1. 8 Angles in Rectilinear Figures
8.1 Angles of Triangles
8.2 Special Triangles
8.3 Sum of Interior Angles of Polygons
8.4 Sum of Exterior Angles of Polygons
8.5 Tessellation of a Plane
8.6 Constructions of Regular Polygons

2. 8.1 Angles of Triangles
A. Interior Angles of a Triangle

3. Example 1T Solution: 8 Angles in Rectilinear Figures
In the figure, QSR is a straight line. Find x and y.
Solution:
In PQR,
PQR  QPR  PRQ  180 ( sum of )
y  90  25 180
In PQS,
PQS  QPS  PSQ  180 ( sum of )
65  75  x  180

4. Example 2T Solution: 8 Angles in Rectilinear Figures
In the figure, RST is a right-angled triangle. QP // ST. RTS = 36 and SRT = 90 . Find x and y.
Solution:
In RST,
RST  SRT  RTS  180 ( sum of )
(y + y)  90  36 180
2y  54
QST  SQP = 180 (int. s, QP // ST)
(27 + 27) + x  180
54 + x  180

5. 8.1 Angles of Triangles
B. Exterior Angles of a Triangle

6. Example 3T Solution: 8 Angles in Rectilinear Figures
In the figure, BDC and EAC are straight lines. Find x and y.
Solution:
In ABD,
ADC  ABD  BAD (ext.  of )
x  62  26
In ABC,
BAE  ABC  ACB (ext.  of )
90  62  y

7. Example 4T Solution: 8 Angles in Rectilinear Figures
In the figure, DCG is a straight line. BF // DE and AD // GF, BCG = 165 and CFG = 120. Find x and y.
Solution:
DBC = GFC (alt. s, DA // FG)
= 120
In BCD,
BDC  DBC  BCG (ext.  of )
x  120  165
DBC  BDE  180 (int. s, BF // DE)
120  (45  y)  180

8. Example 5T Solution: 8 Angles in Rectilinear Figures
In the figure, find z.
Solution:
Produce UV to meet TW at S.
In TUS,
WSV = UTS  SUT (ext.  of )
= z  (4z – 160)
= 5z – 160
In SWV,
UVW = WSV  SWV (ext.  of )
3z = (5z – 160) + (z – 5)
3z = 165

9. 8.2 Special Triangles
A. Isosceles Triangles

10. 8.2 Special Triangles
A. Isosceles Triangles

11. Example 6T Solution: 8 Angles in Rectilinear Figures
In the figure, AC = AD = BD, BDA = 100 and BAC = 90. Find x and y.
Solution:
In ABD,
ABD  BAD  BDA  180 ( sum of )
x  x  100  180

12. Example 6T Solution: 8 Angles in Rectilinear Figures
In the figure, AC = AD = BD, BDA = 100 and BAC = 90. Find x and y.
Solution:
In ACD,

13. Example 7T Solution: 8 Angles in Rectilinear Figures
In the figure, PQRS is a rhombus. The diagonals bisect each other and meet at T. TQR = 32. Find TRS.
Solution:
In QRS,
= 32
 RT  QS (property of isos. )
In RST,

14. Example 8T Solution: 8 Angles in Rectilinear Figures
In ABC, BA = BC, ACD = BCD, AB // EF and ABC = 48. Find EGC.
Solution:
In ABC,
In CFG,

15. 8.2 Special Triangles
A. Isosceles Triangles

16. Example 9T Solution: 8 Angles in Rectilinear Figures
In the figure, TW  UV and WV = 5 cm.
(a) Find x.
(b) Are TU and TV of the same length?
(c) Find UW.
Solution:
(a) In TUV,

17. Example 9T Solution: 8 Angles in Rectilinear Figures
In the figure, TW  UV and WV = 5 cm.
(a) Find x.
(b) Are TU and TV of the same length?
(c) Find UW.
Solution:
(b) In TUV,
(c) In TUV,

18. 8.2 Special Triangles
B. Equilateral Triangles

19. Example 10T Solution: 8 Angles in Rectilinear Figures
In the figure, ABC is an equilateral triangle. AD and BE intersect at F. AD  BC and BE  AC. Find AFB.
Solution:
In ABC,
In AEF,

20. Example 11T Solution: 8 Angles in Rectilinear Figures
In the figure, PQR is an equilateral triangle and SQR is an isosceles triangle with SQ = SR. PQS = 20. Find PSR.
Solution:
Join PS.
 QPS  RPS and PQS  PRS  20
QPR  60 (property of equil. )
 2RPS  60
RPS  30
In PRS,
PSR  PRS  RPS  180 ( sum of )
PSR  20  30  180

