Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
History of geometry wikipedia , lookup
Line (geometry) wikipedia , lookup
Multilateration wikipedia , lookup
Rational trigonometry wikipedia , lookup
History of trigonometry wikipedia , lookup
Integer triangle wikipedia , lookup
Trigonometric functions wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Properties of Quadrilaterals 1 Interior angles add up to 360 2 All interior angles are right angles 3 All sides are equal 4 Both diagonals are equal 5 The diagonals are perpendicular 6 The diagonals bisect each other 7 Both diagonals bisect the angles they run into 8 Only one diagonal bisects the other 9 Both pairs of opposite sides are equal 10 Both pairs of opposite sides are parallel 11 Exactly one pair of sides is parallel 12 Adjacent sides are equal 13 Each diagonal bisects the area of the quadrilateral 14 The diagonals bisect each other perpendicularly 15 Both pairs of opposite angles are equal 16 Exactly one pair of opposite angles is equal 17 Exactly one pair of angles is bisected by a diagonal Trapezium Kite Square Rectangle PROPERTY Rhombus or a in each box to indicate whether the quadrilateral named has the property described Parallelogram Put a FAMILY OF QUADRILATERALS Quadrilateral Trapezium Kite Parallelogram Rectangle Rhombus Square Wynberg Boys’ High School Department of Mathematics Grade 10 – Revision of Grade 8 and 9 Geometry 1. Write down (without reasons) the size of the angles marked x and y in the following: a) b) y x y 32° x 2. Work out the value of x in the following. Reasons must be given. a) b) c) x-24° x 53° 3x x 62° 3. Refer to the figure alongside A B y C AE = BE and AÊB 52 . BCDE is a parallelogram. 52° x Giving reasons: E a) Calculate the value of y. b) Calculate the value of x. D 4. In each one of the following diagrams, find the size of the angle or side marked x or y: a) b) C A 36 32 E B x D y x B A D 20 c) M C 50 P 120 x y O N R 5. In the figure, PQRS is a parallelogram with diagonals PR and SQ intersecting at O. Find the value of x. P Q 6x 4x 80 S O R WYNBERG BOYS’ HIGH SCHOOL GRADE 10 MATHEMATICS GEOMETRY REVISION: GRADE 8 TERMINOLOGY AND RULES Note: This information sheet does not cover congruency and similarity. SOME IMPORTANT TERMINOLOGY 1. 2. Supplementary angles Supplement - two angles which add up to 180o - the difference between 180o and a given angle 3. 4. Complementary angles Complement - two angles which add up to 90o - the difference between 90o and a given angle 5. Adjacent angles - two angles which share a common arm, and have the same vertex SOME “RULES” OF GEOMETRY RULE ABBREVIATED REASON 1. Angles around a point add up to 360o. ( ’s around point ____) Name the point. 2. Angles on a straight line add up to 180o. ( ’s on str. line ____) Name the line. 3. Vertically opposite angles are equal. (vert. opp. ’s) 4. When a transversal cuts a pair of parallel lines, then… (a)… pairs of corresponding angles are equal. (Look for an “F” shape.) AND (b)… pairs of alternate angles are equal. (Look for a “Z” or “N” shape.) AND (c)… pairs of co-interior angles are supplementary. (Look for a “C” or “U” shape.) (corres. ’s; ____ // ____) Name the parallel lines. (alt ’s; ____ // ____) Name the parallel lines. (co-int. ’s; ____ // ____) Name the parallel lines. Two lines, cut by a transversal, are parallel if… 5. (a)... two corresponding angles are equal. (corres. ’s equal; ____ ____) Name the equal angles. (b)…two alternate angles are equal. (alt ’s equal; ____ ____) Name the equal angles. OR OR (c)…two co-interior angles are supplementary. (co-int. ’s suppl.; ___ +___ 180o ) Name the supplementary angles. RULE ABBREVIATED REASON 6. The sum of the interior angles of a triangle is 180o. ( sum of ____) Name the triangle. 7. The exterior angle of a triangle is equal to the sum of the two interior opposite angles. 8. In an isosceles triangle the angles opposite the equal sides are equal to each other. 9. In a triangle, sides that lie opposite equal angles are equal in length. 10. In a quadrilateral (4-sided figure), the sum of the interior angles is 360o. (ext. of ____) Name the triangle. (____ ____; isosc. ____) Name the equal sides. (opp. equal ’s; ____ ____) Name the equal angles. ( sum of quad. ____) Name the quadrilateral. THE THEOREM OF PYTHAGORAS AND ITS CONVERSE AND COROLLARIES RULE Theorem of Pythagoras In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. Converse of the Theorem of Pythagoras If the square on the longest side of a triangle EQUALS the sum of the squares on the other two sides, then the angle opposite the longest side is a RIGHT ANGLE. ABBREVIATED REASON (Pythag. in ____) Name the triangle. (converse of Pythag. in ____) Name the triangle. Corollaries of the Theorem of Pythagoras If the square on the longest side of a triangle is LESS than the sum of the squares on the other two sides, then the angle opposite the longest side is a ACUTE, and the triangle is ACUTE-ANGLED. If the square on the longest side of a triangle is GREATER than the sum of the squares on the other two sides, then the angle opposite the longest side is OBTUSE, and the triangle is OBTUSE-ANGLED. (corollary of Pythag. in ____) Name the triangle. (corollary of Pythag. in ____) Name the triangle. WYNBERG BOYS’ HIGH SCHOOL GRADE 10 MATHEMATICS REVISION (TEST PREP) WORKSHEET GEOMETRY (INCLUDING PYTHAGORAS) Note: This worksheet does not cover congruency and similarity. NAME & SURNAME: ______________________________________________ CLASS: ________ QUESTION 1 Determine the value of x in each of the following diagrams. You must give reasons to justify your statements. (a) ˆ x , AOB ˆ 90o , In the given diagram, COD B ˆ 65o and AOD is a straight line. BOC C 65o A (b) ˆ 70o In the diagram alongside, AC // DF, EBA ˆ 2 x 30o . PBEQ is a straight line. and BEF x D O P A 70 B o C 2 x 30o D F E Q (c) In the given diagram, P̂ 2x , Q̂ 75o , Q ˆ 125o and PRS is a straight line. QRS 75o 125o 2x P (d) S R In the diagram alongside, AOC is a straight line, B ˆ x. ˆ 135o and BOC AOB 135o A O x C (e) The given diagram shows ABC, A with  100o , B̂ 45o , and Ĉ x . 