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Pythagoras ‘Theorem In a right-angled triangle, the square of the length to hypotenuse is equalto the sum of the squares of the lengths of the other two sides.(Pythagoras’ theorem). if the square of one side of a triangle is equal to the sum of the squares ofthe other two sides then the triangle is a right angled triangle (converse toPythagoras’ theorem). 1 Motivation You’re locked out of your house and the only open window is on the second floor, 25 feet above the ground. You need to borrow a ladder from one of your neighbors. There’s a bush along the edge of the house, so you’ll have to place the ladder 10 feet from the house. What length of ladder do you need to reach the window? Figure 1: Ladder to reach the window Brief history: Pythagoras lived in the 500’s BC, and was one of the first Greek mathematical thinkers. Pythagoreans were interested in Philosophy, especially in Music and Mathematics? The statement of the Theorem was discovered on a Babylonian tablet circa 1900−1600 B.C. Professor R. Sullying in his book 5000 B.C. and Other Philosophical Fantasies tells of an experiment he ran in one of his geometry classes. He drew a right triangle on the board with squares on the hypotenuse and legs and observed the fact the square on the hypotenuse had a larger area than either of the other two squares. Then he asked, ”Suppose these three squares were made of beaten gold, and you were offered either the one large square or the two small squares. Which would you choose?” Interestingly enough, about half the class opted for the one large square and half for the two small squares. Both groups were equally amazed when told that it would make no difference. Page 1 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. (c2 = a2 + b2) It is believed that this theorem was known, in some form, long before the time of Pythagoras. A thousand years before Pythagoras, the Babylonians recorded their knowledge of the theorem on clay tablets. The Egyptians used the concept of the theorem in the building of the pyramids. In 1100 B.C. China, Tschou-Gun knew of this theorem. It was, however, Pythagoras who generalized the theorem to all right triangles and is credited with its first geometrical demonstration. Consequently, the theorem bears his name. Geometrical Proof: The Pythagorean Theorem has drawn a good deal of attention from mathematicians. There are hundreds of geometrical proofs (or demonstrations) of the theorem, with even a larger number of algebraic proofs. Geometrically, the Pythagorean Theorem can be interpreted as discussing the areas of squares whose sides are the sides of the triangle (as seen in the picture at the left). The theorem can be rephrased as, "The (area of the) square described upon the hypotenuse of a right triangle is equal to the sum of the (areas of )the squares described upon the other two sides." Converse: Theorem: If a triangle is a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. Converse: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, the triangle is a right triangle. Pythagorean Triples: There are certain sets of numbers that have a very special property in connection to the Pythagorean Theorem. Not only do these numbers satisfy the Pythagorean Theorem, but any multiples of these numbers also satisfy the Pythagorean Theorem. Page 2 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] For example: the numbers 3, 4, and 5 satisfy the Pythagorean Theorem. If you multiply all three numbers by 2 (you will get 6, 8, and 10), these new numbers ALSO satisfy the Pythagorean theorem. The special sets of numbers that possess this property are called Pythagorean Triples. The most common Pythagorean Triples are: 3, 4, 5 5, 12, 13 8, 15, 17 Page 3 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Answer the following questions dealing with the Pythagorean Theorem. 1. Choose: Princess Fiona is locked in the tower of Castle Kronen. You have volunteered to rescue the princess. If the tower window is 36 feet above the ground and you must place your ladder 10 feet from the base of the castle (because of the moat), which choice is the shortest length ladder you will need to reach the tower window? 34feet 35 feet 37 feet 38 feet Choose: 2. A baseball diamond is a square with sides of 90 feet. What is the shortest distance, to the nearest tenth of a foot, between home plate and second base? 90.0 127.3 180.0 180.7 Page 4 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Choose: 3. The sides of a triangle measure 2.4, 3.2, and 4. Is this triangle a right triangle? YES NO Choose: 4. waddle swim Daisy Duck has a nest on the edge of the pond. From her favorite feeding spot, she can either waddle on land around the pond to the nest (80 meters by 60 meters), or she can swim across the pond to the nest. Daisy waddles more quickly than she swims. She waddles at the rate of 30 m/min and she swims at the rate of 20 m/min. Which route is quicker to travel from the feeding spot to the nest? Waddling on land or swimming in the pond? Page 5 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Choose: 5. Using the Pythagorean Theorem, find the area of an equilateral triangle whose side measures 5 units. Find the area to the nearest tenth of a square unit. 4.3 6.5 10.8 12.5 Choose: 6. 5 ft 10 ft 25 ft 35 ft Joe Bean regularly takes a short-cut across Mr. Wilson's lawn instead of walking on the sidewalk on his way home from school. How much distance is saved by Joe cutting across the lawn? Page 6 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Choose: 7. Find x. Choose: 8. 