21. 8.3 Sum of Interior Angles of Polygons

22. 8.3 Sum of Interior Angles of Polygons

23. Example 12T Solution: 8 Angles in Rectilinear Figures
For a regular 16-sided polygon, find
(a) the sum of the interior angles,
(b) the size of each interior angle.
Solution:
(a) Sum of the interior angles
= (16 – 2)  180 ( sum of polygon)
= 14  180
(b) Size of each interior angle

24. Example 13T Solution: 8 Angles in Rectilinear Figures
In the figure, AED = EDC = DCB = 90. Find a.
Solution:
2a  2a  90  90  90  (5 – 2)  180 ( sum of polygon)
4a + 270 = 180
4a = 270

25. Example 14T Solution: 8 Angles in Rectilinear Figures
The sum of the interior angles of a regular polygon is 2880. Find the number of sides of the polygon.
Solution:
Let n be the number of sides of the polygon.
( sum of polygon)

26. Example 15T Solution: 8 Angles in Rectilinear Figures
Each interior angle of a regular polygon is a right angle. Find the number of sides of the polygon.
Solution:
Let N be the number of sides of the polygon.
Sum of the interior angles = ( sum of polygon)

27. 8.4 Sum of Exterior Angles of Polygons
At each vertex, there are two exterior angles. v, w, x, y and z are a different set of exterior angles.

28. Example 16T Solution: 8 Angles in Rectilinear Figures
The figure shows a pentagon ABCDE with its sides produced. FAB = 110, GBA  70, JEF  60 and JDH  135. Find z.
Solution:
Produce DH to K.
JDK  135 = 180 (adj. s on st. line)
JDK = 45
110 + 60 +45 + z + 70 = 360 (sum of ext. s of polygon)

29. Example 17T Solution: 8 Angles in Rectilinear Figures
If the size of an interior angle of a regular polygon is 4 times that of an exterior angle, find the number of sides of the polygon.
Solution:
Let N be the number of sides of the polygon.
Sum of the interior angles = (N – 2)  180 ( sum of polygon)
(sum of ext. s of polygon)

30. 8.5 Tessellation of a Plane

31. 8.5 Tessellation of a Plane

32. 8.6 Constructions of Regular Polygons
A. Using a Protractor to Construct Regular Polygons

33. 8.6 Constructions of Regular Polygons
B. Using a Pair of Compasses to Construct Regular Polygons

34. 8.6 Constructions of Regular Polygons
B. Using a Pair of Compasses to Construct Regular Polygons

35. 8.6 Constructions of Regular Polygons
B. Using a Pair of Compasses to Construct Regular Polygons

36. 8.6 Constructions of Regular Polygons
B. Using a Pair of Compasses to Construct Regular Polygons

37. 8.6 Constructions of Regular Polygons
B. Using a Pair of Compasses to Construct Regular Polygons

38. Follow-up 1 Solution: 8 Angles in Rectilinear Figures
In the figure, PSQ is a straight line. Find x and y.
Solution:
In PQR,
PQR  PRQ  QPR  180 ( sum of )
62  (x  18)  50  180
x  130  180
In PSR,
PSR  PRS  SPR  180 ( sum of )
y  50  50  180

39. Follow-up 2 Solution: 8 Angles in Rectilinear Figures
In the figure, B and C are points on AD and AE respectively such that BC // DE. Find x and y.
Solution:
BCE  CED  180 (int. s, BC // DE)
ACB  CED (corr. s, BC // DE)
 45  40
In ABC,
= 85

40. Follow-up 3 Solution: 8 Angles in Rectilinear Figures
In the figure, ABC and CDE are straight lines. AD and BC intersect at F. Find x and y.
Solution:
In BCE,
ABE = BCE  BEC (ext.  of )
In ABF,
AFE = BAF  ABF (ext.  of )

41. Follow-up 4 Solution: 8 Angles in Rectilinear Figures
In the figure, QP //ST. PS and QT intersect at R. If PQR = 50, find y.
Solution:
RTS = PQR (alt. s, QP // ST)
= 50
In RST,
QRS = RTS  RST (ext.  of )

42. Follow-up 5 Solution: 8 Angles in Rectilinear Figures
In the figure, ABC = 72, BAD = 56 and ADC = 164. Find y.
Solution:
Produce AD to meet BC at E.
In ABE,
DEC = BAE + ABE (ext.  of )
In DEC,
ADC = DEC + DCE (ext.  of )