100o 45o x C B (f) In the diagram alongside, AB // CD, P ˆ 105 and PRD ˆ x. AQP o 105o A B Q PQRS is a straight line. x C D R S (g) In the given diagram AE // BD, BA // CF F and FG FE. F̂ 30 and B̂ 5x 20 . o o 30o A G 3 1 E 2 1 2 5 x 20o B (h) 1 2 D C In the diagram alongside AE // BD, BA // CE and EC ED. E A 2 ˆ 2x and B̂ 4 x 10 . CED 1 2x o Determine the value of x . 4 x 10o B 1 2 C D QUESTION 2 (a) State the theorem of Pythagoras in words. [HINT: In a right-angled triangle ...] (b) In PQR, shown alongside, Q̂ 90o , P PQ 72 cm and QR 21 cm. 72 cm Q (1) Determine the length of PR. (2) Determine the perimeter of the triangle. 21 cm R QUESTION 3 The diagram below shows a film set being constructed. To reach and work on high spots on the construction site, the set-builders used a variety of cherry-picker type cranes, such as the one in the picture. (a) In the diagram, lines have been drawn to indicate what measurements are known. Note that CGFE is a rectangle and that ACD is not a straight line. A A 4m G 4m B B 3m 3m C G C 5m 5m F 2m E 3m D F 2m E 3m D Using your knowledge of the theorem of Pythagoras, calculate the height (AF) at which a person would be working when standing on the platform of the crane. Show all necessary working and give your final answer rounded off correctly to one decimal place. (b) The set builders are concerned that the walls of the building are not perpendicular to the ground. They record the measurements shown on the diagram below. XY is the measurement taken on the wall of the building. YZ is the measurement taken along the ground. X X 4,5 m 4,5 m 4,0 m 4,0 m Y Y Z Z 2,5 m 2,5 m Determine (by calculation) whether the wall of the building is perpendicular to the ground or not. If it is not perpendicular, determine whether it is leaning towards the crane, or away from it, giving a reason for your answer. QUESTION 4 Consider the diagram given below, which shows a circle, with centre O, and two squares. As shown in the diagram, the circle touches each side of square ABCD, and the four corners of PQRS lie on the circle. The area of square ABCD is 256 cm2. Determine the area of square PQRS. Show all your working. (Note that a maximum of two marks will be awarded for the correct answer, if no working is shown.) A B P Q O S D R C A Fair Fence? The Afrika family and Benjamin family have been neighbours for years and for as long as they can remember they have had a crooked fence between their properties. They have finally decided that they want to replace the crooked fence with a single straight fence but obviously they want the areas of each of their properties to remain the same. Benjamin Afrika Can you help them by choosing where to build one straight fence (roughly where the present fence is), but you must be able to show that the area of each property will stay the same? Fair Fence - SOLUTION Join AC to form a triangle Draw DE parallel to AC through B You now have a triangle between parallel lines and its area remains the same as long as its base remains the same. Drag B to D as shown. D A B E C So triangle ADC has the same area as triangle ABC Hence the one straight fence must be built along DC D A C E QUADRILATERALS You need: ruler, pencil, eraser, protractor, scissors, glue Considering the definitions you have been given, and using your ruler, protractor etc, draw one example of each kind of quadrilateral on the coloured paper, as listed. In each case your diagram should be about 9 cm across. You must also draw in the diagonals of each quadrilateral. By the end of the exercise you will be required to cut out your quadrilateral and stick it in the space provided to the left of the grid. One of the reason for cutting it out is so that you can experiment with folding (HINT!), but you will probably find it easier to make measurements from the figure before it is cut out. Make whatever measurements or observations you need to enable you to decide whether the statements given are true or false for your diagram, and record your results with ticks (not crosses) in the appropriate column. Parallelogram = White Rectangle = Blue Trapezium = Grey Rhombus = Pink Square = Green Kite = Brown When you have finished with your own shapes, you will get together with a group of three other boys and share your experience, and you will discuss as a group what you found in your four individual cases; thus you will arrive at a group decision about which sentences must be true. In some cases you will decide that the sentence could be true or false depending on circumstances, so you would answer ‘maybe’. Remember that we are looking for logical certainty here: if you say ‘YES’ then you must be convinced that the property you are talking about is unavoidable, while ‘NO’ means it is impossible. Record your name here: Who else was in your group? Your own case PARALLELOGRAM Paste your parallelogram here Both pairs of opposite side are parallel Both pairs of opposite sides are equal Both pairs of opposite angles are equal Just one pair of opposite angles is equal All angles are right angles Some, but not all, angles are right angles The two diagonals are equal in length The two diagonals bisect each other Just one diagonal is bisected The diagonals are perpendicular Just one of the diagonals bisects the angles it runs into Both diagonals bisect the angles they run into Each diagonal is an axis of symmetry Just one diagonal is an axis of symmetry YES The group decision NO YES NO Maybe Your own case TRAPEZIUM Paste your trapezium here YES NO The group decision YES NO Maybe Both pairs of opposite side are parallel Both pairs of opposite sides are equal Both pairs of opposite angles are equal Just one pair of opposite angles is equal All angles are right angles Some, but not all, angles are right angles The two diagonals are equal