MEP Practice Book ES3. 1. Calculate the size of the angles marked with a letter in each diagram. None to scale

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1 ME rctice ook ES3 3 ngle Geometr 3.3 ngle Geometr 1. lculte the size of the ngles mrked with letter in ech digrm. None to scle () 70 () c 36 (d) (e) (f) d e f 30 g (g) (h) j (i) k 33 h 22 i 35 l (j) (k) (l) m n o p (m) (n) s (o) q r v 33 u t Find the ngles mrked with letter in ech rectngle elow. () () c 25

2 ME rctice ook ES3 3. he frmework of smmetricl roof is illustrted elow. is perpendiculr to. c 40 Find the size of the ngles mrked, nd c. 4. Write down n eqution tht is stisfied in ech of the following digrms. In ech cse, solve the eqution for. 2 () () (d) (e) (f) Q.5 Q nd RS re stright lines. Work out the vlue of. R Q 125 Not drwn ccurtel 34 S (Q) 26

3 ME rctice ook ES3 6. his tringle hs two equl sides. () Wht nme is given to this tpe of tringle? () Find the vlues of nd. 70 Not drwn ccurtel (Q) 7. is n isosceles tringle. E is kite. Work out the vlue of. 40 E Not drwn ccurtel 40 (Q) 3.4 ngles with rllel nd Intersecting Lines 1. lculte the unknown ngles in the following digrms. () () e c d 69 (d) (e) (f) f g h º (g) (h) (i) 63 f p e d 27

4 ME rctice ook ES3 (j) (k) (l) 67 e 25 m 85 e 80 f 57 g (m) (n) (o) s (p) (q) (r) p 69 e c (s) (t) For ech digrm, find n eqution in, nd hence solve for. () ()

5 ME rctice ook ES3 3. Find the vlues of the unknown ngles in ech of the following. () 126 () c (d) (e) 4e 3e 5e f (f) 72 g is rhomus. 27 ngle = 27 he digonls nd cross t. lculte the size of the ngle. 5. he pentgon E is the frme for Irhim's mountin ike. E is n isosceles tringle in which Not to scle = nd ngle = 65. In the qudrilterl E ngle = 70, ngle E = 90 nd is prllel to E. () (i) lculte the size of ngle. (ii) Wht fcts out the ngles of tringle did ou use in our clcultion? () lculte the size of the ngle E (MEG) 29

6 ME rctice ook ES3 6. he lines nd re prllel. r p 40 Not drwn ccurtel q 50 () () Write down the vlue of p. Give reson for our nswer. Write down the vlue of q. Give reson for our nswer. Work out the vlue of r. (Q) 7. QR nd SUV re prllel stright lines. 38 Q R igrm not ccurtel drwn S U V Work out the vlue of the ngle mrked. Give resons for our nswer. (Edecel) 3.5 ngle Smmetr in olgons 1. Find the sum of the interior ngles of () qudrilterl () pentgon. 2. Find the size of ech interior ngle of () regulr hegon () regulr nongon. 3. Find the numer of sides of polgon if the sum of its interior ngles is () 1800 () Ech interior ngle of regulr polgon is 140. Find the numer of sides of the polgon. 5. Ech interior ngle of regulr n-gon is 168. Wht is the vlue of n? 30

7 ME rctice ook ES3 6. Find the vlue of in ech of the following digrms. () 110 () (d) (e) (f) he ngles of qudrilteril re 3, 4, 5 nd 6. () Find. () Wht re the ngles in degrees? 8. () For ech digrm ove, show three different ws of shding prts of the shpes so tht the hve line smmetr ut no rottionl smmetr. () Shde sections of one shpe so tht it hs rottionl smmetr of order 2 ut no lines of smmetr. Is it possile to do this for oth shpes? Repet () for rottionl smmetr of order 3. (d) Repet () for rottionl smmetr of order () regulr polgon hs n interior ngle of 175. How mn sides does it hve? () second regulr polgon hs n interior ngle which is 1 smller. How mn sides does it hve? Is it possile for regulr polgon to hve n interior ngle of 173? 31

8 ME rctice ook ES3 10. () he digrm shows prt of tiling pttern of regulr pentgons nd nother shpe. 108 (i) Which of the following correctl descries the shded shpe: squre, rhomus, trpezium, rectngle, prllelogrm, kite? (ii) lculte the size of the ngle mrked. (iii) regulr pentgon hs rottionl smmetr. Wht is the order of rottionl smmetr of regulr pentgon? () nother tiling pttern is formed using regulr octgons nd squres, s shown. lculte the size of the ngle mrked. rw tiling pttern using regulr hegons onl. You must drw t lest five hegons. (SEG 11. he digrm shows prt of regulr polgon. Ech interior ngle is 144. Not drwn ccurtel 144 lculte the size of the eterior ngle of the polgon. (Q) 32

