Year 7 Angles Dr J Frost jfrosttiffin kingston

Year 7 Angles Dr J Frost (jfrost@tiffin. kingston. sch. uk) www. drfrostmaths. com Objectives: Understand notation for angles. Know basic rules of angles (angles in triangle, on straight line). Recognise alternate, corresponding and vertically opposite angles. Find angles in isosceles triangles. Deal with and introduce algebraic angles. Construct diagrams from written information and form angle proofs. Last modified: 7 th February 2016

For Teacher Use: Recommended lesson structure: Lesson 1: Angle notation. Angle basics. Lesson 2: Z/F/C/v. opposite angles. Angle properties of quadrilaterals. Lesson 3: Isosceles Triangles Lesson 4: Algebraic Angles Lesson 5/6: Constructing diagrams from information. Proof Lesson 7: Consolidation/Mini-assessment

Angle Notation We use capital letters for points. ? ? ? 1 1 3 2 ? 3 ?

Angle Basics ? ? ? reflex acute obtuse Name of angle if: Less than 90: Acute ? Between 90 and 180: Obtuse ? Over 180: Reflex? angle

Test Your Understanding i ? ii ?

Exercise 1 (on provided worksheet) 1 Find the angles marked with letters. ? ? ?

Exercise 1 (on provided worksheet) 2 ? 3 ? 4 ? 5 ? 6 ? 7 ?

Exercise 1 (on provided worksheet) 8 ? 9 ? 10 ? 11 ? 12 ?

Exercise 1 (on provided worksheet) 13 ? 14 ? 15 ? 16 ?

Angles involving parallel lines There are three more laws of angles that you need to know, use and quote: (These arrows indicate the lines are parallel) ! Alternate angles are equal. ? (Sometimes known as ‘Z’ angles) ! Corresponding angles are equal. ? (Sometimes known as ‘F’ angles) ! Vertically opposite angles ? are equal. Bro Note: The word ‘vertically’ here is the adjective form of ‘vertex’, which means a point. So “vertically opposite” means “opposite with respect to a point”.

How to spot them To identify alternate angles: Step 1: Identify a line connecting two parallel lines. Click >> Step 2: At each end on your parallel lines, shoot out along the parallel lines in opposite directions. Your angles are the ones wedged between the lines. Click >> To identify corresponding angles: Step 1: Identify an angle between one line of a parallel pair, and a connecting line. Click >> Step 2: This can be shifted along to the other parallel line of the pair. Click >>

Examples (For extra practise outside of class) 4 2 5 Double arrows allows us to match another pair of parallel lines. 3 1 # 1 2 3 4 5 Angle ? ? Alternate ? ? ? Corresponding ? ? ?

Check Your Understanding i ? ? ii ? Bro Hint: Is there a line in the diagram we could add so we can then use alternate angles?

Cointerior angles We’ve seen ‘Z’ and ‘F’ angles. There’s also ‘C’ angles! ? We can identify them in a similar way to alternate angles: identify a line connecting two parallel lines, but this time go in the same direction at each end rather than opposite directions. ? Application to parallelograms So we can say: ? ? ? ?

Example ?

One further property of quadrilaterals What are the sum of the angles in a quadrilateral? Draw Hint >> ?

Exercise 2 (on provided worksheet) 1 a For each diagram (i) Find the missing angles and (ii) List reasons for each answer. ? ? b ? ?

Exercise 2 2 b a ? ? c d ? ?

Exercise 2 f e ? ?

Exercise 2 3 ?

Exercise 2 4 Find the angles indicated. ? ? ?

Exercise 2 5 ? 6 ?

Exercise 2 7 ? 8 ? 9 ? 10 ?

Exercise 2 11 ? 12 ? 13 ?

Exercise 2 14 ?

Isosceles Triangles What do these marks mean? The lines are of the ? same length. ? ! “Base angles of an isosceles triangle are equal. ” ?

Warning! Sometimes diagrams are drawn in such a way that it’s not visually obvious what the two angles the same are. You can use the ‘finger slide method’ to identify these. Diagram not drawn accurately. Click for Broanimation > Put your fingers on the two marks. Slide your fingers in the same direction but away from each other. These two corners are where the angles are the same.

Test Your Understanding i ? ii ?