in length The two diagonals bisect each other Just one diagonal is bisected The diagonals are perpendicular Just one of the diagonals bisects the angles it runs into Both diagonals bisect the angles they run into Each diagonal is an axis of symmetry Just one diagonal is an axis of symmetry Your own case RECTANGLE Paste your rectangle here Both pairs of opposite side are parallel Both pairs of opposite sides are equal Both pairs of opposite angles are equal Just one pair of opposite angles is equal All angles are right angles The two diagonals are equal in length The two diagonals bisect each other Just one diagonal is bisected The diagonals are perpendicular Just one of the diagonals bisects the angles it runs into Both diagonals bisect the angles they run into Each diagonal is an axis of symmetry Just one diagonal is an axis of symmetry YES NO The group decision YES NO Maybe Your own case RHOMBUS Paste your rhombus here YES NO The group decision YES NO Maybe Both pairs of opposite side are parallel Both pairs of opposite sides are equal Both pairs of opposite angles are equal Just one pair of opposite angles is equal All angles are right angles Some, but not all, angles are right angles The two diagonals are equal in length The two diagonals bisect each other Just one diagonal is bisected The diagonals are perpendicular Just one of the diagonals bisects the angles it runs into Both diagonals bisect the angles they run into Each diagonal is an axis of symmetry Just one diagonal is an axis of symmetry Your own case SQUARE Paste your square here Both pairs of opposite side are parallel Both pairs of opposite sides are equal Both pairs of opposite angles are equal Just one pair of opposite angles is equal All angles are right angles Some, but not all, angles are right angles The two diagonals are equal in length The two diagonals bisect each other Just one diagonal is bisected The diagonals are perpendicular Just one of the diagonals bisects the angles it runs into Both diagonals bisect the angles they run into Each diagonal is an axis of symmetry Just one diagonal is an axis of symmetry YES NO The group decision YES NO Maybe Your own case KITE Paste your kite here Both pairs of opposite side are parallel Both pairs of opposite sides are equal Both pairs of opposite angles are equal Just one pair of opposite angles is equal All angles are right angles Some, but not all, angles are right angles The two diagonals are equal in length The two diagonals bisect each other Just one diagonal is bisected The diagonals are perpendicular Just one of the diagonals bisects the angles it runs into Both diagonals bisect the angles they run into Each diagonal is an axis of symmetry Just one diagonal is an axis of symmetry YES NO The group decision YES NO Maybe QUADRILATERALS – Definitions A quadrilateral is a plane, closed, four-sided figure. (‘Plane’ means it can be drawn on a flat sheet of paper, ‘closed’ means that if the sides were fences, sheep could not get out.) The sum of the interior angles is 360. Some quadrilaterals have special features, and have special names. For the purposes of this exercise you can assume only what is explicitly given in the definitions here: Parallelogram: the opposite sides are parallel Trapezium: just one pair of sides is parallel Rectangle: all the interior angles are 90 Rhombus: all four sides are equal Square: all the interior angles are 90 and the four sides are all equal Kite: Two adjacent sides are equal to each other with one length, and the other two adjacent sides are equal to each other but with a different length Use the coloured paper to make your figures: Parallelogram Trapezium Rectangle Rhombus Square Kite = = = = = = White Grey Blue Pink Green Brown Study the rubric overleaf to see how you will be assessed. Note particularly what you will have to comment on about the people who work with you in the group session – which is also what they will be commenting on about you! RUBRIC for QUADRILATERALS PEER ASSESSMENT In the grid below, list the boys who were in your group, putting your own name at the top of the list. Alongside each name, including your own, put a ranking 1 to 4 according to the rubric given below. You may be asked by your teacher to give some verbal justification for the codes you record. Level 4 PARTICIPATION MANNER CONTRIBUTION Level 3 Worked hard with the group at all times Contributed well enough Polite at all times, willing to help those whose understanding was weak, enjoying the discussions Contributed to discussions in the interests of the group as a whole A pivotal member of the team, without whom the task could not have been done An active participant with useful contributions Learner’s Name Level 2 Level 1 Made some contributions, but not many Only interested in determining answers, but not involved in proper discussions Provided some help or insight, but generally passive Participation Did not participate or help at all Dismissive of other boys’ problems or ideas, not interested in the work Made no meaningful contribution to the task, either through laziness or because of a lack of understanding Manner Contribution Your own name goes in this block TEACHER ASSESSMENT Par’m Trapezium Rectangle Rhombus Square Kite FIGURE: Correct shape/colour (1) Appropriate size (1) Care/neatness (1) Evidence of investigation (1) Individual GRID: correspondence with pasted figure is Poor/Satisfactory/Good (3) Group GRID: general accuracy is Poor/Satisfactory/Good (3) TOTAL: SUMMARY PEER /12 CODE: 0-22 TEACHER /60 23-29 30-59 TOTAL /72 60-72 1 Not Achieved 2 Partially Achieved 3 Achieved 4 Excellent Quadrilaterals and Isosceles Triangles 1. ABCD is a square, with BF̂D = 125. Find, giving reasons clearly, the size of BT̂C A B T F 125 C D 2. Determine the size of the angles marked x. Show all steps and give reasons for you answers. 