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KS3/KS4: Angles Skipton Girlsโ High School 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. 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 ๐ด ๐ต We use capital letters for points. ๐ถ ! We can refer to this angle using: โข ๐ด๐ต๐ถ (with a โhatโ on the ? middle letter) โข โ ๐ด๐ต๐ถ ? โข Or in words: โAngle?๐ด๐ต๐ถโ Quick Starter: How would we refer to each of the following angles? (Use โ ๐ด๐ต๐ถ notation) ๐ 1 ๐ 3 ๐ถ 2 ๐ ๐ 1 โ ๐๐๐ or? โ ๐ถ๐๐ 2 โ ๐๐ถ๐ ? 3 โ ๐๐๐ ? Angle Basics 30° 128° ๐๐°? ! โAngles on a straight line ? sum to 180°.โ ๐๐๐° ? (These wordings will be important for angle proofs later) ! โAngles around a point ? sum to 360°.โ 85° ๐๐° ? 50° ! โAngles in a triangle?sum to 180°.โ reflex acute obtuse Name of angle if: Less than 90: Acute ? Between 90 and 180: Obtuse ? Over 180: Reflex?angle Test Your Understanding i [JMC 2008 Q4] In this diagram, what is the value of ๐ฅ? A 16 B 36 C 64 D 100 E 144 Solution: C ? ii [JMC 2003 Q12] What is the size of the angle marked ๐ฅ? A 42° B 67° C 69° D 71° E 111° Solution: A ? Exercise 1 1 (on provided worksheet) Find the angles marked with letters. 71° ๐ ๐ 45° ๐ 150° ๐ ๐ 105° ๐ ๐ 52° 60° 110° ๐ = ๐๐°, ? ? ๐ = ๐๐°, ? ๐ = ๐๐°, ? ๐ = ๐๐๐°, ? ๐ = ๐๐° 30° Exercise 1 (on provided worksheet) 2 [JMC 2000 Q3] What is the value of ๐ฅ? Solution: 28 3 [JMC 2015 Q6] What is the value of ๐ฅ in this triangle? Solution: 50 ? ? 4 [JMC 2011 Q6] What is the sum of the marked angles in the diagram? Solution: ๐๐๐° ? 5 [JMC 1997 Q17] How big is angle ๐ฅ? Solution: ๐๐๐° ? 6 [IMC 2014 Q3] An equilateral triangle is placed inside a larger equilateral triangle so that the diagram has three lines of symmetry. What is the value of ๐ฅ? Solution: ๐๐๐° ? 7 [IMC 2011 Q9] In the diagram, ๐๐ is a straight line. What is the value of ๐ฅ? Solution: 170 ? Exercise 1 8 (on provided worksheet) [JMC 2010 Q8] In a triangle with angles ๐ฅ°, ๐ฆ°, ๐ง° the mean of ๐ฆ and ๐ง is ๐ฅ. What is the value of ๐ฅ? Solution: 60 ? 9 [JMC 1999 Q13] In the diagram, โ ๐ ๐๐ = 20° and โ ๐๐๐ = 70°. What is โ ๐๐ ๐? Solution: ๐๐๐° ? 10 [JMC 2001 Q18] Triangle ๐๐๐ is equilateral. Angle ๐๐๐ = 40°, angle ๐๐๐ = 35°. What is the size of the marked angle ๐๐๐? Solution: ๐๐๐° ? 11 [JMC 2007 Q16] What is the sum of the six marked angles? Solution: ๐๐๐๐°. The sum of the angles at the 7 points is ๐ × ๐๐๐ = ๐๐๐๐. Weโve excluded the angles in the 3 triangles, so ๐๐๐๐ โ ๐ × ๐๐๐ = ๐๐๐๐. ? 12 [JMC 2015 Q16] The diagram shows a square inside an equilateral triangle. What is the value of ๐ฅ + ๐ฆ? Solution: 150 ? Exercise 1 13 [JMO 2001 A3] What is the value of ๐ฅ in the diagram alongside? Solution: 15 ? 14 [Kangaroo Pink 2010 Q9] In the diagram, angle ๐๐๐ is 20°, and the reflex angle at ๐ is 330°.The line segments ๐๐ and ๐๐ are perpendicular. What is the size of angle ๐ ๐๐? Solution: ๐๐° ? 15 [Kangaroo Pink 2006 Q8] The circle shown in the diagram is divided into four arcs of length 2, 5, 6 and ๐ฅ units. The sector with arc length 2 has an angle of 30° at the centre. Determine the value of ๐ฅ. (Hint: An arc is a part of the line that makes up the circle, i.e. part of the circumference. The arc length grows in proportion to the angle at the centre) Solution: 11 ? 16 [SMC 2010 Q3] The diagram shows an equilateral triangle touching two straight lines. What is the sum of the four marked angles? Solution: ๐๐๐°. Angles at two points some to ๐๐๐°, but we exclude two angles of ๐๐°. ? 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. ? as โZโ angles) (Sometimes known ! Corresponding angles are equal. ? as โFโ angles) (Sometimes known ! Vertically opposite angles ? are equal. Bro Note: The word โverticallyโ here is the adjective form of โvertexโ, which means a point. So โvertically oppositeโ means โopposite with respect to a pointโ. 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) Identify the alternate and corresponding angle for each indicated angle (use โ notation). ๐ ๐ ๐ ๐ด ๐ ๐ท ๐ถ ๐ต ๐ ๐ธ ๐บ 5 4 ๐น 2 ๐ป Double arrows allows us to match another pair of parallel lines. 