Using angles to give you information about sides You’ve so far used sides which are equal to find angles. But we can do the opposite too! If two angles are equal, then two sides are equal. 1. Use information provided. 2. Work out some initial angles. 3. Use angles to give us information about side lengths. 4. Use new knowledge of side lengths to work out more angles… Go >

Angle Wall Challenge How far can you get down the angle challenge wall? (do in order, and draw the diagram first) Hint: You may want to add extra lines. ? ? ?

Exercise 3 1 (on provided worksheet) ? 2 ? 3 ? 4 ? 5 ?

Exercise 3 (on provided worksheet) 6 ? 7 ? 8 ? 9 ? 10 ?

Exercise 3 (on provided worksheet) 11 ? 12 ? 13 ?

Exercise 3 14 (on provided worksheet) ?

STARTER : Algebraic Angles What are the angles in each case, in terms of the variables given? (Hint: Just think what you’d do usually – it’s no different here!) i ? ii iii ? ?

Overview ! There are two types of problems you’ll have which involve algebraic angles: 1. Angles given …and you have to find an expression for a given angle. Example: 2. No angles given …and you have to introduce variables yourself, either so that you can prove two angles have some relationship, or so you can form an equation and hence find an angle. Example:

Finding remaining angle in triangle/on line Angle 1 Angle 2 Quickfire Questions: ? ? (Expand brackets) ? (Simplify) Angle 1 Angle 2 ? ? ? ?

Finding remaining angle in triangle/on line Angle 2 Angle 1 ? Angle 1 Angle 2 Remaining ? ? ?

Full example Two possible ways: ? ?

Check Your Understanding How far can you get down the angle challenge wall? (do in order, and draw the diagram first) ? ? ?

Modelling restrictions on angles ? ? Bro Note: To ‘bisect’ means to cut in half.

Modelling restrictions on angles ? ?

Exercise 4 1 (on provided worksheet) ? ? 2 ? ? ?

Exercise 4 3 (on provided worksheet) a c ? b ? d ? ? ?

Exercise 4 (on provided worksheet) h e ? f i ? ? ? g ? ? ? ?

Exercise 4 4 a (on provided worksheet) ? b ? ? ? c ? ? ?

Exercise 4 5 (on provided worksheet) 6 ? ? ? ? ?

Exercise 4 7 (on provided worksheet) 8 ? ? ?

Exercise 4 9 (on provided worksheet) ? ? ?

Exercise 4 10 ? 11 ? 12 ? (on provided worksheet)

Exercise 4 13 (on provided worksheet) ?

Proof Now that we have a number of angle skills, including introducing algebraic angles, we now have all the skills to form a ‘proof’. A ‘proof’ is a sequence of justified statements that results in the desired conclusion. ! Examples (which we’ll do later)

STARTER: Constructing diagrams Sometimes you are not given a diagram, but have to construct one given information. Can you form a suitable diagram given each of the descriptions? Make sure that you use marks/arrows to indicate when sides are the same length or parallel, or where angles are equal. ? ? Bro Note: For parallelograms/ rhombuses, we need not indicate parallel sides because it is implied by the lengths.

A harder one for discussion… Bro Note: To ‘bisect’ an angle is to cut it in half. So a ‘bisector’ of an angle is a line which bisects the angle. ?

Simple Proofs The simplest proofs just require you to find an angle, but you need to give a reason for each step. ?

Test Your Understanding i ? ii (if you finish) ?

Exercise 5 a (on provided worksheet) 1 a ? b ? c ?

Exercise 5 a (on provided worksheet) 2 Draw diagrams which satisfy the following criteria, ensuring you note where lines are of equal length or parallel. You do NOT need to find any angles. a ? b ? c ? d ?

Exercise 5 a 3 a (on provided worksheet) ? b ?

Other Types of Proof ? Proof ? Bro Note: You need a conclusion so it’s clear that your proof is complete. . Refer to the provided ‘cheat sheet’. Similarly, what would we need to do in order to: ?

Test Your Understanding i Proof ? ii (if you finish) Proof ?

Exercise 5 b 1 a (on provided worksheet) ? b ?

Exercise 5 b (on provided worksheet) 2 ?

Exercise 5 b 3 (on provided worksheet) ?

Exercise 5 b 4 (on provided worksheet) ?

Exercise 5 b 5 (on provided worksheet) ?

Exercise 5 b 6 (on provided worksheet) ?

Exercise 5 b 7 (on provided worksheet) ?

Exercise 5 b (on provided worksheet) 8 ? Proof ? Diagram