2·1 2·2 P Q B A x 42° x 85° S (4) R D M 2·3 N E C (5) AD=DE=EC 3x O L x+10° (5) K LM=MN 3. ABCD is a parallelogram, with BA extended to P so that PA = PD. A P B Calculate the value of x. 52 D x C B A 4. x ABCD is a parallelogram, and TB bisects ABˆ C . Calculate the value of x and of z. 72 D 5. C T A ABCD is a parallelogram. AB = AE and DC = EC DAˆ E 40 z D 40 1 Calculate, with reasons, the sizes of: 1 6. a) Ê1 b) Â1 c) d) Ĉ Ê3 AE = BE and AÊB 52 . BCDE is a parallelogram. Calculate the value of y. b) Calculate the value of x. 3 C E Figure 2 B a) 2 A B y C 52° x E 7. D Find the value of a, b, etc. in the following diagrams: a) B b) F E 130° D a c 35° A H b C G EFGH is a parallelogram J c) L P 60° K M O KLPM is a parallelogram. KLMN is a kite. d N d) V e 10° R S 70° QRST is a rhombus. QRVT is a kite Q e) T X f) Y 3f C 2g+50° A W f+10° Z B 3g+20° 7g+30° D E 8. E A In the figure, AED and BFC are straight lines. D 115 65 Prove that ABCD is a parallelogram. 65 B F C QUESTION 6 In the following diagram, AD BC and AD // BC . a) D A Prove that ABC CDA . C B (4) b) Hence prove that ABCD is a parallelogram. Page 3 of 15 QUESTION 2 A a) In the sketch alongside, AC // KL and DC BC . Aˆ 38 and CBˆ L 68 . C 1 38 2 2 D K 1 1 2 68 B Determine each of the following angles. Give a reason for each of your answers. 1) B̂1 _________________________________________________ L 2) B̂ 2 _________________________________________________ 3) Ĉ 2 _________________________________________________ 4) Ĉ1 _________________________________________________ b) In the diagram alongside, HONEYB is a regular hexagon and BACH is a rhombus. Calculate, giving reasons, the value of y . N (4) O H E C y Y A B QUESTION FIVE A kite ABCD drawn on a rectangular page PQRS is represented alongside. A P Q The length of line segment AC is given by x and the length of line segment BD is given by y. 5.1 Using the formula for the area of a triangle, show that the area of a kite can be given as: Area (kite) = ½xy (3) D y B M x 5.2 If you were to cut out the shape of the kite from the page, what percentage of the paper would you discard? Show all working. (3) 5.3 If D and B were the midpoints of PS and QR respectively, what would be the shape formed? (2) S C 5.4 Your Maths Teacher made a statement in class, “A kite that has equal diagonals is a rhombus.” Is their statement true or false? Justify your answer by either using a proof if the statement is true or a counter example if the statement is false. (3) [10] 1.1 Complete the following sentences by writing down the missing word: 1.1.1 A quadrilateral whose diagonals are equal but which is not a square must be a …………… (1) 1.1.2 If a parallelogram has perpendicular diagonals then it must be a R ……….. (1) C D For quadrilateral ABCD it is given that B̂ D̂ 1.2 and that AC bisects BĈD What kind of quadrilateral is ABCD? (1) [3] A B y x x In the above isosceles triangle, each base angle is x and the apex angle is y . If the apex angle is 15 larger than the base angles, find the size of each angle. [6] P 7. In PQR , XY is parallel to QR, PQ = 6 = PY, 2YR = 6 QR = 2PY Note: PQR not drawn to scale X Q a) Calculate: XQ ____________ (2) Y R b) Calculate XY ____________ (2) 5.1 State whether the following conjectures are true or false. If false, give a reason for your answer. 5.1.1 5.1.2 5.1.3 5.1.4 All regular pentagons have 5 sides. All octagons are similar. Triangles are polygons. All polygons are convex. (6) 9.1 In the sketch below, BA // QT and AP = AC. BCˆ Q 45 and ACˆ T 55 A x T 55 B P 45 C Q Calculate the size of x , showing all working and giving reasons. (8) 9.2 Given that AD = DB and AE = EC, what can be said about DE and BC in the sketch below? (2) A D B >> >> E C 9.3 In the sketch below AD = AS = SR, AE // SM and RE // PM. Prove: 10.3.1 DR = RP (4) 10.3.2 AD 1 DP (2) 6 D A S R P [16] E >> >> M 7.1 Calculate the value of x and y, giving reasons: W x S 10 24 T 25 y 2,8 R V (5) 7.2 Find the values of a, b and c, giving reasons: P b Y 38° X 9cm 74° a R c (7) Q 7.3 reasons: If AE Prove that ΔABC is isosceles, giving BC and E is the mid-point of BC. A B 1.1 1.1.1 E Find the values of the variables giving reasons: C (6) 50 70 b c (4) 1.1.2 A B 50 d e C O D OB // DC Ô is the centre of the circle. (4) 1.1.3 G 13 f H 12 J (3) 1.1.4 h g (4) [15] 1.2 Are these lines parallel or not? Give reasons. 1.2.1 AB and CD A B 30 D C 150 (2) 1.2.2 DE and NR M 65 D E 140 N R (2) 1.2.3 QR and ST P 3 Q 4 R 5 S 8 T (2) [6] 1.3 The following pairs of triangles are similar, find the unknown sides. Show all working. MNO PQR M P 10 5 a 8 N b O Q 7 R [4] 1.4 A P O R B M C If AR = 8 units, BP // MR, AO = OM and BM = MC ,find the length of RC. Give reasons for your statements. [4] 1.5 M N P Q 1.5.1 There are many ways to prove a quadrilateral is a parallelogram. Four of these are 1. both pars of opp. sides are equal. 2. one pair of opp. sides are equal and parallel. 3. both pairs of opp. Sides are parallel. 4. diagonals bisect. With this given information, explain why MNQP will form a parallelogram in the above sketch if AM=MB, AN=NC, OQ=QC and OP=PB, (4) 1.5.2 Explain with reasons why? 1 NO = NB. 3 (2) QUESTION 1 1.1 Refer to Figure 1. A y y AB ∥ PQ M AM bisects BÂP and PM bisects AP̂Q x 1.1.1 Write down, with a reason, the numerical value B P x Q Figure 1 of 2x 2 y 1.1.2 (1) Calculate the size of AM̂P (2) QUESTION 2 Refer to Figure 3. The circle with centre O has a chord AB. O OP is drawn perpendicular to AB. Figure 3 A 2.1 2.2 Prove that ∆OAP ∆OBP (4) If AB = 8 cm and OP = 3 cm, determine the radius of the circle. [7] B P (3) QUESTION 1: Geometry 1.1 Write down (without reasons) the size of the angles marked x and y in the following: (a) (b) y x y 32° x (4) 1.2 (a) Work out the value of x in the following. Reasons must be given. (b) (c) x x-24° 53° x 3x 62° (6) QUESTION 1. In each one of the following diagrams, find the size of the angle or side marked x or y (give reasons): a) A b) C 32 36 B E x D y A x B D 20 (4) c) (8) M 50 P 120 x y (4) O N [16] R QUESTION 2. In the figure, PQRS is a parallelogram with diagonals PR and SQ intersecting at O. Find the value of x (giving reasons). P Q 6x 4x 80 O S R [5] QUESTION 3. From the given diagram: C a) prove that ABC is similar to DEC ; 2,5 cm 1,5 cm D b) calculate x, a and b, giving reasons; E a c) prove that DEC DEA . (3) (7) (4) 2,5 cm x A 1. b a) B name a six sided polygon. [14] (1) b) c) 1. Find the size of an interior angle of a regular octagon. If the two shorter sides of a right angled triangle are 5 cm and 12 cm, find the length of the hypotenuse. (3) (3) Decide whether or not the following are true: a) If two quadrilaterals have two equal opposite sides and two equal opposite angles, then it is a parallelogram. b) If a quadrilateral has two right angles and two equal diagonals, then it is a rectangle. 