side of 32 or 16
                                                                A1

      162 + x 2 = 342
                                                                M1

      x = 30
                          2 × ‘8, 15, 17’ triangle get M1, A1
                                                                A1

      64
                                                                A1
                                                                     [6]




                                                                       Page 15 of 25
Page 16 of 25
M5.         BAF = 120°
                     This can just be stated
                     or exterior angle of hexagon = 60°
                      or reflex     FAB = 240°
                                                                      B1

      360 – (120 + 90 + 90) = 60°
                       oe
                         HAJ must be shown to be 60° by calculation
                                                                      B1




                                                                           Page 17 of 25
AH = AJ
              This can just be stated or shown on diagram
                                                              B1

  AJH =   AHJ = (180 – 60) ÷ 2
              Dep on first B2
                                                            B1dep
                                                                    [4]




                                                                      Page 18 of 25
M6.   (a)      (i)        180
                                                          B1

        (ii)     18
                           Their (a)(i) ÷ 10
                           Not ft 280 or 040 or > 360
                                                        B1 ft

       (iii)         06
                           Do not allow 6
                                                          B1




                                                                Page 19 of 25
(b)   210
            Allow 208 – 212
                                B1

      21
            SC1 15
            Not 21.0
                              B1 ft




                                      Page 20 of 25
(c)   300 – 180
                  180 ÷ 10
                              M1

      120
                  30 – 18
                               A1

      12
                             A1 ft
                                     [8]




                                       Page 21 of 25
M7.   Breaks down into areas of rectangles and areas of (quarter) circles
                 Any combination of rectangles and circles
                     or 12.56... or       is enough evidence for area of circles
                   NB 12.56 from 2 × π × 2, if seen is M0
                   NB 3.14 on its own is not evidence of the area
                      of a quarter circle as it is π
                                                                                   M1




                                                                                        Page 22 of 25
Uses an ‘addition’ method (method 1) and finds

Area of one (or 5) ‘external’ quadrants




or

or
                  Uses a ‘subtraction’ method (methods 2 and 3) and finds
                  5 × area one quadrant


                  or
                                                                            M1 dep

= 0.8584..., {0.9, 0.86, 0.858}

or = (× 5) 4.292... {4.3, 4.29}
                   15.71, 15.7
                                                                               A1

52.3 or 52.29... 68 – 5π
                   Allow 52 if 52.3 or 52.29... or a full method seen
                                                                               A1
                                                                                     [4]




                                                                                       Page 23 of 25
M8.         (a)   Valid explanation
                          eg allied angles (add up to 180)
                              inside parallel lines (add up to 180)
                              y + y + 2x + 2x = 360 so y + 2x = 180
                              2y + 4x = 360
                              In a C add up to 180
                          Condone interior angles (add up to 180)
                                                                      B1

      (b)    3x + y = 230
                        oe
                                                                      B1




                                                                           Page 24 of 25
(c)   Attempt to eliminate a variable
      (with 2x + y = 180)
                  eg 6x + 2y = 460 and 6x + 3y = 540 and subtraction
                  Note: Full marks can be awarded for this part on
                        follow through
                                                                        M1

      x = 50
                                                                       A1 ft

      y = 80
                                                                       A1 ft
                 3x + y = 130 → x = –50, y = 280
                 3x = y –50 → x = 26, y = 128
                 3x + y = 410 → x = 230, y = –280
                                                                               [4]




                                                                                 Page 25 of 25

				
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