Level 5 NZ Curriculum Achievement Objectives for Geometric Reasoning
In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:
GM5-5 Deduce the angle properties of intersecting and parallel lines and the angle properties of polygons and apply these properties.
GM5-5 Deduce the angle properties of intersecting and parallel lines and the angle properties of polygons and apply these properties.
In this unit you will revise:
1) Naming, Classifying and Measuring Angles 2) Angle Properties 3) Angles in Triangles and Quadrilaterals 4) Angles on Parallel Lines 5) Pythagorean Theorem (finding the long side) |
In this unit you will learn:
6) Interior Angles in Polygons 7) Exterior Angles in Polygons 8) Angles in a Circle 9) Pythagorean Theorem (finding the short side) |
1)Naming, Classifying and Measuring angles
An angle is formed when two lines called arms meet at a point called a vertex. When naming angles, three letters are usually used. The letters are positioned at the end points of the arms. When naming the angle, the vertex must be in the middle.
The angle to the right, can be named as ABC or CBA.
The angle to the right, can be named as ABC or CBA.
Angles are classified according to their size. The diagram to the right shows the six different names.
Angles can be measured manually with a protractor.
Click the videos below to show you how to do this.
Click the videos below to show you how to do this.
|
|
2) Angle Properites
There are three basic angle properties:
3) Angles in Triangles and Quadritlaterals
Triangles can be described by considering side lengths and internal angles. Sides that have the same length are marked with similar dashes and equal angles are indicated with the same markings.
All four-sided two-dimensional shapes are quadrilaterals. Quadrilaterals are described according to their properties of their sides and angles, in a similar way to the way in which triangles are described.
- sides that are equal in length are shown with the same number of small dashes.
- sides marked with the same number of arrow heads are parallel.
|
|
The four angles of a quadrilateral always add up to 360 degrees.
4) Angles on parallel lines
|
|
5) Pythagorean Theorem (finding the long side)
In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
|
|
6) Interior Angles in a Polygon
The sum of interior angles of a polygon change with the number of sides in the shape. We already know that each triangle has 180 degrees in it. We can use this relationship to find out the sum in any shape. Use the following rule:
The number of triangles in a polygon is 2 less than the number of sides.
Find the number of triangles and the multiply by 180 to get the total amount of degrees inside a polygon.
The number of triangles in a polygon is 2 less than the number of sides.
Find the number of triangles and the multiply by 180 to get the total amount of degrees inside a polygon.
The shape to the right has 5 sides. Therefore we can fit 3 triangles into the shape. Since each triangle has 180 degrees, 3 triangles have 3 X 180 = 540 degrees.
If the shape is regular (all sides equal) we then divide the 540 degrees by the 5 angles to calculate that each angle is 108 degrees. Click on the link below for more help. http://www.mathsisfun.com/geometry/interior-angles-polygons.html |
7)Exterior Angles in a Polygon
All exterior angles in any polygon add up to 360 degrees.
8) Angle Properties in Circles
The first thing we need to learn is the parts of a circle. Use the diagram below to learn the name of each part.
In year 10, we will learn three angle properties of circles.
|
|
|
9) Pythagorean Theorem (finding the short side)
In order to find one of the shorter sides of a right angled triangle, we use the Pythagorean Theorem as in section 5 above.
Below are three example of how to use the Pythagorean Theorem. The first one calculates the long side (hypotenuse). The other two calculate one of the shorter sides. Note how to use the formula is rearranged to find one of the shorter sides.
Check out this website for more practice on Pythagorean Theorem.
http://www.mathwarehouse.com/geometry/triangles/how-to-use-the-pythagorean-theorem.php
http://www.mathwarehouse.com/geometry/triangles/how-to-use-the-pythagorean-theorem.php
Check out these cool sites...
https://www.youtube.com/user/LearnCoach
http://www.mathsisfun.com/angle180.html
http://www.mathsisfun.com/geometry/complementary-angles.html
http://www.mathplayground.com/alienangles.html
http://www.bbc.co.uk/bitesize/ks2/maths/shape_space/angles/play/popup.shtml
https://www.goconqr.com/flashcard/38042384/year-10-geometric-reasoning
http://www.mathsisfun.com/angle180.html
http://www.mathsisfun.com/geometry/complementary-angles.html
http://www.mathplayground.com/alienangles.html
http://www.bbc.co.uk/bitesize/ks2/maths/shape_space/angles/play/popup.shtml
https://www.goconqr.com/flashcard/38042384/year-10-geometric-reasoning