2. In quadrilateral ABCD , E is the midpoint of AB , F is the midpoint of CD and BC AD . What can be said about ABCD ? EF 2 Investigate parallelogram ABCD where E , F , G and H are the midpoints of the sides of the parm. E A B H D Investigate (Cross’s Theorem): F G C If two rectangles have the same diagonal, must the rectangles be congruent? Answer: No QUESTION 5: After the studying the following diagram, complete the statements below. (no reasons necessary). F 12 B H 3 2 11 1 G 9 I 6 4 7 C 5 5.1 2 + 6 + 4 = . . . . . degrees 5.2 5=2+ ..... 5.3 1 + 7 + 9 + 11 = . . . . . degrees 5.4 3 + 5 + 8 + 10 + 12 = . . . . . degrees 5.5 1 + 3 + 5 + 7 + 8 = . . . . . degrees 8 J 10 E A (10) QUESTION 6: 6.1 Prove the theorem that states that the internal angles of a triangle are supplementary. 6.2 D Find the value of x in the diagram below if AB=AC and BC=BE. A 34 F (5) (6) QUESTION 7: 7.1 List three properties of a rhombus that are not properties of a parallelogram. (3) 7.2 PQRS is a rhombus with PS = PR = x mm. Express the following lengths in terms of x . No reasons are required. 7.2 7.2.2 7.2.3 .1 QT QS PT P Q T S 7.3 R (5) In the figure, KMQR is a rhombus. The bisector of MKˆ Q cuts MQ at P. Prove that MPˆ K 3MKˆ P K M P (6) QUESTION 8: Q R 8.1 List 5 ways in which you could prove that a quadrilateral is a parallelogram. (5) 8.2 Prove any one of these, except the one that is the definition of a parallelogram. (5) 8.3 ABCD is a quadrilateral with AD//BC and E on BC such that AB = AE and EC =CD. A D 2x x 8.3.1 Prove that ABCD is a parallelogram. (6) 8.3.2 Prove that EA=EC. (2) 8.3.3 Calculate x if AEˆ D 105 (4) QUESTION 2: 2.1 State the four cases of congruency. (4) 2.2 In the diagram ABCD is a parallelogram and AD DE . Prove that DF FC B A D C F (7) QUESTION 3: E 3.1 Redraw these sketches of a rhombus on your answer sheet and, using appropriate symbols, show all the properties of a rhombus. (6) 3.2 List five ways in which one can show that a quadrilateral is a parallelogram. 3.3 (5) F In the diagram alongside, FGHJ is a rhombus. FO JK and OG KH . G O J H K 3.3.1 Prove that JOHK is a parallelogram. (5) 3.3.2 Now show that JOHK is a rectangle. (2) QUESTION 4: 4.1 Using the diagram below, prove the theorem that the diagonals of a parallelogram bisect each other. M N O Q 4.2 P (6) In the diagram RSTV is a parallelogram Y R S O V W T 4.2.1 Prove that ROY TOW . (5) 4.2.2 Prove that RYTW is a parallelogram. (4) MATHEMATICS CYCLE TEST GRADE 10 Date: April 2006 Time: 60 min Total: 70 Start question 1 and question 2 on separate exam pad sheets. Question 1 1.3 Find the values of the variables giving reasons: 1.1.1 50 70 b c (4) 1.1.2 A B 50 d e C O D OB // DC Ô is the centre of the circle. (4) 1.1.3 G 13 f H 12 J (3) 1.1.4 h g (4) [15] 1.4 Are these lines parallel or not? Give reasons. 1.2.1 AB and CD A B 30 D C 150 (2) 1.2.2 DE and NR M D 65 140 N E R (2) 1.2.4 QR and ST P 3 Q 4 R 5 S 8 T (2) [6] 1.3 The following pairs of triangles are similar, find the unknown sides. Show all working. MNO PQR M P 10 5 a 8 N b O Q 7 R [4] 1.4 A P O R B M C If AR = 8 units, BP // MR, AO = OM and BM = MC ,find the length of RC. Give reasons for your statements. [4] 1.5 M N P Q 1.5.3 There are many ways to prove a quadrilateral is a parallelogram. Four of these are 1. both pars of opp. sides are equal. 2. one pair of opp. sides are equal and parallel. 3. both pairs of opp. Sides are parallel. 4. diagonals bisect. With this given information, explain why MNQP will form a parallelogram in the above sketch if AM=MB, AN=NC, OQ=QC and OP=PB, (4) 1.5.4 Explain with reasons why? 1 NO = NB. 3 [6] Question 2 (2) y B (3; 7) 1 x A (2;3) Show all working and use above sketch to: 2.1.1 Find the distance between A and B. (3) 2.1.2 Find the gradient from A to B. (2) 2.1.3 Find the midpoint between A and B. (3) 2.1.4 Write down the equation of the straight line AB. (2) 2.1.5 Which of the following points will be on the line AB. Show all working. a) (4;9) b) (2; 5) (2) (2) [14] 2.2 Monique works in the post office and can stamp 750 letters every 5 minutes. 2.2.1 Calculate the average rate of change at which she stamps the letters. (2) 2.2.2 Work out and show in a table what amount of letters she will stamp in 2 min, 3 min, 7 min, 8 min. (2) 2.2.3 Find an equation that links the number of stamps (y) to the minutes (x) and draw this graph on a suitable axis. (4) [8] 2.3 Draw sketch graphs if: 2.3.1 m is negative and c is positive. (2) 2.3.2 m is positive and c is negative. (2) (No values are to be put onto these graphs, indicate only the x and y axis.) [4] 2.4 P (2; 5) Q (5; 7) T S R (4;1) PQRS is a parallelogram. Remember that in a parallelogram, diagonals bisect. 2.4.1 Find T, the intersection of the two diagonals. (2) 2.4.2 Find S, one of the vertices of the parallelogram. (2) [4] 2.5 Mr Boon is currently driving a Toyota Prado which cost him R690 000 new, in January 2004. By the end of each year the car’s value depreciates such that it forms a linear equation. By the end of 2006 the value of the car is R640 000. A new Toyota Verso increases in price each year also such that it forms a linear equation. The price of a Verso was R240 000 in the beginning of 2006 and will cost R270 000 at the beginning of 2007. 