3 ๐ฝ ๐ผ ๐พ 1 ๐ ๐ ๐ฟ ๐ ๐ ๐ ๐ ๐ # Angle Alternate Corresponding 1 โ ๐ฝ๐๐ โ ๐๐ท๐ or โ ๐บ๐ฟ๐ 2 ? โ DJC ? โ ๐ผ๐ถ๐ท ? โ ๐ป๐บ๐ฟ ? ? โ ๐น๐๐ ? โ BAC ? โ ๐๐๐ผ ? โ ๐ท๐ฝ๐ผ ? 3 4 5 โ ๐๐น๐ป โ ๐ถ๐ท๐ฝ ? โ ๐ฟ๐๐ ? โ ๐๐ด๐ ? โ ๐๐ฟ๐ ? โ ๐๐ฝ๐พ ? Check Your Understanding ANB is parallel to CMD. LNM is a straight line. Angle LMD = 68° (i) Work out the size of the angle marked ๐ฆ. ๐๐๐° ? (ii) Give reasons for your answer. โAlternate angles are equalโ OR โCorresponding angles ? line sum to 180โ are equalโ, โAngles on straight i ii [JMC 2011 Q11] The diagram shows an equilateral triangle inside a rectangle. What is the value of ๐ฅ + ๐ฆ? A 30 B 45 C 60 D 75 E 90 ๐ฆ ๐ฅ Solution:?C Hint: Is there a line in the diagram we could add so we can then use alternate angles? Cointerior angles Weโve seen โZโ and โFโ angles. Thereโs also โCโ angles! ! Cointerior angles sum to?๐๐๐° 180 โ ๐ฅ ? ๐ฅ 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: 110° ? 70° ?70° ! Cointerior angles in a parallelogram add to?๐๐๐° ! Opposite angles in a parallelogram are equal. ? Example Can you come up with an alternative explanation for why ๐ฆ = 112°? โ ๐ด๐ต๐ฉ = ๐๐๐° (cointerior angles sum to 180) ๐ = ๐๐๐° (vertically opposite ? angles are equal) One further property of quadrilaterals What are the sum of the angles in a quadrilateral? Draw Hint >> ! Angles in a quadrilateral sum to ๐๐๐° ? 50° ๐ฅ 35° 45° ? ๐ฅ = ๐๐๐° Exercise 2 1 a For each diagram (i) Find the missing angles and (ii) List reasons for each answer. i) ๐๐° ? ii) Corresponding ? equal. angles are b ? ๐๐° i) ๐ = ๐๐๐°, ๐ = ii) For x: Corresponding angles are equal. For y: Angles on straight line ? add to 180. Alternate angles are equal. Exercise 2 2 b a ๐ฆ ๐ฅ 115° 122° 80° ๐ = ๐๐°? ๐ = ๐๐° ? c 30° d ๐ 57° ๐ฅ ๐ ๐ 82° ๐ = ๐๐°, ๐ =?๐๐°, ๐ = ๐๐° 41° ๐ = ๐๐° ? Exercise 2 f e 85° 110° ๐ฅ 112° 75° ๐ฆ ? ๐ = ๐๐° ๐ = ๐๐° ? Exercise 2 3 For rhombuses are evenly spaced around a point. Given the angle shown, find ๐ฅ. Solution: ๐๐๐° ? 30° ๐ฅ Exercise 2 4 Find the angles indicated. 100° 41° ๐ ๐ ๐ ๐ ๐ 96° 136° ๐ = ๐๐°? ๐ = ๐๐°? ๐ = ๐๐๐° ? ๐ = ๐๐°? ๐ = ๐๐° ? Exercise 2 5 42° Find the angle ๐ผ Solution: ๐๐° ๐ผ ? 29° ๐ท 6 ๐ ๐ถ The line ๐๐ต bisects the angle ๐ด๐ต๐ถ (this means it cuts it exactly in half). Determine the angle ๐ต๐๐ถ. Solution: ๐๐° ? 70° ๐ด ๐ต Exercise 2 7 [JMC 2006 Q7] What is the value of ๐ฅ? Solution: ๐๐° ? 8 [JMC 2007 Q9] In the diagram on the right, ๐๐ is parallel to ๐๐. What is the value of ๐ฅ? Solution: 86 ? 9 [JMC 2009 Q19] The diagram on the right shows a rhombus ๐น๐บ๐ป๐ผ and an isosceles triangle ๐น๐บ๐ฝ in which ๐บ๐น = ๐บ๐ฝ. Angle ๐น๐ฝ๐ผ = 111°. What is the size of angle ๐ฝ๐น๐ผ? Solution: ๐๐° ? 10 [IMC 2005 Q7] In the diagram, what is the sum of the marked angles? Solution: ๐๐๐°. Sum of angles in 4 triangles is ๐๐๐° × ๐ = ๐๐๐°. The four angles in the quadrilateral sum to 360, thus because of vertically ? opposite angles, the innermost angles of the triangles sum to 360. Thus sum of angles is ๐๐๐ โ ๐๐๐ = ๐๐๐°. Exercise 2 11 [JMO 2003 A9] Find the value of ๐ + ๐ + ๐. Solution: 210 ? 12 [SMC 2006 Q3] The diagram shows overlapping squares. What is the value of ๐ฅ + ๐ฆ? Solution: 270 ? 13 [Cayley 2012 Q2] In the diagram, ๐๐ and ๐๐ are parallel. Prove that ๐ + ๐ + ๐ = 360. Suppose we add a line horizontal with ๐น as shown, where ๐ is split into ๐๐ and ๐๐ . ๐ and ๐๐ are cointerior thus ๐ + ๐๐ = ๐๐๐°. ๐๐ and ๐ are cointerior thus ๐๐ + ๐ = ๐๐๐° ? Thus ๐ + ๐ + ๐ = ๐ + ๐๐ + ๐๐ + ๐ = ๐๐๐° + ๐๐๐° = ๐๐๐° ๐ ๐๐ ๐๐ Exercise 2 14 [Kangaroo Grey 2005 Q20] Five straight lines intersect at a common point and five triangles are constructed as shown. What is the total of the 10 angles marked in the diagram? A 300° B 450° C 360° D 600° E 720° Answer: ๐๐๐°. Sum of angles in 5 triangles is ๐๐๐° × ๐ = ๐๐๐°. Since angles at the central point add to ๐๐๐°, but each angle in a triangle at the centre can be paired with a vertically opposite angle not in a triangle, the central angles in the triangles sum to ๐๐๐°. ๐๐๐° โ ๐๐๐° = ๐๐๐° ? Isosceles Triangles What do these marks mean? The lines are of the ? same length. ๐๐° ? ๐๐° ? 