Math unit31 angles and symmetry
-
Upload
elearningja -
Category
Education
-
view
112 -
download
1
Transcript of Math unit31 angles and symmetry
![Page 1: Math unit31 angles and symmetry](https://reader033.fdocuments.us/reader033/viewer/2022042707/58f30c991a28aba35a8b457b/html5/thumbnails/1.jpg)
Unit 31Functions
Presentation 1 Line and Rotational SymmetryPresentation 2 Angle PropertiesPresentation 3 Angles in Triangles Presentation 4 Angles and Parallel Line: ResultsPresentation 5 Angles and Parallel Lines: Example Presentation 6 Angle Symmetry in Regular Polygons
![Page 2: Math unit31 angles and symmetry](https://reader033.fdocuments.us/reader033/viewer/2022042707/58f30c991a28aba35a8b457b/html5/thumbnails/2.jpg)
Unit 3131.1 Line and Rotational
Symmetry
![Page 3: Math unit31 angles and symmetry](https://reader033.fdocuments.us/reader033/viewer/2022042707/58f30c991a28aba35a8b457b/html5/thumbnails/3.jpg)
An object has rotational symmetry if it can be rotated about a point so that it fits on top of itself without completing a full turn. The number of times this can be done is the order of rotational symmetry.Shapes have line symmetry if a mirror could be placed so that one side of the shape is an exact reflection of the order.Example
Rotational symmetry of order 2
2 lines of symmetry (shown with dotted lines)
Rotational symmetry of order 3
3 lines of symmetry (shown with dotted lines)
![Page 4: Math unit31 angles and symmetry](https://reader033.fdocuments.us/reader033/viewer/2022042707/58f30c991a28aba35a8b457b/html5/thumbnails/4.jpg)
(a) 1
(b) 2
(a) 0
(b) 1
An object has rotational symmetry if it can be rotated about a point so that it fits on top of itself without completing a full turn. The number of times this can be done is the order of rotational symmetry.Shapes have line symmetry if a mirror could be placed so that one side of the shape is an exact reflection of the order.
Exercises What is (a) the order of rotational symmetry,
(b) the number of lines of symmetry of each of these shapes
(a) 2
(b) 2
(a) none
(b) 1
?
?
?
?
?
?
?
?
![Page 5: Math unit31 angles and symmetry](https://reader033.fdocuments.us/reader033/viewer/2022042707/58f30c991a28aba35a8b457b/html5/thumbnails/5.jpg)
Unit 3131.2 Angle Properties
![Page 6: Math unit31 angles and symmetry](https://reader033.fdocuments.us/reader033/viewer/2022042707/58f30c991a28aba35a8b457b/html5/thumbnails/6.jpg)
Angles at a Point
The angles at a point will always add up to 360°.It does not matter how many angles are formed at the point – their total will always be 360°
Angles on a lineAny angles that form a straight line add up to 180°
![Page 7: Math unit31 angles and symmetry](https://reader033.fdocuments.us/reader033/viewer/2022042707/58f30c991a28aba35a8b457b/html5/thumbnails/7.jpg)
Angles in a TriangleThe angles in a triangle add up to 180°
Angles in an Equilateral TriangleIn an equilateral triangle each interior angle is 60° and all the sides are the same length
![Page 8: Math unit31 angles and symmetry](https://reader033.fdocuments.us/reader033/viewer/2022042707/58f30c991a28aba35a8b457b/html5/thumbnails/8.jpg)
Angles in a Isosceles TriangleIn an isosceles triangle two sides are the same length and the two angles opposite the equal sides are the same
Angles in a quadrilateralThe angles in any quadrilateral add up to 360°
![Page 9: Math unit31 angles and symmetry](https://reader033.fdocuments.us/reader033/viewer/2022042707/58f30c991a28aba35a8b457b/html5/thumbnails/9.jpg)
Unit 3131.3 Angles in Triangles
![Page 10: Math unit31 angles and symmetry](https://reader033.fdocuments.us/reader033/viewer/2022042707/58f30c991a28aba35a8b457b/html5/thumbnails/10.jpg)
Note that the angles in any triangle sum to 180°ExampleIn this figure, ABC is an isosceles trianglewith and (a) Write an expression in terms of p for the
value of the angle at C.(b) Determine the size of EACH angle in the triangle.Solution(a)as ABC is an isosceles triangle,
(b)for triangle ABC,
?
?
??
?
?
? ?
??
Hence the angles are 58°, 61° and 61°.
![Page 11: Math unit31 angles and symmetry](https://reader033.fdocuments.us/reader033/viewer/2022042707/58f30c991a28aba35a8b457b/html5/thumbnails/11.jpg)
Unit 3131.4 Angles and Parallel Lines:
Results
![Page 12: Math unit31 angles and symmetry](https://reader033.fdocuments.us/reader033/viewer/2022042707/58f30c991a28aba35a8b457b/html5/thumbnails/12.jpg)
Results• Corresponding angles are equal e.g. d = f, c = e• Alternate angles are equal e.g. b = f, a = e• Supplementary angles sum to 180° e.g. a + f = 180°Thus• If corresponding angles are equal, then the two lines are parallel.• If alternate angles are equal, then the two lines are parallel.• If supplementary angles sum to 180°, then the two lines are parallel e.g. a + f = 180°
![Page 13: Math unit31 angles and symmetry](https://reader033.fdocuments.us/reader033/viewer/2022042707/58f30c991a28aba35a8b457b/html5/thumbnails/13.jpg)
Unit 3131.5 Angles and Parallel Lines:
Example
![Page 14: Math unit31 angles and symmetry](https://reader033.fdocuments.us/reader033/viewer/2022042707/58f30c991a28aba35a8b457b/html5/thumbnails/14.jpg)
?
ExampleIn this diagram AB is parallel to CD. EG is parallel to FH, angle IJL=50° and angle KIJ=95°.Calculate the values of x, y and z, showing clearly the steps in your calculations.
SolutionAngles BIG and END are supplementary angles, so
but angles END and FMD are corresponding angles so
x
?
?
?
zy
?
Angles BCD and ABC are alternate angles, so
In triangle BIJ
So
?
?
?
?
? ?
Angles AKH and FMD are alternate angles, so?
?
![Page 15: Math unit31 angles and symmetry](https://reader033.fdocuments.us/reader033/viewer/2022042707/58f30c991a28aba35a8b457b/html5/thumbnails/15.jpg)
Unit 3131.6 Angle Symmetry in Regular
Polygons
![Page 16: Math unit31 angles and symmetry](https://reader033.fdocuments.us/reader033/viewer/2022042707/58f30c991a28aba35a8b457b/html5/thumbnails/16.jpg)
?
Example 1 Find the interior angle of a regular dodecagonSolutionThe dodecagon has 12 sidesThe angle marked x, is given by
The other angle in each of theisosceles triangle is
The interior angle is ?
?
?
??
?
![Page 17: Math unit31 angles and symmetry](https://reader033.fdocuments.us/reader033/viewer/2022042707/58f30c991a28aba35a8b457b/html5/thumbnails/17.jpg)
Example 2 Find the sum of the interior angles of a regular heptagonSolutionYou can split a regular heptagon into 7 isosceles trianglesEach triangle contains three angles that sum to 180°
We need to exclude the angles round the centre that sum to 360°
Note: Is the result the same for an irregular heptagon?
?
??
?
??
?