2.5.1 In each case find suitable equations and plot them on one Cartesian plane. (2) 2.5.2 Mr. Boon wants to drive his Prado has long as possible but want to buy a new verso without having to pay in. Find the point at which Mr Boon can sell his Prado and buy a brand new Verso without having any money left. (3) [5] Total: [70] Sw/rl/24.04.06 1. Identify the following polygons, stating: a) the name b) whether it is convex or concave i) ii) YIELD iii) iv) (Use one of the white polygons on the ball) (8) 2a) H THINK is a regular pentagon. T I Calculate the size of x. 1 2 1 K x 2 N F (4) b) X T A B EXTEND is a regular. hexagon. AN // BP. Calculate the value of a. E E 1 2 D N a P c) How many sides does a polygon have if the sum of its angles is 2700? (3) (3) 3a) A What is wrong with this picture? Explain. B 72 72 E 74 74 C D (3) b) State whether the following are true or false. If false, explain why. i) If Pˆ Qˆ , then the supplement of P̂ is less than the supplement of Q̂ . (2) ii) In ABC, Aˆ 90, AB = 3, BC = 4 and AC = 5. (2) c) If ABC XYZ, then Aˆ Xˆ , Bˆ Yˆ , Cˆ Zˆ . i) Write down the converse of this statement. ii) Is this converse true or false? If false, give a counter example. (1) (2) 4. A 1 2 G 1 2 H F 1 56 2 66 1 2 3 1 B C 3 D E Aˆ1 Dˆ 1 x ; Cˆ 2 y ; Dˆ 3 z. AC CD and CH HD. Calculate the values of x, y and z. 5a) L Ŝ L̂ ; Ĝ Â. T is the midpoint of SL. G N Is SG = TA? Show all reasoning. T R Y (7) b) A B C 2,42 AB = AC and  = 90. Calculate AB. (3) 6. 9,45 cm 10,5 cm Above is a picture of the Arc de Triomphe in Paris. Alongside is a plan of the front, with the measurements of the height and width. If the Arc de Triomphe is 50m high, how wide is it? Show all working. (3) 7a) A AB and AD are each divided into four equal parts. P S Ĉ D̂. Determine the length of: i) BD Q T ii) PS R U B D (6) 48 C b) T is the centre of the circle. PQ // ST. If PS = 3cm, determine the length of SR. R S P T Q (4) QUESTION 4 ABCD is a rectangle. O is any point inside ABCD. B A Prove that : OAB + OCD = OAD + OBC O D C QUESTION 1 Complete the following statements: 1.1 A trapezium is … (2) 1.2 A rhombus is … (2) 1.3 A kite is … (2) 1.4 RSTU is a quadrilateral with R = 4x, S = 5x, T = x and U = 2x. 1.4.1 Calculate the size of S. 1.4.2 Prove that RSTU is a trapezium. (3) (5) [14] QUESTION 2 A ABCD is a parallelogram with A = 68 . EC bisects C. Calculate the values of p and q. E o B q o 68 p [5] C D QUESTION 3 3.1 Write down the definition of a parallelogram. 3.2 PS // QR. T is the midpoint of PR. Prove that PQRS is a parallelogram. STQ is a straight line. (2) P Q T S (6) [8] R QUESTION 4 U V 4.1 Prove the theorem that states that the opposite angles of a parallelogram are equal. Redraw the diagram. X 4.2 DEFG is a parallelogram with GH // IE. H Prove that DI = FH. D W (6) E I G F (7) [13] QUESTION 5 Parallelogram ABCD is shown with AX bisecting A and DX bisecting D. Prove that AXD is a right angle. A B X D C [5] 1. Find a , b, c and d , giving reasons : A M L B 1.1 1.2 a E C Q 130 20 D O N P (2) (5) /7/ QUESTION 2 2. F Find value of x , giving reasons. A 2.1 D 190x x 2.2 I B x x x G x 3x 20 C H (3) (5) x /8/ E QUESTION 3 C x 3 Find, giving reasons : 3.1. 2 angles each equal to x (2) D 3.2 2 angles each equal to 2x 3.3 (2) Hence calculate the value of x . (2) B /6/ A QUESTION 4 4 Find angle AOC , giving reasons. /4/ C D QUESTION 5 5 5.1 Prove ABC AED (3) 5.2 Prove BCD = EDC (4) /7/ Grade 10 GEOMETRY REVISION EXERCISE March 1998 NOTE: Draw diagrams on the right hand side of the page - it gives more writing space. Add all information given in the question to the diagram. Support all statements with reasons - using the accepted abbreviations Present your argument succinctly (in a brief, concise, logical manner) Parallel Lines. [a = b (vert opp ‘s)] a = d (corr ‘s, PQ // RS) b = d (alt ‘s, PQ // RS) b + e = 180o (co-int ‘s, PQ // RS) P a c Q b R d e S Calculate, with reasons, the value of x in each of the following cases: 1 A E B 3x C 2 A D F x + 50o G AB // EF & BD // FG 3 B x + 14o 2x + 10o C D A AB // CD B 4 A B o o 125 C 50 x C D 160o E D AB // DE Solve questions 3 and 4 in two different ways! x E AB // DE Triangles The angle sum of any triangle is 180o The exterior angle of any triangle is equal to the sum of the two interior opposite angles. An isosceles triangle has two equal sides, and the angles opposite these sides are also equal. Find, with reasons, the value of x and y in each of the following: 5 A 64o 6 A F x y 3x y B C AB // FE AC = AD o x 84 D 38o E B D C A E 7 8 34o Prove, with reasons, that AC = CD if AB = BC 0 136 E B AB // DC. DAB = 120o , DBC = 85o, DCB = 65o Prove that ADB is isosceles. A D C B D C 9 PQRS is a parallelogram. Prove that P = R. P Q S 10 In PQR, P = Q = x RT // QP, QR is produced to S. Prove that RT bisects PRS. R P T Q 11 R BD = DC = DA ABC = x, BAC = y Find the size of ACB in degrees. S A D B y x C A 12 AB = BC. AD = AC. BAD = 30o and ABD = x. Calculate the size of x in degrees. 30o x B 13 D SPR = SRP = x QRP = 90o Prove that S is the midpoint of PQ. P S x x Q 14* C EC = BC. ABC = BDC = 90o. Prove that BE bisects ABD. R A E D B C Congruency Two triangles are congruent if they are identical in shape and size. (Two triangles are similar (///) if they are have the same shape - angles correspondingly equal and sides in proportion - but not necessarily equal in size. Tested using the “ test”) Two triangles can be proven to be congruent by proving one of the following sets of conditions. To ensure equal size, at least one length must be the same for each triangle. 1 S.S.S 2 S. included . S 3 S. . . 