Angles 2
You will learn how to:
• Derive and use the formula for the sum of the interior angles of any polygon.
• Know that the sum of the exterior angles of any polygon is 360°.
• Use properties of angles, parallel and intersecting lines, triangles and quadrilaterals to calculate missing angles.
Starting point
Do you remember…
• that corresponding, alternative and vertically opposite angles are equal?
For example, find the sizes of angles marked x and y
• that angles in a triangle add up to 180° and that angles in a quadrilateral add up to 360°? For example, find the sizes of the angles marked a and b
• how to use properties of the exterior angle of a triangle? For example, calculate the size of angle u.
This will also be helpful when…
• you learn to solve problems that involve bearings.
Worked example 2
Martha wants to collect data on the wingspans of greenfinches (a type of bird). She expects the birds to have a wingspan of between 24 cm and 28 cm. Design a data collection sheet with four class intervals that would be suitable for collecting Martha’s data. All class intervals should have equal width.
The data collection sheet needs four rows, so split the interval between 24 cm and 28 cm into four classes:
The data being collected are continuous so each class interval should be written with inequality symbols.
Worked example 3
The table shows information about the number of cars for sale in a garage.
Show this information as a Venn diagram.
Begin by drawing the circles for the Venn diagram.
Three circles are needed as there are three categories:
• type of engine (petrol or diesel)
• age (less than 3 years or 3 or more years)
• air conditioning (with or without)
This Venn diagram has eight sections where numbers can be written. These match with the eight cells in the table.
3 Write as an ordinary number:
a) 6 × 105 b) 3.8 × 104 c) 7 × 10–6 d) 8.03 × 10–3 e) 1 × 106 f) 9.09 × 10–4 g) 6.261 × 107 h) 5.077 × 10–5 i) 9.2 × 103
4 Write in standard form:
a) 20 000 b) 0.000082 c) 356 000
d) 0.03801 e) 1 500 000 f) 560 g) 0.000571 h) 17 i) 0.1
5 Benigno, Melinda and Sonto want to write the number 1 in standard form.
Benigno says it is 1 × 101
Melinda says it is 1 × 100
Sonto says it is just 1
Explain which answer is correct. Explain why the other two answers are incorrect.
6 Write each number in standard form.
a) 50 × 103 b) 0.8 × 104 c) 306 × 105 d) 44 × 10–3 e) 0.1 × 10–4 f) 10 × 106
7 Technology question Most scientific calculators have a standard form mode, or ‘Sci’ mode. In this mode, all calculated numbers are shown in standard form. Find out how to put your calculator into this mode.
Use your calculator to convert these numbers to standard form.
a) 720 000 b) 0.0000081 c) 50 800 000 d) 0.00004105 e) 9 346 415 f) 0.0000004
Tip
You may find it easier to convert to ordinary numbers first.
Tip In standard form mode on many calculators, if you type an ordinary number and press ‘=’ or ‘enter’, the calculator displays the number in standard form.
8 Technology question To enter a number in standard form, such as 2.7 × 10–11, into a calculator, either:
type 2.7 × 10, then the index key (which may be marked ^ or xy), and then −11, or type 2.7, then the key labelled ‘× 10x’, ‘EXP’ or ‘EE’ (which means ‘× 10 to the power’), and then −11.
Use a calculator to do these calculations.
a) (3 × 106) × (2 × 107) b) (6.8 × 107) ÷ (4 × 10–12) c) (5 × 1014) + (7 × 1013) d) (3.7 × 10–10) – (5.8 × 10–11)
Discuss
In question 8, in which calculation(s) do the brackets affect the result? Explain your answers.
In part a) of the Worked example, the direction of the 180° rotation was not stated. Should it have been?