9 ME rctice ook ES3 12. he digrm shows 6-sided shpe, EF. ll the sides of the shpe re equl in length. F E igrm not ccurtel drwn () (i) Find the vlue of. (ii) Give reson for our nswer. () Work out the vlue of. (Edecel) 13. () is qudrilterl. he side is etended to E. Work out the vlue of Not drwn ccurtel 125 E () lculte the size of the eterior ngle of regulr hegon. Not drwn ccurtel (Q) 3.6 Smmetr roperties of 3 Shpes 1. he following solids hve rottionl smmetr. For ech of them, stte the order of rottionl smmetr out the given is. () () 33

10 ME rctice ook ES3 2. For ech of the following prisms, drw n is so tht the order of rottionl smmetr out tht is is 2. () () 3. In the given prism, the cross-section is in the shpe of regulr pentgon. rw () n is S so tht the order of rottionl smmetr out S is 2; () n is XY so tht the order of rottionl smmetr out XY is Stte the order of rottionl smmetr out ech of the es shown. ll the 12 edges of the solid re equl in length. () () 5. R S For the solid ove, find the order of its rottionl smmetr out () Q () RS. Q 34

11 ME rctice ook ES3 6. () cue hs 9 plnes of smmetr. rw digrms to show these plnes. () cue hs 3 es of rottionl smmetr of order 4. rw digrms to show these es. he digrm of cue opposite shows one is of rottionl smmetr of order 3. here re 3 other es with the sme order. rw digrms to show these es. (d) here re 6 es with smmetr of order 2. rw digrms to show these es. 7. rw solid tht hs one is of smmetr nd rottionl smmetr of order 5 out the is. 3.7 ompss erings 1. he mp elow shows the positions of some villges. rgfoot Hrcome N W E S Sheepwsh ove West Leigh rtton Scle: 2 miles to 1 cm () () (d) (e) Which villge is due north of Sheepwsh? Which villge is due west of ove? Wht is the compss direction of Sheepwsh from West Leigh? How mn miles is (i) rtton from ove (ii) Hrcome from rgfoot? Mke trcing of the mp nd mrk the positions of (i) rle, which is 3 miles due south of Hrcome, (ii) Lee, which is 4 miles south est of rgfoot. 35

12 ME rctice ook ES3 2. For ech of the following, write down the ering of from. () North () North North (d) North Wht is the ering of () Q from () from Q? North 106 North Q 4. Wht is the ering of () from S () S from? S rw digrm with 4 towns mrked, so tht tht three of the towns re equidistnt from the fourth town,, nd hve erings from of () 036 () North Z 135 field is in the shpe of squre, with corners W, X, Y nd Z. he ering of Y from Z is 135. Find the ering of W Y () Y from X () W from Z. X 7. Wht is the ering of () Q from () from from (d) from? 25 North 55 36

13 ME rctice ook ES3 8. he figure shows the positions of, Q nd R. North Wht is the ering of () Q from () from Q R from (d) from R (e) Q from R (f) R from Q? 9. point is 280 m due North of the point. mn wlks from in the direction 050. lculte how fr he wlks efore he is Q 100 R () equidistnt from nd, () s close s possile to, due est of. 10. he digrm shows scle drwing of one side,, of tringulr field,. N Scle: 1 cm represents 50 m () Use the digrm to clculte the ctul distnce fron to. () Mesure nd write down the three figure ering of from. he ering of from is 130. he ctul distnce from to is 350 metres. Mrk the point on cop of the digrm. (Q) 37

14 ME rctice ook ES3 3.8 ngles nd ircles 1 1. Find the ngles mrked with letter in ech of the following digrms. (In ech cse is the centre of the circle.) () 55 () c (d) e d c d e 72 c 57 (e) (f) c c 2. Find the ngles mrked with letter in ech digrm elow. (In ech cse is the centre of the circle.) () 30 is tngent 38

15 ME rctice ook ES3 () 70 nd re tngents 3. Find the ngles mrked with letters in ech of the following digrms. (In ech cse is the centre of the circle.) () () (d) Find the dimeter of ech circle elow. (In ech cse is the centre of the circle. () 8 6 ()

16 ME rctice ook ES3 5. In the digrm, lines nd E re prllel. E is dimeter of the circle, centre. ngle E = E () Find the size of (i) ngle z ii) ngle () Write down the size of ngle z. Give reson for our nswer. Not to scle (R) 3.9 ngles nd ircles 2 1. In ech of the following circles, find the ngles mrked with letters. () () 22 d 28 f c 48 h g 2. In ech of the following circles, is the centre. Find the ngles mrked with letter. () () p In ech of the following figures, find the vlue of. In ech cse, is the centre of the circle. () ()