50° 70° ? ๐๐° ! โBase angles of an isosceles triangle are equal.โ Warning! Sometimes diagrams are drawn in such a way that itโs not visually obvious what the two angles the same are. You can use the โfinger slide methodโ to identify these. Diagram not drawn accurately. Click for animation > 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 [JMC 2013 Q3] What is the value of ๐ฅ? A 25 B 35 C 40 D 65 E 155 Solution: C ? ii [IMC 2012 Q11] In the diagram, ๐๐๐ ๐ is a parallelogram; โ ๐๐ ๐ = 50°; โ ๐๐๐ = 62° and ๐๐ = ๐๐. What is the size of โ ๐๐๐ ? A 84° B 90° C 96° D 112° E 124° Solution: ?C Using angles to give you information about sides Youโve 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. [Kangaroo Pink 2004 Q6] In the diagram ๐๐ = ๐๐. What is the size of โ ๐๐๐ ? 75° 65° 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 anglesโฆ Go > Go > Go > 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. ๐๐° 60° ๐๐° ๐ ๐๐° ๐๐° ๐๐° ๏ ๐๐° ๐๐° ? โ ๐ท๐ด๐ถ = 30° ? โ ๐ด๐ท๐ต = 45° (ฮ๐ด๐ท๐ถ is?isosceles) โ ๐ด๐ท๐ถ = 75° (ฮ๐ด๐ท๐ถ is?isosceles) โ ๐ต๐ท๐ถ = 30° (75° โ 45°) ? โ ๐ท๐ต๐ถ = 15° (angles within ? ฮ๐ท๐ถ๐ต) ๏ โ ๐ต๐ด๐ถ = 60° ๐๐° ๏ This was a Junior Maths Olympiad problem! (to prove that โ ๐ต๐ท๐ถ = 2 × โ ๐ท๐ต๐ถ) Exercise 3 1 Find the value of each variable. Solution: ๐ = ๐๐°, ๐ = ๐๐°, ๐ = ๐๐° ? 2 [JMC 2014 Q10] An equilateral triangle is surrounded by three squares, as shown. What is the value of ๐ฅ? Solution: 30 ? 3 [JMC 2005 Q13] The diagram shows two equal squares. What is the value of ๐ฅ? Solution: 140 ? 4 [IMC 2010 Q6] In triangle ๐๐๐ , ๐ is a point on ๐๐ such that ๐๐ = ๐๐ = ๐๐ and โ ๐๐๐ = 20°. What is the size of โ ๐๐ ๐? Solution: ๐๐° ? 5 [IMC 1999 Q8] In the diagram ๐๐ = ๐๐ = ๐ ๐. What is the size of angle ๐ฅ? Solution: ๐๐๐° ? Exercise 3 (on provided worksheet) 6 [JMC 2008 Q19] In the diagram on the right, ๐๐ = ๐๐ = ๐๐, ๐๐ = ๐๐ , โ ๐๐๐ = 20°. What is the value of ๐ฅ? Solution: 35 ? 7 [Kangaroo Grey 2010 Q9] The diagram shows a quadrilateral ๐ด๐ต๐ถ๐ท, in which ๐ด๐ท = ๐ต๐ถ, โ ๐ถ๐ด๐ท = 50°, โ ๐ด๐ถ๐ท = 65° and โ ๐ด๐ถ๐ต = 70°. What is the size of โ ๐ด๐ต๐ถ? Solution: ๐๐° ? 8 [Cayley 2004 Q1] The โstarโ octagon shown in the diagram is beautifully symmetrical and the centre of the star is at the centre of the circle. If angle ๐๐ด๐ธ = 110° (as indicated), how big is the angle ๐ท๐๐ด? (i.e. ๐ฅ) Solution: ๐๐° ? 9 [IMC 2003 Q8] Lines ๐ด๐ต and ๐ถ๐ท are parallel and ๐ต๐ถ = ๐ต๐ท. Given that ๐ฅ is an acute angle not equal to 60°, how many other angles in this diagram are equal to ๐ฅ? Solution: 4 ? 10 [IMC 1997 Q11] In the quadrilateral ๐ด๐ต๐ถ๐ท, โ ๐ด๐ต๐ถ = 90°, โ ๐ต๐ด๐ท = 70°, and ๐ด๐ต = ๐ต๐ท = ๐ต๐ถ. What is the size of โ ๐ต๐ท๐ถ? Solution: ๐๐° ? ๐ ๐ฅ ๐ ๐ท ๐ด ๐ถ ๐ต ๐ 110° ๐ธ Exercise 3 11 (on provided worksheet) [Kangaroo Pink 2013 Q9] The diagram shows an equilateral triangle ๐ ๐๐ and also the triangle ๐๐๐ obtained by rotating triangle ๐ ๐๐ about the point ๐. Angle ๐ ๐๐ = 70°. What is angle ๐ ๐๐? Solution: ๐๐° ? 12 [IMC 2000 Q8] In the triangle ๐ด๐ต๐ถ, ๐ด๐ท = ๐ต๐ท = ๐ถ๐ท. What is the size of angle ๐ต๐ด๐ถ? Solution: ๐๐° ? 13 In the diagram ๐ด๐ต is parallel to ๐ถ๐ท and ๐ด๐ธ = ๐ด๐ถ. Determine โ ๐ถ๐ด๐ธ. ๐ด ๐ต 30° Solution: ๐๐° ? ๐ธ 54° ๐ถ ๐ท Exercise 3 14 (on provided worksheet) [TMC Regional 2008 Q9] If ๐ด๐ต = ๐ต๐ถ = ๐ถ๐ท = ๐ท๐ธ = ๐ธ๐น and angle ๐ด๐ธ๐น = 75°, what is the size of angle ๐ธ๐ด๐น? Solution: ๐๐° ? STARTER : Algebraic Angles What are the angles in each case, in terms of the variables given? (Hint: Just think what youโd do usually โ itโs no different here!) i 180?โ ๐ฅ ๐ฅ ii iii 90 ?โ ๐ฆ ๐ฆ ๐ง 180 โ?2๐ง ! Overview There are two types of problems youโll have which involve algebraic angles: 1. Angles given โฆand you have to find an expression for a given angle. Example: 2. No angles given โฆand 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: Given that ๐ด๐ต = ๐ด๐ถ and ๐ง < 90, which of the following expressions must equal ๐ง? A ๐ฅโ๐ฆ B ๐ฅ+๐ฆ C ๐ฅ + ๐ฆ โ 180 D 180 + ๐ฅ โ ๐ฆ E 180 โ ๐ฅ + ๐ฆ [JMO 2010 A10] In the diagram, ๐ฝ๐พ and ๐๐ฟ are parallel. ๐ฝ๐พ = ๐พ๐ = ๐๐ฝ = ๐๐ and ๐ฟ๐ = ๐ฟ๐ = ๐ฟ๐พ. Find the size of angle ๐ฝ๐๐. Finding remaining angle in triangle/on line Angle 1 Angle 2 ? If Angle 1 is 90 + ๐ฅ, what is angle 2? 