3 4 Find the value of x. 5 6 Page 7 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] 9. A spider has taken up Choose: residence in a small cardboard box which measures 2 inches by 4 inches by 4 inches. What is the length, in inches, of a straight spider web that will carry the spider from the lower right front corner of the box to the upper left back corner of the box? Page 8 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 9 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 10 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 11 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 12 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Theorems Related with Area Figures on the Same Base and Between the Same Parallels Page 13 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 14 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 15 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 16 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 17 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] 1. If BC=QR=10cm AB=4cm find AREA of PQR 2. The area of ∆ABC is given to be 18 cm2. If the altitude DL equals 4.5 cm, find the base of the ∆BCD. Page 18 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] 3. If perimeter of triangle BCD=12cm ,BD=CD=3cm and altitude of triangle ABC =4cm find area of triangle ABC 4. ABCD is parallelogram. The area of ∆ABC is 14cm2, AD=7cm Find DL? 5. If the area of ∆ABD is 20cm2 and DE= 4cm find the perimeter of ∆PQR? If ∆PQR is equilateral triangle. 6. The area of triangle BCD=44cm,BC =11cm a. Find altitude of triangle b. Find area of parallelogram BCED Page 19 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Theorem: Parallelograms on the same base and between the same parallelsare equal in area. Page 20 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 21 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] 9. Page 22 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 23 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 24 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Chords of a Circle one and only one circle can pass through three non-collinear points a straight line drawn from the center of a circle to bisect a chord which isnot a diameter is perpendicular to the chord perpendicular from the centre of a circle on a chord bisects it if two chords of a circle are congruent then they will be equidistant fromthe centre two chords of a circle which are equidistant from the centre are congruent Page 25 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] 28.1.2 draw a circle passing through three given non – collinear points; Circle through 3 Points Printable instructions worksheet. After doing this Your work should look like this We start with three given points. We will construct a circle that passes through all three. 1. (Optional*) Draw straight lines to create the line segments AB and BC. Any two pairs of the points will work. * We draw the two lines to make it clear when we later draw their perpendicular bisectors, but it is not strictly necessary for them to actually be there to do this. 2. Find the perpendicular bisector of one of the lines. See Constructing the Perpendicular Bisector of a Line Segment. Page 26 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] After doing this Your work should look like this 3. Repeat for the other line. 4. The point where these two perpendiculars intersect is the center of the circle we desire. 5. Place the compass point on the intersection of the perpendiculars and set the compass width to one of the points A,B or C. Draw a circle that will pass through all three. 6. Done. The circle drawn is the only circle that will pass through all three points. Page 27 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Activity 2 •A straight line drawn from the centre of a circle to bisect a chord which is not a diameter is perpendicular to the chord converse •perpendicular from the centre of a circle on a chord bisects it • Step1. Drown of circle of any radius. Step 2. Draw a chord of any measurement on this circle . Step 3. Find the midpoint of chord Step 4. Join a line segment from centre to mid point. Page 28 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Activity 3 If two chords of a circle are congruent then they will be equidistant from The centre Converse •Two chords of a circle which are equidistant from the centre are congruent Step1. Drown of circle of any radius. Step 2. Draw two chord of same Step 3.Join measurement on this circle . line segment OA and OB from centre to these two chords . B OA=OB 0 A Page 29 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Worksheet 7.2: Chord Properties Math 11 1. For each item, find the required lengths, to the nearest tenth. a. Determine AO and AB. b. Determine EF and CD. c. Determine OA and CD. d. Determine OC and AB. e. Determine OC and AB. f. Determine AB and FD. g. Determine OA and CD. h. Determine OE. Page 30 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] 2. A chord that is 10 cm long is 12 cm from the centre of a circle. Find the length of the radius. 3. The diameter of a circle is 20 cm. A chord is 8 cm from the centre. What is the length of the chord? 4. A chord is 12 cm long and the diameter of the circle is 16 cm. What is the distance between the chord and the centre of the circle? 