43. Follow-up 6 Solution: 8 Angles in Rectilinear Figures
In the figure, D is a point on BC such that BD = DC = AD and ACD = 28. Find x.
Solution:
In ACD,
In ABC,
In ABD,
x  (x  28)  28  180
2x  124

44. Follow-up 7 Solution: 8 Angles in Rectilinear Figures
In the figure, ABCD is a square. Its diagonals bisect each other and intersect at E (that is, AE = BE = CE = DE). Find BEC and EBC.
Solution:
In ABC,
EBC  EBA  90
2EBC  90

45. Follow-up 8 Solution: 8 Angles in Rectilinear Figures
In the figure, RS is an angle bisector of PRQ and intersects PQ at S. PQ = PR and SPR = 72. Find QSR.
Solution:
In PQR,
PQR = PRQ (base s, isos. )
QPR  PQR  PRQ = 180 ( sum of )
In PRS,
QSR = SPR  PRS (ext.  of )

46. Follow-up 9 Solution: 8 Angles in Rectilinear Figures
In the figure, RSQ is a straight line. PS = RS and PSQ = 130.
(a) Find a and b.
(b) Is PQS an isosceles triangle? Explain your answer.
Solution:
(a)
RPS = PRS (base s, isos. )
= a
RPS  PRS = PSQ (ext.  of )
In PSQ,
SPQ  PQS  PSQ = 180 ( sum of )

47. Follow-up 9 Solution: 8 Angles in Rectilinear Figures
In the figure, RSQ is a straight line. PS = RS and PSQ = 130.
(a) Find a and b.
(b) Is PQS an isosceles triangle? Explain your answer.
Solution:
(b)
In PQS,
QPS = 65 – 35 = 30
PQS = 20
PSQ = 130
 All the angles in the triangle are unequal.
 PQS is not an isosceles triangle.

48. Follow-up 10 Solution: 8 Angles in Rectilinear Figures
In the figure, XW  YZ. XY = XZ = 6 cm and XYZ = 60.
Find y and z.
Solution:
In XYW,
XYW  XWY  YXW  180 ( sum of )
60  90  y  180

49. Follow-up 10 Solution: 8 Angles in Rectilinear Figures
In the figure, XW  YZ. XY = XZ = 6 cm and XYZ = 60.
Find y and z.
Solution:
In XYZ,
 XZY  XYZ (base s, isos. )
 60
WZ  WY (property of isos. )
 z cm
and ZXW  YXW (property of isos. )
 30
XYZ  YXZ  XZY  60
 XYZ is an equilateral triangle.
 YXZ  30  30  60

50. Follow-up 11 Solution: 8 Angles in Rectilinear Figures
In the figure, ABC is an equilateral triangle. BC is produced to D such that CA = CD. Is BAD a right angle? Explain your answer.
Solution:
In ABC,
ACB = BAC = 60 (property of equil. )
In ACD,
CAD = CDA (base , isos. )
(ext.  of )

51. Follow-up 12 Solution: 8 Angles in Rectilinear Figures
Find the size of each interior angle of a regular octagon.
Solution:
Sum of the interior angles
= (8 – 2)  180 ( sum of polygon)
= 6  180
= 1080

52. Follow-up 13 Solution: 8 Angles in Rectilinear Figures
In the figure, PQR = PUT = 90 and UTS = 140. Find x.
Solution:

53. Follow-up 14 Solution: 8 Angles in Rectilinear Figures
The sum of the interior angles of a regular polygon is 1980.
Find the number of sides of the polygon.
Solution:
Let n be the number of sides of the polygon.
( sum of polygon)

54. Follow-up 15 Solution: 8 Angles in Rectilinear Figures
Each interior angle of a regular polygon is 162. Find the number of sides of the polygon.
Solution:
Let N be the number of sides of the polygon.
Sum of the interior angles = ( sum of polygon)

55. Follow-up 16 Solution: 8 Angles in Rectilinear Figures
The figure shows a pentagon ABCDE with its sides produced. BAE  100. Find y.
Solution:
FAB  100 = 180 (adj. s on st. line)
FAB = 80
80  2y  3y  3y  2y = 360 (sum of ext. s of polygon)

56. Follow-up 17 Solution: 8 Angles in Rectilinear Figures
If the sum of the interior angles and the sum of the exterior angles of a polygon are equal, find the number of sides of the polygon.
Solution:
Let n be the number of sides of the polygon.
Sum of the interior angles = (n – 2)  180 ( sum of polygon)
(sum of ext. s of polygon)