180 โ ?90 + ๐ฅ (Expand brackets) = 180? โ 90 โ ๐ฅ = 90 โ ๐ฅ? (Simplify) Bro Tip: An easier way to do the subtraction is to think what we have to do to 90 + ๐ฅ to get to 180. Adding 90 gets us from 90 to 180, and subtracting ๐ฅ cancels out the ๐ฅ. Quickfire Questions: Angle 1 Angle 2 ๐ฅ 180?โ ๐ฅ ๐ฅ + 20 160?โ ๐ฅ ๐ฅ โ 10 190?โ ๐ฅ 2๐ฅ + 60 120?โ 2๐ฅ ๐ฅโ๐ฆ 180 โ?๐ฅ + ๐ฆ 180 โ 2๐ฅ 2๐ฅ ? 160 โ ๐ฅ 20?+ ๐ฅ Finding remaining angle in triangle/on line Angle 2 Angle 1 ? Angle 1 Angle 2 Remaining ๐ฅ ๐ฅ 180?โ 2๐ฅ ๐ฅ + 20 ๐ฅ + 50 110?โ 2๐ฅ 2๐ฅ 90 โ ๐ฅ 90?โ ๐ฅ 90 โ 3๐ฅ 90 โ 2๐ฅ 5๐ฅ ? ๐ฅโ๐ฆ 160 + ๐ฅ 20 โ ? 2๐ฅ + ๐ฆ 30 + 2๐ฅ โ ๐ฆ 50 + ๐ฅ + ๐ฆ 100?โ 3๐ฅ Full example [JMC 2002 Q21] Given that ๐ด๐ต = ๐ด๐ถ and ๐ง < 90, which of the following expressions must equal ๐ง? A ๐ฅโ๐ฆ B ๐ฅ+๐ฆ C ๐ฅ + ๐ฆ โ 180 D 180 + ๐ฅ โ ๐ฆ E 180 โ ๐ฅ + ๐ฆ Try to gradually work out angles in the diagram (in terms of ๐ฅ and ๐ฆ) ๐ท Two possible ways: โ ๐ด๐ต๐ถ = ๐ฆ (ฮ๐ด๐ต๐ถ is isosceles) โ ๐ด๐ท๐ต = 180 โ ๐ฅ (angles on straight line sum to 180) Therefore using angles in ฮ๐ด๐ต๐ท: ๐ง = 180 โ ๐ฆ โ 180 โ ๐ฅ = 180 โ ๐ฆ โ 180 + ๐ฅ =๐ฅโ๐ฆ ? โ ๐ด๐ต๐ถ = ๐ฆ (ฮ๐ด๐ต๐ถ is isosceles) โ ๐ต๐ด๐ถ = 180 โ 2๐ฆ (angles in triangle ๐ด๐ต๐ถ sum to 180) โ ๐ท๐ด๐ถ = 180 โ ๐ฅ โ ๐ฆ (angles in triangle ๐ด๐ท๐ถ sum to 180) Therefore ๐ง = โ ๐ต๐ด๐ถ โ โ ๐ท๐ด๐ถ = 180 โ 2๐ฆ โ 180 โ ๐ฅ โ ๐ฆ =๐ฅโ๐ฆ ? Check Your Understanding How far can you get down the angle challenge wall? (do in order, and draw the diagram first) ๐ถ ๐ท ๏ โ ๐ด๐ถ๐ต = 180 โ 2๐ฅ° ? ๏ โ ๐ต๐ถ๐ท = 2๐ฅ โ 90° ? ๏ โ ๐ด๐ต๐ท = 180 โ ๐ฅ ๐ฅ ๐ด ๐ต ? Modelling restrictions on angles We have already seen we can represent the two base angles of an isosceles triangle both as ๐ฅ to ensure they have the same value. But how can we model other descriptions? ๐ถ 2๐ฅ ? ๐ฅ? ๐ด โAngle ๐ถ๐ต๐ด is twice the value of angle ๐ถ๐ด๐ต.โ ๐ต ๐ท โThe line ๐ด๐ถ bisects the angle ๐ท๐ด๐ต.โ ๐ถ ?๐ฅ ๐ด ๐ฅ? Bro Note: To โbisectโ means to cut in half. ๐ต Modelling restrictions on angles ๐ด In a triangle ๐ด๐ต๐ถ, โ ๐ถ๐ด๐ต = 2 × โ ๐ต๐ถ๐ด and โ ๐ด๐ต๐ถ = 7 × โ ๐ต๐ถ๐ด ๐ต a) Suggest suitable expressions for each of the angles. ๐, ๐๐,?๐๐ b) Hence determine โ ๐ต๐ถ๐ด. ๐ + ๐๐ + ๐๐ = ๐๐๐ ๐๐๐ = ๐๐๐ ? So ๐ = ๐๐° ๐ถ Exercise 4 1 (on provided worksheet) Given the value of โ ๐ท๐ต๐ด indicated, work out the value of โ ๐ท๐ต๐ถ. โ ๐ซ๐ฉ๐จ โ ๐ซ๐ฉ๐ช ๐ค 180 โ ๐ค ๐ฅ + 10 170?โ ๐ฅ 2๐ฅ โ 30 160?โ 2๐ฅ 90 + 2๐ฅ 90 ? โ 2๐ฅ 130 + ๐ฅ โ 2๐ฆ ๐ท ? ๐ด ๐ต ๐ถ 50 โ ? ๐ฅ + 2๐ฆ Given the values of โ ๐ด๐ต๐ถ and โ ๐ต๐ด๐ถ indicated, work out the value of โ ๐ด๐ถ๐ต. 2 ๐ด ๐ต ๐ถ โ ๐จ๐ฉ๐ช โ ๐ฉ๐จ๐ช โ ๐จ๐ช๐ฉ ๐ฅ ๐ฅ 180?โ 2๐ฅ ๐ฅ 10 170?โ ๐ฅ ๐ฅ + 10 150 20?โ ๐ฅ 2๐ฅ + 30 20 โ 5๐ฅ 130?+ 3๐ฅ 4๐ฅ โ 20 7๐ฅ โ 30 230 ? โ 11๐ฅ Exercise 4 3 (on provided worksheet) Determine all the angles in each diagram in terms of ๐ฅ and/or ๐ฆ. a c ๐ฆ 180?โ 2๐ฆ ๐ฅ ?๐ฆ ๐ฆ b ๐๐๐ โ?๐ โ ๐ d 90?โ ๐ฅ 180 โ 2๐ฅ ๐ฅ ๐๐ ? ๐ +?๐ Exercise 4 (on provided worksheet) h e ๐๐๐ โ?๐ 20 + ๐ฅ 150 โ ๐ฆ ๐๐ โ ๐?+ ๐ ๐ฅ f i ๐ฅ ? ๐ โ ๐๐๐ ? ๐๐๐ โ ๐๐ ๐๐๐ ?โ๐ ?๐ ๐ฅ g ๐ฆ โ 10 ๐ฅ + 20 ๐๐๐ โ?๐ โ ๐ ? ๐๐๐ โ ๐๐ ๐๐๐ โ ๐ ? Exercise 4 4 Write suitable expressions in terms of ๐ฅ for each missing angle. Hence determine the smallest angle in the diagram in each case. a Angle ๐ด๐ต๐ท is 3 times bigger than angle ๐ท๐ต๐ถ. ๐ท ? 3๐ฅ b ๐ฅ 4๐ = ๐๐๐ ? โด ๐ = ๐๐ ? ๐ด ๐ต ๐ถ Angle ๐ต๐ถ๐ด is twice as large as angle ๐ต๐ด๐ถ. ๐ด ๐ฅ? ๐๐° + ๐๐ = ๐๐๐° ๐ = ๐๐°? ? 2๐ฅ ๐ต c The angles ๐ด๐ถ๐ต, ๐ด๐ต๐ถ and ๐ต๐ด๐ถ are in the ratio 3: 4: 5. ๐ด ๐๐๐ = ๐๐๐ โด ๐ = ๐๐°? Smallest angle = ๐๐° 5๐ฅ ? 4๐ฅ? ๐ต ๐ถ ?3๐ฅ ๐ถ Exercise 4 5 [Based on JMO 2005 B4] In this figure ๐ด๐ท๐ถ is a straight line and ๐ด๐ต = ๐ต๐ถ = ๐ถ๐ท. Also, ๐ท๐ด = ๐ท๐ต. We wish to find the size of โ ๐ต๐ด๐ถ. Let โ ๐ท๐ด๐ต = ๐ฅ. ? ๐ฅ ? 2๐ฅ or 180 โ 3๐ฅ ? 180 โ 2๐ฅ ๐ฅ ? 