1. Given: Circle O CD = 16 AB = 16 OB = 10 marked perpendiculars Choose: 4.5 6 7.5 Find OF. 2. 10 Choose: Given: Circle O, diameter, marked perpendicular Find x. 2.5 5 6.5 10 Page 31 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] 3. Choose: Given: Circle O, marked perpendiculars, hash marks indicating congruent Find x. 4 8 10 16 Choose: 4. Given: Circle O, AB = CD, marked perpendiculars Find x. 6 9 15 16 Choose: 5. Given: Circle O, marked perpendicular 26 Find AB. 40 32 64 Page 32 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 33 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 34 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Tangent to a Circle if a line is drawn perpendicular to a radial segment of a circle at its outer end point it is tangent to the circle at that point. the tangent to a circle and the radial segment joining the point of contact and the Centre are perpendicular to each other. the two tangents drawn to a circle from a point outside it are equal in length. if two circles touch externally or internally the distance between their centers is respectively equal to the sum or difference of their radii Page 35 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point. If you spin an object in a circular orbit and release it, it will travel on a path that is tangent to the circular orbit. Theorem: If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency. Tangent segments to a circle from the same external point are congruent. Theorem: (You may think of this as the "Hat" Theorem because the diagram looks like a circle wearing a pointed hat.) This theorem can be proven using congruent triangles and the previous theorem. The triangles shown below are congruent by the Hypotenuse Leg Postulate for Right Triangles. The radii (legs) are congruent and the hypotenuse is shared by both triangles. By using Corresponding Parts of Congruent Triangles are Congruent, this theorem is proven true. Page 36 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Common Tangents: Common tangents are lines or segments that are tangent to more than one circle at the same time. 4 Common Tangents 3 Common Tangents 2 Common Tangents (2 completely separate circles) (2 externally tangent circles) (2 overlapping circles) 2 external tangents (blue) 2 internal tangents (black) 2 external tangents (blue) 1 internal tangent (black) 2 external tangents (blue) 0 internal tangents 1 Common Tangent (2 internally tangent circles) 0 Common Tangents (2 concentric circles) Concentric circles are circles with the same center. 1 external tangent (blue) 0 internal tangents 0 external tangents 0 internal tangents (one circle floating inside the other, without touching) 0 external tangents 0 internal tangents Page 37 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Step-by-step Instructions After doing this Your work should look like this We start with a point P somewhere on a given circle, with center point O. If the center is not given, you can use: "Finding the center of a circle with compass and straightedge or ruler", or "Finding the center of a circle with any right-angled object". 1. Draw a straight line from the center O, through the given point P and on beyond P. . 2. Put the compass point on P and set it to any width less than the distance OP. Then, on the line just drawn, draw an arc on each side of P. This creates the points Q and R as shown. Page 38 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] After doing this Your work should look like this 3. Set the compass on Q and set it to any width greater than the distance QP. 4. Without changing the compass width, draw an arc approximately in the position shown on one side of P. 5. Without changing the compass width, move the compass to R and make another arc across the first, creating point S. Page 39 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] After doing this Your work should look like this 6. Draw a line through P and S. 7. Done. The line PS just drawn is the tangent to the circle O through point P. Page 40 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] After doing this Your work should look like this We start with a given circle with center O, and a point P outside the circle. 1. Draw a straight line between the center O of the given circle and the given point P. 2. Find the midpoint of this line by constructing the line's perpendicular bisector. The midpoint may be inside or outside the circle, depending on the circle size and the location of the given point. Page 41 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] After doing this Your work should look like this 3. Place the compass on the midpoint just constructed, and set it's width to the center O of the circle. 4. Without changing the width, draw an arc across the circle in the two possible places. These are the contact points J, K for the tangents. 5. Draw the two tangent lines from P through J and K. Page 42 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] After doing this Your work should look like this 6. Done. The two lines just drawn are tangential to the given circle and pass through P. Page 43 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] ` 𝐼𝑛 𝑓𝑖𝑔𝑢𝑟𝑒 1 Page 44 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Work sheet Page 45 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 46 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 47 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 48 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Work sheet Page 49 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 50 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 51 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 52 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Answer the following questions dealing with the tangents and circles. (Do not assume that diagrams are drawn to scale.) 