17 ME rctice ook ES3 (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r)

18 ME rctice ook ES3 4. In the digrm, is dimeter of the circle. Given tht ngle = 24 nd ngle = 35, 24 find ngle QX. Q X In the digrm, nd Find X ˆ. ˆ = 25 ˆ = X In the digrm, ˆ = 54 ˆ = nd ˆ = 80. Find ˆ In the digrm, is the centre of the circle. d intersect outside the circle t. Given tht ˆ = nd ˆ = 32, find ˆ. 8. In the digrm, is the centre of the circle. hords nd intersect inside the circle t. is perpendiculr to. Given tht ˆ = 81 nd ˆ = 59, find ˆ. 42

19 ME rctice ook ES3 9. In the digrm, = = 54 nd ˆ = 54. Find the vlue of ˆ. 10. In the digrm, ˆ = 65 ˆ = nd Find ˆ. ˆ = Q R In the digrm, is the centre of the circle 40 S nd RS ˆ = 40. lculte QR ˆ nd RS ˆ. 12. In the figure, is the centre of the circle,. Given tht ˆ = 20 nd ˆ = 30, find ˆ ,, nd re points on circle. is equl in length nd prllel to. Lines nd intersect t E. ngle E = 35 () Not to scle Write down the size of ngle E. Give reson for our nswer. E 35 () (i) Find the size of ngle E. Show ll our working clerl. (ii) Wht does this tell ou out point E? Give reson for our nswer. (R) 43

20 ME rctice ook ES ircles nd ngents 1. Given tht is tngent to the circle with centre, find the vlues of, nd z. () () z (d) z 36 (e) (f) In the digrm, is the tngent to the circle t nd X is dimeter. Given tht Q ˆ = 42, find ˆQX, XQ ˆ nd XQ ˆ. 30 Q 42 44

21 ME rctice ook ES3 3. In the digrm, is the centre of the circle. is the tngent to the circle t X, X ˆ = 60 nd X ˆ = 22. Wht is the size of X ˆ? 22 X In the digrm, is the tngent to the circle t point. Given tht NM ˆ = 30 nd M ˆ = 97, find M ˆ. N M 5. Given tht is tngent t to the circle with the centre, find the vlue of nd of in ech cse. () () 25 (d)

22 ME rctice ook ES3 (e) (f) (g) (h) (i) Given tht nd re tngents to the circle with centre, find the vlue of nd of in ech cse. () 20 46

23 ME rctice ook ES3 () (d) 3 cm cm 50 (e) 72 (f)

24 ME rctice ook ES3 7. Find the length in ech cse. () () 2.4 cm 3.2 cm 1.7 cm 5.2 cm 1.8 cm 3.8 cm 6 cm is tngent. 8 cm (d) 7 cm 3 cm is tngent. 8. If = 8 cm, = 7 cm nd = 9 cm, clculte the lengths of () chord () tngent. 48

25 ME rctice ook ES3 9. S is tngent to the circle, centre. Q is point on the circumference of the circle. Q is stright line. = 26 cm nd = 24 cm. S Q igrm not ccurtel drwn () () ngle = 90 Give reson wh. Work out the rdius Q of the circle. Work out the re of the circle. Give our nswer correct to 3 significnt figures. (Edecel) 10. In the digrm,, nd re points on the circumference of circle, centre. nd re tngents to the circle. ngle = 50. () (i) Work out the size of ngle. (ii) Give reson for our nswer. () (i) Work out the size of ngle. (ii) Give reson for our nswer. 11., nd re points on the circumference of circle with centre. nd re tngents. ngle = 40. q 50 p igrm not ccurtel drwn (Edecel) Not drwn ccurtel 40 () (i) Work out the vlue of p. (ii) Hence write down the vlue of q. () he tngent is etended to. he line is dded to the digrm. ngle = Not drwn ccurtel (i) Work out the vlue of. (ii) Work out the vlue of. 40 (Q) 49

26 ME rctice ook ES3 12. () 72 Not drwn ccurtel is the centre of the circle. lculte the vlue of. () is the centre of the circle.,, nd re points on the circumference. ngle = 126 Not drwn ccurtel 126 (i) lculte the vlue of. (ii) lculte the vlue of., Q nd R re points on the circumference of the circle. N is the tngent to the circle t. Q Not drwn ccurtel 70 R z 52 N lculte the vlue of z. Give reson for ech step of our working. (Q) 50

3 Angle Geometry. 3.1 Measuring Angles. 1. Using a protractor, measure the marked angles.

3 Angle Geometry. 3.1 Measuring Angles. 1. Using a protractor, measure the marked angles. 3 ngle Geometry MEP Prtie ook S3 3.1 Mesuring ngles 1. Using protrtor, mesure the mrked ngles. () () (d) (e) (f) 2. Drw ngles with the following sizes. () 22 () 75 120 (d) 90 (e) 153 (f) 45 (g) 180 (h)