180 โ 3๐ฅ ? ๐ฅ a) By using the information provided, determine all the remaining angles in the diagram in terms of ๐ฅ. b) By considering the three angles in ฮ๐ท๐ถ๐ต, hence determine โ ๐ต๐ด๐ถ. Solution: ๐๐° 6 [Based on JMC 2011 Q23] The points ๐, ๐, ๐ lie on the sides of the triangle ๐๐๐ , as shown, so that ๐๐ = ๐๐ and ๐ ๐ = ๐ ๐. โ ๐๐๐ = 40°. a) Let โ ๐๐๐ = ๐ฅ° and โ ๐๐๐ = ๐ฆ°. What is the value of ๐ฅ + ๐ฆ? Solution: ๐๐๐° b) Using the information provided, find expressions for โ ๐๐ ๐ and โ ๐๐๐. (See diagram) c) Hence find an expression for โ ๐ ๐๐. (See diagram) d) Using your answer to part (a), hence find the value of โ ๐ ๐๐. ๐๐ + ๐๐ โ ๐๐๐ = ๐๐๐ โ ๐๐๐ = ๐๐๐° ? ? ? 2๐ฅ + 2๐ฆ โ 180 ? 180 โ 2๐ฅ ๐ฅ ๐ฆ ? 180 โ 2๐ฆ Exercise 4 7 [Based on TMC Final 2014 Q4] The triangle ๐ด๐ต๐ถ is isosceles with ๐ด๐ถ = ๐ต๐ถ as shown. The point ๐ท lies on the line ๐ต๐ถ such that the triangle ๐ด๐ต๐ท is isosceles with ๐ด๐ต = ๐ต๐ท as shown. โ ๐ต๐ด๐ถ = 2 × โ ๐ต๐ถ๐ด and we wish to work out the value of โ ๐ด๐ท๐ถ. Suppose we let โ ๐ต๐ถ๐ด = ๐ฅ. Hence determine: a) โ ๐ต๐ด๐ถ ๐๐ (as specified) b) โ ๐ถ๐ต๐ด ๐๐ c) Hence by considering the triangle ๐ถ๐ด๐ท, determine ๐ฅ. ๐๐ = ๐๐๐° therefore ๐ = ๐๐° d) Use the remaining information to determine โ ๐ด๐ท๐ถ. ๐๐๐° ? ? ? 8 [JMO 2004 B1] In the rectangle ABCD, M and N are the midpoints of AB and CD respectively; AB has length 2 and AD has length 1. Given that โ ๐ด๐ต๐ท = ๐ฅ°, calculate โ ๐ท๐๐ in terms of ๐ฅ. ? 45° (because ฮ๐๐ต๐ถ is isosceles and right-angled) ? ๐ฅ Solution: ๐๐๐ ? โ๐ Exercise 4 9 [Based on JMO 2006 B3] In this diagram, Y lies on the line AC, triangles ABC and AXY are right angled and in triangle ABX, AX = BX. The line segment AX bisects angle BAC and angle AXY is seven times the size of angle XBC. Let โ ๐๐ต๐ถ = ๐ฅ. In terms of ๐ฅ, find expressions for: a) โ ๐ด๐๐ (see diagram) ? b) โ ๐ต๐๐ (hint: the lines ๐๐ and ๐ต๐ถ are parallel) (see diagram) ? c) โ ๐๐ด๐ (see diagram) ? d) โ ๐ต๐ด๐ (see diagram: AX?bisects โ ๐ฉ๐จ๐ช) e) โ ๐ต๐๐ด (using the fact that ฮ๐ต๐๐ด is isosceles) (see diagram) ? f) Optional: By considering angles around a suitable point, hence find โ ๐ด๐ต๐ถ. ๐๐๐ + ๐๐ + ๐๐๐ โ ๐ = ๐๐๐ ๐๐๐ + ๐๐๐ = ๐๐๐ ๐๐๐ = ? ๐๐๐ ๐ = ๐° 14๐ฅ โ ๐จ๐ฉ๐ช = ๐ + ๐๐ = ๐๐° 7๐ฅ 180 โ ๐ฅ ๐ฅ 90 โ 7๐ฅ Exercise 4 10 [JMC 2003 Q23] In the diagram alongside, ๐ด๐ต = ๐ด๐ถ and ๐ด๐ท = ๐ถ๐ท. How many of the following statements are true for the angles marked? ๐ค=๐ฅ ๐ฅ + ๐ฆ + ๐ง = 180 ๐ง = 2๐ฅ A all of them B two C one D none of them E it depends on ๐ฅ Solution: A ? [JMC 2005 Q23] In the diagram, triangle ๐๐๐ is isosceles, with 11 ๐๐ = ๐๐. What is the value of ๐ in terms of ๐ and ๐? 1 A ๐โ๐ 2 D ๐+๐ Solution: B 1 B ๐+๐ C ๐โ๐ 2 E Impossible to determine ? 12 [JMC 2015 Q25] The four straight lines in the diagram are such that ๐๐ = ๐๐. The sizes of โ ๐๐๐, โ ๐๐๐ and โ ๐๐๐ are ๐ฅ°, ๐ฆ° and ๐ง°. Which of the following equations gives ๐ฅ in terms of ๐ฆ and ๐ง? A ๐ฅ =๐ฆโ๐ง B ๐ฅ = 180 โ ๐ฆ โ ๐ง ๐ง ๐ฆโ๐ง C ๐ฅ=๐ฆโ D ๐ฅ = ๐ฆ + ๐ง โ 90 E ๐ฅ = 2 2 Solution: E ? Exercise 4 13 [JMO 2008 B3] In the diagram ABCD and APQR are congruent rectangles. The side PQ passes through the point D and โ ๐๐ท๐ด = ๐ฅ°. Find an expression for โ ๐ท๐ ๐ in terms of x. ๐ ๐ Solution: ๐ 1 ๐ฅ 2 ๐ฅ° 1 90 โ ๐ฅ 2 1 90 โ ๐ฅ 2 ? Proof Now that we have a number of angle skills, including introducing algebraic angles, we now have all the skills to form a โproofโ. A โproofโ is a sequence of justified statements that results in the desired conclusion. ! Examples (which weโll do later) ๐ด Using the diagram, prove that โ ๐ด๐ถ๐ท = โ ๐ด๐ต๐ถ + โ ๐ต๐ด๐ถ ๐ต ๐ถ ๐ท [JMO 2015 B2] The diagram shows triangle ๐ด๐ต๐ถ, in which โ ๐ด๐ต๐ถ = 72° and โ ๐ถ๐ด๐ต = 84°. The point ๐ธ lies on ๐ด๐ต so that ๐ธ๐ถ bisects โ ๐ต๐ถ๐ด. The point ๐น lies on ๐ถ๐ด extended. The point ๐ท lies on ๐ถ๐ต extended so that ๐ท๐ด bisects โ ๐ต๐ด๐น. Prove that ๐ด๐ท = ๐ถ๐ธ. 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. ๐ด โ๐ด๐ต๐ถ is an equilateral triangle.โ ๐ต โ๐ด๐ต๐ถ is a triangle such that ๐ด๐ต = ๐ต๐ถ. A point ๐ lies outside the triangle such that ๐๐ด = ๐๐ต. ๐ถ ๐ ๐ต ? ๐ด ๐ด โ๐ด๐ต๐ถ is a triangle where โ ๐ด๐ต๐ถ = 90°. A point ๐ท lies on ๐ด๐ถ such that โ ๐ด๐ท๐ต = 90°.