4 90o. H. S. Approach: Always state the triangles in which you are working - labelled in corresponding order. State the 3 conditions with reasons, keeping information under the correct triangle. State that the triangles are congruent ( ) and add the condition for congruency used. Then use the congruency to draw any further conclusions required. eg Prove that AB = CD given AB // CD and BO = OC. A B O In ABO and DCO 1 BO = CO 2 A = D 3 O = O ABO DCO AB = CD (given) (alt ‘s, AB // CD) (vert opp ‘s) (S. . .) C D B A 15 O is the centre of the circle. AOB = COD Prove that AB = CD. O C D 16 AD = BC and EDA = FCB Prove that ACD = BDC. A B O E 17* x D x C AB = AC and EB = EC Prove that AD BC F A E P B C D 18* PS = PT and PQ = PR. Prove that QPO = RPO. T S O Q R 2.2 Write down the letters a to e and then match the quadrilateral to the property: PROPERTY QUADRILATERAL (a) two pairs of adjacent sides equal 1 kite (b) four axes of symmetry 2 parallelogram (c) only one pair of sides parallel 3 rhombus (d) diagonals equal in length, but sides not 4 rectangle necessarily equal in length (e) diagonals cross at 90o and bisect eadh other 5 square 6 trapezium 5 Find the value of x and y in the following figures. Give your reasons clearly. A B C D 5.1 145 o x 115 o y E (4) 5.2 A 65 o 25 o D 5.3 y B C M 54 o A x N x 2x C B y 2y O 5.4 (4) P D (4) A 40 o 70 o E x C 6 D Write down the 4 reasons for congruency. (4) (2) QUESTION 1 1.1 Write down the definition of a parallelogram. (2) 1.2 For each of 1.2.1 to 1.2.3 below, write down one property of a parallelogram which has to do with its ... 1.2.1 sides (1) 1.2.2 angles (1) 1.2.3 diagonals (1) 1.3 Sketch a rhombus on your answer sheet. Make markings on your sketch which indicate at least 5 properties of a rhombus. (Think of sides, angles, diagonals, lines of symmetry, etc.) (5) /10/ QUESTION 2 Use the properties of these quadrilaterals to find your answers. Reasons need not be given in this question. 2.1 2.2 2.1.1 Name the quadrilateral. (1) 2.1.2 Write down the sizes of angles a and b . (2) 2.2.1 Name the quadrilateral. (1) 2.2.2 Write down the sizes of angles x and y . (2) /6/ a b x y 95 QUESTION 3 From this point on, you should give clear reasons for all your steps. A 3.1 In ABC, A = 75 and C = 60. DEFB is a parallelogram. Calculate the angles of parallelogram DEFB. D (6) E B C F 3.2 Calculate the angles of parallelogram PQRS. P 6x 12 (4) 2x + 40 3.3 The drawing shows parallelogram R ABCD and parallelogram ABEC. Prove S that C is the midpoint of DE. (‘Midpoint of DE’ means it lies in the middle of DE.) (4) D C E Q Do not re-draw the diagrams - Use the test question diagrams and hand them in stapled to the back of your answers. Show full proofs, giving reasons wherever necessary. There are 8 questions. K J Find the size of a. 1 F a H G L 30o M [4] T 2 TR = TS. QRS is a straight line. Find the size of b. b = = 110o Q R S [4] D 3 o EF // LK // GHJ. Find the sizes of c, d and e. 50 E F L K 3d – 20o e c d G J [6] H In ABC, P is any point on AB. BT bisects ABC. PT // BC. Prove, with reasons, that PT = PB. 4 A P o T o B C [4] P. T. O. X XYZ is drawn together with four other triangles. Just name, in correct order, the one triangle that is congruent to XYZ and give the reason for the congruency. Lengths are in cm. 5 3 38o Y A D 92o 38o 50o C B F 4 G K 3 38o 50o 3 E 92o 2,5 50o Z J 4 50o 4 2,5 [2] H M 3 L A \ 6 AB = DC and ABC = DCB. Prove, using congruency, that ABC DBC and that A = D. B C / D [5] P 7 PQT and PSR are straight lines. PS = PQ. = Prove that TS = RQ. Q T x = x S R [5] 1. P In the diagram alongside Q = 63°, P = 85°, and TSRU is a parallelogram. Calculate the sizes of the angles of the parallelogram, giving reasons for all statements. T S Q R U (8) 2. K In parallelogram KLMN K = x+60° and M = 3x-30°. L Find the sizes of the angles of the parallelogram. N M (6) 3. In parallelogram DEFG D = 4x and E = 2x-30°. Find the sizes of the angles of DEFG. E D G F (6) Grade 10 SG 1 GEOMETRY: CONGRUENCY RIDERS Prove that AO = OD and BO = OC if AB = CD and AB // CD. A 1999 B O C D A 2 B Prove that AB = CD and AB // CD. O C 3 O is the centre of the circle. Prove that AC = CB. D O A C B A 4 D Prove that AC = BD = A 5 AB = AE. Prove that CE = BD. B = x O B = x = C E O C 6 D AB = AE, AD = AC and BAE = DAC Prove that BC = DE. A / x = = B 7 Prove that AB = DC and AB // DC. A = = D x / E B C D C In each question, redraw the diagram in your workbook, and give reasons for all the statements that you make. P 1. PQRS is a parm with P = 60°, PQ = PS and QS = 50 mm. 1.1 1.2 Q Prove that PQRS is a rhombus. Find the perimeter of the rhombus. S R A 2. In ABC, A = 80° and C = 55°. Calculate the angles of the parallelogram DEFB. D E B 3. K Calculate the angles of the parm KLMN, if N = 5x – 12° and L = 3x + 18° N L M A 4. ABCD is a parm, with EF AB, GH BC, and B = 54. Calculate the sizes of i) D ii) A iii) AGK iv) EKH E G D K B 5. ABCD is a trapezium with AD BC, and A = D = 50°. If FC = DC, prove that ABCF is a parallelogram. C F H C F F A B D C 1. ABDE is a parm. AF = CD. A F Prove: 1. ABF DEC G 2. FG = GC 2. E B D C A AO = OC; BAC = ACD; ABD = BDC D O Prove that ABCD is a parm. B C F 3. BA = AF = CD; FE = EC; AE = ED Prove that ABCD is a parm. E A D C B A 4. F ACDF is a parm. Prove 1. BF = CE 2. BCEF is a parm 3. BCE = ABF B E C D A 5. F D AGEF and ABCD are parms. Prove that GEF = BCD G E B C 1. Complete the following sentences, writing the words needed to complete the sentence P grammatically in the space provided on the answer sheet: 1.1 S The definition of a parallelogram is as a quadrilateral with 55 mm opposite sides …..…. V A rhombus is a ……. with ……….. sides equal T A trapezium is a quadrilateral with ………… If one pair of opposite sides of a quadrilateral are equal x 60° and parallel then the quadrilateral is a ……………… x A rectangle is a ……… with interior angles equal to ……….. R If a quadrilateral has two pairs of adjacentQ sides equal and opposite sides not equal it is a 1.2 1.3 1.4 1.5 1.6 ……… The diagonals of a rhombus ………… each other and are …………… (10) 1.7 2. From the sketch alongside, name (giving the vertices in alphabetical order) 2.1 2.2 2.3 3. one rhombus three parallelograms two trapezia (6) For each of the sketches find the size of the angles marked with a small letter. Fill in your answers (no reasons required) on the answer sheet. C 3.1 3.2 D B A F E (7) h A 4. a PQRS is a rhombus. 