1. Choose: 7.5 12 15 30 Page 53 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] 2. Choose: 21 16 8 4 3. Choose: 15 14 12 9 4. Choose: 12 8 4 2 Page 54 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] 5. Choose: 5 6 7 9 6. Choose: 5 10 12 15 Choose: 7. YES NO (OB = 15) Page 55 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] 8. a = OA; b = OB; c = CB Find a, b, c. Choose: a= Choose: b= Choose: 9 16 12 12 18 14 14 20 16 16 24 20 9. c= Choose: YES NO Page 56 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 57 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 58 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Chords and Arcs If two arcs of a circle (or of congruent circles) are congruent then the corresponding chords are equal. If two chords of circle (or of congruent circles) are equal, then their corresponding arcs (minor, major or semi – circular) are congruent. Equal chords of a circle (or of congruent circles) subtend equal angles atthe centre (at the corresponding centers). If the angles subtended by two chords of a circle (or congruent circles) atthe centre (corresponding centers) are equal, the chords are equal. Page 59 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Arcs in Circles An arc is part of a circle's circumference. Definition: In a circle, the degree measure of an arc is equal to the measure of the central angle that intercepts the arc. In a circle, the length of an arc is a portion of the circumference. Definition: Remembering that the arc measure is the measure of the central angle, a definition can be formed as: Example: In circle O, the radius is 8, and the measure of minor arc Find the length of minor arc is 110 degrees. to the nearest integer. Solution: = 15.35889742 = 15 Understanding how an arc is measured makes the next theorems common sense. Page 60 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Theorem: In the same circle, or congruent circles, congruent central angles have congruent arcs. Theorem: In the same circle, or congruent circles, congruent arcs have congruent central angles. (converse) Remember: In the same circle, or congruent circles, congruent arcs have congruent chords. Knowing this theorem makes the next theorems seem straight forward. Theorem: In the same circle, or congruent circles, congruent central angles have congruent chords. Theorem: In the same circle, or congruent circles, congruent chords have congruent central angles. (converse) Solve the following problems dealing with angles in circles. (Do not assume diagrams are drawn to scale.) Choose: 1. Given the labeled diagram at the left, with diameter . Find x. 59 44 43 34 Page 61 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Choose: 2. Given circle with center indicated. Find x. 55 70 110 290 Choose: 3. Given circle with center indicated. 36 Find x. 90 54 108 Choose: 4. Given diameter. Find x. 28 56 62 124 Page 62 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] 1. Page 63 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 64 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 65 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 66 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 67 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Page 68 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Angle in a Segment of a Circle . Circle Theorems Student Sheet: Angle in centre is twice the angle at circumference 1 2 C B A 38 O Angle BAC is 52º. Find angles BOC and BCO Angle AOB is 130º Find angles ACB and OBC 3 4 c O 71 b a OB=BC. Calculate the size of angle CAB. Find the angles a, b and c. Page 69 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] 5 6 Q P O 56 R Find the angle ADC. S Find angle QSR. Circle Theorems Student Sheet: Angles in the same segment are equal 1 2 A C B 44 A 33 O 45 B 52 D C D Find The Angles CBD and ADB. Find the angles ACD, AOD and ADC. 3 4 N L 56 C 132 b O 42 P M a Page 70 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Find the angles MNO and LMN. 5 Calculate the angles a, b and c. 6 Calculate the angles BCD and CAD. Calculate the angles PQR and QRS. Page 71 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] Circle Theorems Student Sheet: Angle in a semi-circle is 90° 1 2 Z X Y 46 O Find the value of y Find angles XOZ, OZY, OZX and show that XZY is 90º 3 4 A M O C Find angles QPT and RTP in terms of x 26 B M is the midpoint of AC. Find the angle MOC Page 72 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] 5 6 D 23 A C B O Find the angles e and d Calculate the size of the angle ADC. Circle Theorems Students Sheet: Cyclic Quadrilaterals 1 2 D X 61 100 Y E 109 81 38 81 Z V W C 80 Calculate the sizes of angles XWV and ZXW. 100 B A Calculate the sizes of angles CBE and EBA. Page 73 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected] 3 4 Q A R 108 44 D P 44 108 72 T 80 S 56 B C Calculate the sizes of angles TPR and TQR. 5 Calculate the sizes of angles ACB and CAB. 6 Z Z 88 63 20 Y A 64 28 116 20 X 28 234 44 O 126 24 Y W 117 V W Calculate the sizes of angles ZVY, XZY and YWV. X Calculate the sizes of angles WOY, WXY and OYX. Page 74 of 74 MADE BY ARSLAN SHAIKH AGA KHAN SCHOOL HYDERABAD [email protected]