โ ๐ถ ๐ท ? ๐ถ ๐ต ๐ด โThe point ๐น lies inside the regular pentagon ๐ด๐ต๐ถ๐ท๐ธ so that ๐ด๐ต๐น๐ธ is a rhombus.โ ๐ธ ? ๐น ๐ท ๐ถ ๐ต Bro Note: For parallelograms/ rhombuses, we need not indicate parallel sides because it is implied by the lengths. Bro Tip: We name โverticesโ (i.e. corners) using capital letters, and go either clockwise or anticlockwise around the shape. e.g. If โ๐ด๐ต๐ถ๐ท is a squareโ, then the first two diagrams are OK, but the last is not. ๐ด ๐ต ๐ถ ๐ต ๏ผ ๏ผ ๐ท ๐ท ๐ถ ๐ด ๐ด ๐ต ๏ป ๐ถ ๐ท A harder one for discussionโฆ Two of the angles of triangle ๐ด๐ต๐ถ are given by โ ๐ถ๐ด๐ต = 2๐ผ and โ ๐ด๐ต๐ถ = ๐ผ, where ๐ผ < 45°. The bisector of angle ๐ถ๐ด๐ต meets ๐ต๐ถ at ๐ท. The point ๐ธ lies on the bisector, but outside the triangle, so that โ ๐ต๐ธ๐ด = 90°. When produced, ๐ด๐ถ and ๐ต๐ธ meet at ๐. Construct the diagram. Note: To โbisectโ an angle is to cut it in half. So a โbisectorโ 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. ๐ด๐ต is parallel to ๐ถ๐ท and ๐ฟ๐๐ is a straight line. Prove that ๐ฆ = 112°. โ ๐ฟ๐๐ต = 68° (corresponding angles are equal) ? โ ๐ฟ๐๐ด = 112° (angles on a straight line sum to 180°) Tip: A helpful way to write out each step of the proof (when referring to angles) is in the form: โ ๐๐๐๐๐ = ๐๐๐๐๐ (reason) Test Your Understanding i ๐๐ = ๐๐ and ๐๐๐ is a straight line. Prove that โ ๐๐๐ = 40° โ ๐๐๐ = 70° (angles on straight line sum to 180°) โ ๐๐๐ = 70° ? triangle are equal) (base angles of isosceles โ ๐๐๐ = 40° (angles in triangle sum to 180°) ii (if you finish) ๐ด ๐ต๐ถ๐ท is a straight line. Let โ ๐ด๐ต๐ถ = ๐ฅ and โ ๐ต๐ด๐ถ = ๐ฆ. Prove that โ ๐ด๐ถ๐ท = ๐ฅ + ๐ฆ. ๐ฆ ๐ฅ ๐ต ๐ถ ๐ท โ ๐ด๐ถ๐ท = 180 โ ๐ฅ โ ๐ฆ° (angles in triangle sum to 180°) โ ๐ด๐ถ๐ท = ๐ฅ + ๐ฆ (angles on a ? straight line sum to 180°) Exercise 5a (on provided worksheet) 1 a Prove โ ๐ ๐๐ถ = 55°. Your proof should only require one line and be in the form โโ ๐๐๐๐๐ = ๐ฃ๐๐๐ข๐ ๐๐๐๐ ๐๐ โ โ ๐น๐ธ๐ช = ๐๐° (corresponding angles are equal) ? ๐ด b 85° ๐ต ๐ธ ? Prove that โ ๐ธ๐น๐ป = 85°. (Your proof should consist of two lines) ๐ท ๐ถ (Lots of possible proofs, but hereโs oneโฆ) โ ๐ช๐ญ๐ฎ = ๐๐° (corresponding angles are equal) โ ๐ฌ๐ญ๐ฎ = ๐๐° (vertically opposite angles are equal) ? ๐บ ๐น ๐ป c Prove that โ ๐ต๐ถ๐ท = 2๐ฅ. (Your proof should consist of three lines) ๐ต โ ๐จ๐ฉ๐ช = ๐ (base angles of isosceles triangle ๐จ๐ฉ๐ช are equal) โ ๐ฉ๐ช๐จ = ๐๐๐ โ ๐๐ (angles in triangle sum to ๐๐๐°) โ ๐ฉ๐ช๐ซ = ๐๐ (angles on straight line sum to ๐๐๐°) ? ๐ฅ ๐ด ? ๐ถ ๐ท Exercise 5a (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. ๐ถ a ๐ด๐ต๐ถ is a right-angled triangle such that โ ๐ถ๐ด๐ต = 90°. ๐ท is a point on ๐ต๐ถ such that ๐ด๐ท = ๐ต๐ท. ๐ท ? b ๐ต ๐ต ๐ด ๐ด๐ต๐ถ is an isosceles triangle where ๐ด๐ต = ๐ด๐ถ. ๐ท is a point that lies inside the triangle such that ๐ต๐ถ๐ท is equilateral. ?๐ท ๐ด ๐ถ ๐ท c 25° ? ๐ด d ๐ต ๐ถ Points ๐ด, ๐ต and ๐ถ lie in that order on a straight line. A point ๐ท, not on the line, is placed such that ๐ด๐ท = ๐ด๐ต, ๐ต๐ท = ๐ต๐ถ, and โ ๐ต๐ท๐ถ = 25°. ๐ ๐ Points ๐ and ๐ lie outside a parallelogram ๐๐๐ ๐, and are such that triangles ๐ ๐๐ and ๐๐ ๐ are equilateral and lie wholly outside the parallelogram. ๐ ? ๐ ๐ ๐ Exercise 5a 3 a (on provided worksheet) ๐ต ๐ด ๐ถ Prove that โ ๐ต๐ถ๐ด = 20° 40° โ ๐ฉ๐จ๐ซ = ๐๐° (base angles of isosceles ๐ซ are equal) โ ๐จ๐ฉ๐ซ = ๐๐๐° (angles in triangle sum to 180) โ ๐ช๐ฉ๐ซ = ๐๐° (angles on straight line sum to 180) โ ๐ฉ๐ซ๐ช = ๐๐° (base angles of isosceles ๐ซ are equal) โ ๐ฉ๐ช๐ซ = ๐๐° (angles in triangle sum to 180) ? ๐ท b ๐ต ๐ด 70° ๐ท ๐ถ ๐น In the diagram, ๐ด๐ต and ๐ถ๐ธ are parallel, and ๐ท๐น = ๐ท๐ธ. Prove that โ ๐ท๐ธ๐น = 35° ๐ธ โ ๐ฉ๐ซ๐ฌ = ๐๐° (alternate angles are equal) โ ๐ญ๐ซ๐ฌ = ๐๐๐° (angles on a straight line sum to 180) ? of isosceles ๐ซ๐ซ๐ฌ๐ญ are โ ๐ซ๐ฌ๐ญ = ๐๐° (as base angles equal and angles in triangle sum to 180) Other Types of Proof ๐ต ๐ต๐ถ๐ท is a straight line and ๐ด๐ต = ๐ด๐ถ. Prove that ๐ด๐ถ bisects the angle ๐ต๐ด๐ท. 70° ๐ถ ๐ด Before we start this proof, what specifically are we trying to show? That โ ๐ฉ๐จ๐ช?