58° In the figure, D 22° X RT bisects QR̂P . Give, with reasons and on the answer sheet, B b 4.1 the size of SV̂R 4.2 the size of VR̂T 4.3 the length of SQ c 80° d C (6) U k m V 122° 31° E W 5. Calculate the areas of each of the quadrilaterals shown. Show your working. 5.1 5.3 B A C 900 m 2 20 m F E D 5.2 5.4 A B 17 mm 30 mm (11) 40 mm C D N M P 35 mm QN = 20 cm S Q QK = 30 cm 10 mm R Q P K F. For each of the following figures, make an equation involving x, giving reasons, and then solve it to find the value of x. B A x + 21 O D 17 3x – 18 C Parallelogram 48 65 Parallelogram 63 Rhombus 53 Rectangle 37 27 Kite 1. State whether the following sentences are TRUE or FALSE. For those which are FALSE, give a corrected version. 1.1 1.2 1.3 1.4 1.5 The sum of all interior angles of a quadrilateral is 360 A trapezium has all sides of equal length A kite has opposite sides parallel The diagonals of a rectangle bisect the angles into which they run If the diagonals of a quadrilateral cross at 90, the quadrilateral must be a square (9) 2.1 Give four properties of a quadrilateral such that each one on its own would guarantee that the quadrilateral is a parallelogram. (4) 2.2 What extra properties do you need to prove that a parallelogram is a square? (2) 2.3 Name two quadrilaterals whose diagonals are perpendicular (2) A 3. ABCD is a square, with BF̂D = 125. Find, giving reasons clearly, the size of BT̂C B T F 125 4. In the figure, AED and BFC are straight lines. (5) C D E A D 115 65 Prove that ABCD is a parallelogram. 65 B C F (6) X 5. The diagonals of parallelogram ABCD meet in O. DB is produced to X and YC is drawn parallel to AX to meet BD produced at Y. A B O D C Y 5.1 Prove that AXO CYO (4) 5.2 If CX and AY are joined, prove that AXCY is a parallelogram. (3) Question 1. State whether the following statements are true or false: 1.1 In a trapezium both pairs of opposite sides are parallel. 1.2 The diagonals of a kite bisect each other at 900. 1.3 If a parallelogram has equal diagonals then it must be a square. 1.4 A square is always a rhombus, but a rhombus is only sometimes a square. 1.5 If the diagonals of a quadrilateral are equal then it must be a rectangle. [5] 14] Question 3. STRV and PURV are parallelograms. U P S T V R Prove that PSTU is a parallelogram. [6] Question 4. 4.1 Write down the ratio of the areas of the figures specified. Reasons and steps are not required. ΔBDE : ΔABC ΔAEC : parmABCD A 4·1·1 4·1·2 E D A D E B C B C (4) 4.2 The area of parallelogram ABCD=1500mm2 and BC=30mm. Calculate: B A G E D F C 4.2.1 the length of EF 4.2.2 the area of ΔGDC 4.2 (6) ABCD is a parallelogram. Prove that Area ΔBCP Area ΔABQ (5) D A B P C Q 1. State whether the following sentences are TRUE or FALSE. For those which are FALSE, give a corrected version. 1.1 1.2 1.3 1.4 1.5 The sum of all interior angles of a quadrilateral is 360 A trapezium has all sides of equal length A kite has opposite sides parallel The diagonals of a rectangle bisect the angles into which they run If the diagonals of a quadrilateral cross at 90, the quadrilateral must be a square (9) 2.1 Give four properties of a quadrilateral such that each one on its own would guarantee that the quadrilateral is a parallelogram. (4) 2.2 What extra properties do you need to prove that a parallelogram is a square? (2) 2.3 Name two quadrilaterals whose diagonals are perpendicular (2) B A 1. 51 In the figure ABCD is a parallelogram. Prove that it must be a rhombus M 39 D C B A 2. In the figure ABCD is a parallelogram. Prove that it must be a rhombus. 43 94 D 3. C B A In the figure ABCD is a parallelogram. ˆC Prove that ABˆ C BM D C 6. ABCD is a rhombus, and DC is extended to P so that BP = BC. If DBˆ C = x, find the following in terms of x: 4.1 DAˆ C 4.2 P̂ D M B A x M C P 7. ABCD is a parallelogram. B A Prove P 5.1 ADM CBP 5.2 DP = BM 5.3 MAˆ B PCˆ D M D C Q A 8. B ABCD is a parallelogram. Prove 9. 6.1 ADP CBQ 6.2 AQ = PC D C P Is AQCP a parallelogram? Give reasons for your answer A ABCD is a parallelogram. DP = BQ. B Q Prove that AQCP is a parallelogram. P D C B A P 10. ABCD is a parallelogram, and DP AC with BQ AC. Q Prove that BPDQ is a parallelogram. D C B A 11. ABCD is a parallelogram, and DP QB. Prove that DPBQ is a parallelogram. P Q B A 12. ABCD is a parallelogram, and DP = PV while BD = QV. Q V Prove that APCQ is a parallelogram. P D 13. ABCD is a parallelogram. (HARD) C U B A Given that DP = PU and AP = PV, prove that BV and CU bisect each other. Q P V D C QUESTION 4 [3] Refer to the diagram below, and solve for x (you must show all reasons). QUESTION 5 [8] Refer to the diagram below, and answer the following questions (with reasons). 5.1) Determine the size of P (in terms of x). (3) 5.2) If T 2 90 x , determine the size of R (in terms of x). (2) 5.3) If PTR=110 , determine the value of x. (2) 5.4) What type of triangle is QTR ? (1) QUESTION 6 Prove ACD CAB [4] QUESTION 7 [4] Based on the information provided, make a conjecture about the quadrilaterals below. Then prove your conjecture. You must give the best classification possible: If a quadrilateral is a rhombus, it is not good enough to classify it as a parallelogram. If a quadrilateral is a rectangle, it is not good enough to classify it as a parallelogram. If it is a square it should not be classified as a rectangle. 7.1) (2) 7.2) QUESTION 8 (2) [11] In the diagram below AD=AG and EF||DG 8.1) If ADG = x , State (with reasons) three other angles equal to x. (3) 8.2) Prove that ED = FG. (2) 8.3) Prove that EDG FGD . (3) 8.4) Name 2 other pairs of congruent triangle (do not prove they are congruent) (2) 8.5) Name 1 pair of similar triangles which are not congruent (do not prove they are similar) (1) QUESTION 9 [2] Do not attempt this completed the other 8 questions. In the diagram, AD is an angle bisector of CAB perpendicular to AD. Prove that AE AC FB CD question until you have