= โ ๐ช๐จ๐ซ 30° ๐ท Bro Note: You need a conclusion so itโs clear that your proof is complete.. Refer to the provided โcheat sheetโ. โ ๐ต๐ถ๐ด = 70° (base angles of isosceles triangle are equal) โ ๐ด๐ถ๐ท = 110° (angles on straight line sum to 180°) โ ๐ต๐ด๐ถ = 40° (angles in triangle sum to 180°) โ ๐ถ๐ด๐ท = 40° (angles in triangle sum to 180°) โ ๐ต๐ด๐ถ = โ ๐ถ๐ด๐ท therefore ๐ด๐ถ bisects โ ๐ต๐ด๐ท. Proof ? ๐ต Similarly, what would we need to do in order to: ๐ท โฆprove that ๐ด๐ต๐ถ is a straight line? โฆprove that ๐ด๐ต๐ถ is isosceles? ? Show that โ ๐ต๐ด๐ถ = โ ๐ต๐ถ๐ด Show that โ ๐ท๐ต๐ด + โ ๐ท๐ต๐ถ = 180° ๐ด ๐ต ๐ถ ๐ด ? ๐ถ Test Your Understanding i ๐ถ๐ท๐ต is a straight line and ๐ด๐ถ = ๐ด๐ท = ๐ท๐ต. โ ๐ท๐ด๐ต = 72° as shown. Prove that triangle ๐ด๐ต๐ถ is isosceles. ๐ถ ๐ท โ ๐ซ๐ฉ๐จ = ๐๐° (base angles of isosceles triangle ๐จ๐ซ๐ฉ are equal) ๐ต 36° โ ๐ช๐ซ๐จ = ๐๐° (exterior angle of triangle is sum of two other interior angles) Proof ? โ ๐ซ๐ช๐จ = ๐๐° (base angles of isosceles triangle ๐จ๐ช๐ซ are equal) ๐ด โ ๐ช๐จ๐ซ = ๐๐° (angles in triangle sum to ๐๐๐°) โ ๐ช๐จ๐ฉ = ๐๐° + ๐๐° = ๐๐° ii (if you finish) โ ๐ฉ๐ช๐จ = โ ๐ช๐จ๐ฉ = ๐๐° โด triangle ๐จ๐ฉ๐ช is isosceles. ๐ท In the diagram, ๐ด๐ต = ๐ต๐ถ = ๐ต๐ท and โ ๐ด๐ท๐ถ = 90°. Prove that ๐ด๐ต๐ถ is a straight line. ๐ถ ๐ต ๐ด Hint: You should choose a suitable angle to be ๐ฅ and start by writing โLet โ ๐๐ฆ๐๐๐๐๐ = ๐ฅโ Let โ ๐ซ๐จ๐ฉ = ๐. โ ๐จ๐ฉ๐ซ = ๐ (base angles of isosceles triangle are equal) โ ๐จ๐ฉ๐ซ = ๐๐๐ โ ๐๐ (angles in triangle sum to ๐๐๐°) โ ๐ฉ๐ซ๐ช = ๐๐ โ ๐ (as โ ๐จ๐ซ๐ช = ๐๐°) โ ๐ฉ๐ช๐ซ = ๐๐ โ ๐ (base angles of isosceles triangle are equal) โ ๐ซ๐ฉ๐ช = ๐๐ (angles in triangle ๐ฉ๐ช๐ซ sum to ๐๐๐°) โ ๐ซ๐ฉ๐จ + โ ๐ซ๐ฉ๐ช = ๐๐๐ โ ๐๐ + ๐๐ = ๐๐๐° therefore ๐จ๐ฉ๐ช is a straight line. Proof ? Exercise 5b (on provided worksheet) ๐ท ๐ต 1 a ๐ด ๐ด๐ท๐ถ and ๐ด๐ต๐ถ are right-angled triangles where โ ๐ด๐ถ๐ท = 10° and โ ๐ท๐ถ๐ต = 70°. Prove that triangle ๐ด๐ต๐ถ is isosceles. 10° โ ๐ช๐จ๐ซ = ๐๐° (angles in triangle ๐จ๐ช๐ซ add to ๐๐๐°). โ ๐จ๐ช๐ฉ = โ ๐จ๐ช๐ซ + โ ๐ซ๐ช๐ฉ = ๐๐° โ ๐ซ๐จ๐ช = โ ๐จ๐ช๐ฉ therefore ๐ซ๐จ๐ฉ๐ช is isosceles. 70° ? ๐ถ ๐ธ b ๐ต ๐ด๐ถ = ๐ต๐ถ, ๐ด๐ถ๐ท is a straight line, โ ๐ต๐ถ๐ด = 40° and โ ๐ต๐ถ๐ธ = 70° as per the diagram. Prove that ๐ด๐ต and ๐ถ๐ธ are parallel. 40° 70° ๐ด ๐ถ ๐ท โ ๐ฉ๐จ๐ช = ๐๐° (base angles in isosceles triangle ๐จ๐ฉ๐ช are equal) โ ๐ฌ๐ช๐ซ = ๐๐° (angles on straight line at ๐ช sum to ๐๐๐°) โ ๐ฉ๐จ๐ช = โ ๐ฌ๐ช๐ซ which are corresponding angles, therefore ๐จ๐ฉ and ๐ช๐ฌ are parallel. ? Exercise 5b 2 As before, ๐ด๐ถ = ๐ต๐ถ. Suppose also that ๐ถ๐ธ bisects the angle ๐ต๐ถ๐ท, but unlike before, no angles are known. By introducing a suitable variable (you may wish to start your proof โLet โ ๐๐๐๐๐ = ๐ฅโ) prove that ๐ด๐ต and ๐ถ๐ธ are parallel. ๐ธ ๐ต ๐ด ๐ถ ๐ท Let โ ๐ฌ๐ช๐ซ = ๐. โ ๐ฉ๐ช๐ฌ = ๐ (as ๐ช๐ฌ bisects โ ๐ฉ๐ช๐ซ). โ ๐ฉ๐ช๐จ = ๐๐๐ โ ๐๐ (angles on straight line at ๐ช sum to ๐๐๐°) ? ๐๐๐โ ๐๐๐โ๐๐ โ ๐ฉ๐จ๐ช = = ๐ (base angles of ๐ isosceles triangle ๐จ๐ฉ๐ช are equal) โ ๐ฉ๐จ๐ช = โ ๐ฌ๐ช๐ซ = ๐, and ๐จ๐ช๐ซ is a straight line, therefore ๐จ๐ฉ and ๐ช๐ฌ are parallel. Exercise 5b 3 [JMO 2001 B3] In the diagram, B is the midpoint of AC and the lines AP, BQ and CR are parallel. The bisector of โ ๐๐ด๐ต meets BQ at Z. Draw a diagram to show this, and join Z to C. (i) Given that โ ๐๐ด๐ = ๐ฅ°, find โ ๐๐ต๐ถ in terms of x. (ii) Show that CZ bisects โ ๐ต๐ถ๐ . ? Exercise 5b 4 [JMO 2015 B2] The diagram shows triangle ๐ด๐ต๐ถ, in which โ ๐ด๐ต๐ถ = 72° and โ ๐ถ๐ด๐ต = 84°. The point ๐ธ lies on ๐ด๐ต so that ๐ธ๐ถ bisects โ ๐ต๐ถ๐ด. The point ๐น lies on ๐ถ๐ด extended. The point ๐ท lies on ๐ถ๐ต extended so that ๐ท๐ด bisects โ ๐ต๐ด๐น. Prove that ๐ด๐ท = ๐ถ๐ธ. ? KS3/4: Polygons Skipton Girlsโ High School Objectives: Be able to classify shapes, reason about sides and angles, and find interior/exterior angles of polygons. STARTER: Identifying 2D polygons ! A polygon is a 2D shape with?straight sides. Sides: 3 Triangle ? Equilateral 4 Quadrilateral ? ? Square ? Parallelogram 5 6 7 Pentagon ? Hexagon ? Heptagon ? 8 9 10 ? ? Isosceles Scalene ? Rectangle ? Trapezium Octagon ? Nonagon ? Decagon ? ? Rhombus ? Arrowhead 12 20 Dodecagon ? Kite ? Icosagon ? Key Properties of Polygons Shape Description 2D Shape Polygon A 2D shape with straight edges. Ellipse (Also known as an oval) Circle Quadrilateral Polygon with four edges. Square Regular polygon with four edges. Rectangle Two pairs of parallel sides, all angles equal. Oblong All rectangles which are not squares. Trapezium Quadrilateral with at least one pair of parallel sides. Parallelogram Quadrilateral with two pairs of parallel sides. Rhombus Quadrilateral with all edges the same length. Kite Quadrilateral where sides in adjacent pairs are equal in length. Arrowhead Kite with a reflex angle. Properties of quadrilaterals Shape Name Lines of Num pairs of symmetry parallel sides Diagonals always equal in length? Diagonals perpendicular? Square 4 ? 2 ? Yes? Yes? Rectangle 2 ? 2 ? Yes? No ? Kite 1 ? 0 ? No ? Yes? Rhombus 2 ? 2 ? No ? Yes? Parallelogram 0 ? 2 ? No ? No ? Arrowhead 1 ? 0 ? No ? Yes? RECAP: Interior angles of quadrilateral The interior angles of a quadrilateral add up to 360๏ฐ. ? 1 Parallelogram 2 y 100° x x 50° x = 130° ? 3 ? y = 80° ? x = 100° Trapezium x x = 120° ? 4 Kite x 95° 55° 60° x = 105° ? Sum of interior angles n=3 Total of interior angles = 180° n=4 Total of interior angles = 360° Can you guess what the angles add up to in a pentagon? How would you prove it? Sum of interior angles Click to animate We can cut a pentagon into three triangles. The sum of the interior angles of the triangles is: 3 x 180° = 540° ! For an n-sided shape, the sum of the interior angles is: 180(n-2) ? Test Your Understanding A regular decagon (10 sides). 160° x x 80° 130° 120° x = 140° ? x = 144° ? 120° 40° x = 240° ? x 100° 40° Exercise 1 1a b c b = 260๏ฐ ? x = 75๏ฐ ? x = 25๏ฐ ? d e a = 100๏ฐ ? f x = 222๏ฐ ? x = 309๏ฐ ? Exercise 1 g h x = 54๏ฐ ? 2 i x = 120๏ฐ ? x = 252๏ฐ ? The total of the interior angles of a polygon is 1260°. How many sides does it have? ? N1 The interior angle of a regular polygon is 179°. How many sides does it have? ? Exercise 1 N2 If a n-sided polygon has exactly 3 obtuse angles (i.e. 90๏ฐ < ๏ฑ < 180๏ฐ), then determine the possible values of (Hint: determine the possible range for the sum of the interior angles, and use these inequalities to solve). ? Interior Angles An exterior angle of a polygon is an angle between the line extended from one side, and an adjacent side. Which of these are exterior angles of the polygon? ? NO ? YES NO? Interior Angles To defeat Kim Jon Il, Matt Damon must encircle his pentagonal palace. What angle does Matt Damon turn in total? 360° ? ! The sum of the exterior angles of any polygon is 360°. Click to Start animation Interior Angles If the pentagon is regular, then all the exterior angles are clearly the same. Therefore: Exterior angle of pentagon = 360 / 5?= 72° Interior angle of pentagon = 180 โ 72 ? = 108° Angles in Regular Polygons Num Sides Name of Exterior Regular Polygon Angle 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon ? 90° ? 72° ? 60° ? 51.4° ? 45° ? 40° ? 36° ? 120° Interior Angle ? 90° ? 108° ? 120° ? 128.6°? 135° ? 140° ? 144° ? 60° Bonus Question: What is the largest number of sides a shape can have such that its interior angle is an integer? 360 sides. The interior ? angle will be 179°. Test Your Understanding GCSE question The diagram shows a regular hexagon and a regular octagon. Calculate the size of the angle marked x. You must show all your working. x = 105° ? GCSE Question Hint: Fill in what angles you do know. You can work out what the interior angle of Tile A will be. Question: The pattern is made from two types of tiles, tile A and tile B. Both tile A and tile B are regular polygons. Work out the number of sides tile A has. Sides = 12 ? Test Your Understanding Q1 Q2 50 85 a 75 80 80 80 c a = 110° ? Q3 What is the exterior angle of a 180-sided regular polygon? 360 ÷ 180 ? = 2๏ฐ c = 70°? Q4 The interior angle of a regular polygon is 165. How many sides does it have? Interior angle = 180 โ 165 = 15 n = 360 ÷ 15 = 24 ? Alternative method: Total interior angle = 165n Then solve 180(n โ 2) = 165n