Name of Lecturer: Mr. J.Agius Course: HVAC1 Lesson 46 Chapter 9: Angles and Shapes Quadrilaterals A quadrilateral is any four-sided shape. Any quadrilateral can be split up into two triangles by drawing in a diagonal, like this: y The sum of the four angles in any quadrilateral is 360 i.e. o x w + x + y + z = 360o w z You can find this by just splitting the quadrilateral into two triangles. i.e. ABC = 180o and D ADC = 180o A So the sum of the two triangles is equal to 180o + 180o = 360o which is equal to the sum of the angles of a quadrilateral since these two triangles form a quadrilateral. B C Types of Quadrilaterals There are special types of quadrilateral : the rectangle the rhombus the square (those are all parallelograms), and there is also: the trapezoid the kite If it isn't one of those it is an irregular quadrilateral. 9 Angles & Shapes Page 1 Name of Lecturer: Mr. J.Agius Course: HVAC1 Here are the details: The Rectangle means "right angle" and show equal sides A rectangle is a four-sided shape where every angle is a right angle (90°). Also opposite sides are parallel and of equal length. The Rhombus A rhombus is a four-sided shape where all sides have equal length. Also opposite sides are parallel and opposite angles are equal. Another interesting thing is that the diagonals (dashed lines in second figure) of a rhombus bisect each other at right angles. The Square means "right angle" show equal sides A square has equal sides and every angle is a right angle (90°) Also opposite sides are parallel. A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length). 9 Angles & Shapes Page 2 Name of Lecturer: Mr. J.Agius Course: HVAC1 The Parallelogram Opposite sides are parallel and equal in length, and opposite angles are equal (angles "a" are the same, and angles "b" are the same) NOTE: Squares, Rectangles and Rhombuses are all Parallelograms! Example: if a parallelogram has all sides equal and angles "a" and "b" are right angles, then it is also a square. The Trapezoid (or Trapezium) Trapezoid Regular Trapezoid A trapezoid (UK: trapezium) has one pair of opposite sides parallel. It is called a regular trapezoid if the sides that aren't parallel are equal in length and both angles coming from a parallel side are equal, as shown. A trapezoid is not a parallelogram because only one pair of sides is parallel. The Kite Hey, it looks like a kite. It has two pairs of sides. Each pair is made up of adjacent sides that are equal in length. The angles are equal where the pairs meet. Diagonals (dashed lines) meet at a right angle, and one of the diagonal bisects (cuts equally in half) the other. ... and that's it for the special quadrilaterals; if it doesn't fit one of those it is an Irregular Quadrilateral: 9 Angles & Shapes Page 3 Name of Lecturer: Mr. J.Agius Course: HVAC1 Irregular Quadrilaterals Any quadrilateral that doesn't match one of the previous types. Polygon A quadrilateral is a polygon. In fact it is a 4-sided polygon, just like a triangle is a 3-sided polygon, a pentagon is a 5-sided polygon, and so on. Example 1 Work out the value of x. 136o Answer Rule: The sum of interior angles of a quadrilateral is equal to 360o. So x + 136o + 45o + 100o = 360o x + 281o = 360o so x = 360o – 281o = 79o 9 Angles & Shapes 45o xo 100o Page 4 Name of Lecturer: Mr. J.Agius Course: HVAC1 Exercise 1 Find the angles of each marked angle and give an appropriate reason. 1) 2) 112o t 124o y 80o s 70 o Ans. y = ___________ Ans. t = ___________ s = _____________ Reason:_________________________ Reason:_____________________________ Reason:_____________________________ 3) 82o 135o Ans. a = _____ Reason:________________ Ans. b = _____ Reason:________________ 124o 4) a b c Ans. c = _____ Reason:________________ 133o x Ans. x = _____ Reason:________________ Ans. y = _____ Reason:________________ y 100o 5) g o 70 h 120o Ans. g = _____ Reason:________________ Ans. h = _____ Reason:________________ j i Ans. i = _____ Reason:________________ Ans. j = _____ Reason:________________ 9 Angles & Shapes Page 5 Name of Lecturer: Mr. J.Agius Course: HVAC1 6) Ans. c = _____ Reason:________________ c f Ans. d = _____ Reason:________________ e Ans. e = _____ Reason:________________ 105o d Ans. f = _____ Reason:________________ 7) 116o Ans. c = _____ Reason:________________ Ans. d = _____ Reason:________________ e d c 55o 8) 87o k Ans. e = _____ Reason:________________ Ans. j = _____ Reason:________________ l Ans. k = _____ Reason:________________ 58o j 9 Angles & Shapes 73o Ans. l = _____ Reason:________________ Page 6 Name of Lecturer: Mr. J.Agius Course: HVAC1 Exercise 2 1) 1 2 3 4 6 5 a) Write down the special name for each of these four-sided shapes. b) For each shape say which statements in the list below are true. i. ii. iii. iv. v. vi. 2. The sides are equal Both pairs of opposite sides are parallel Only opposite angles are equal There is just one pair of parallel sides The four sides are made up of just two distinct lengths. The diagonals make an angle of 90o where they meet. In the following questions, some of the diagrams contain more than one quadrilateral. Name each quadrilateral with the appropriate letters and find the size of each marked angle. a) b) s o 86o o p xo o 35 ro 9 Angles & Shapes to qo zo 57o yo Page 7 Name of Lecturer: Mr. J.Agius Course: HVAC1 c) d) qo o 66 po zo 25o xo yo 3. A 3xo ro B 4xo 2xo D a) b) 50o xo C Find the value of x. Prove that ABCD is a trapezium. 4. ABCD is a trapezium with AB parallel to DC. A = 127o and C =76o. Work out B and D. (Hint: draw a diagram first.) 5. JKLM is a rectangle. JK is the longer side. N is a point on JK such that KN = KL. Given MNL = 79o work out JMN. 6. PQRS is a rhombus with the diagonal QS equal to PQ. Work out P. 7. ABCD is a rectangle. E is a point on AB. a) What type of quadrilateral is BCDE? b) Given ADE = 50o work out DEB. c) If DE = EB work out EBD. 9 Angles & Shapes Page 8

© Copyright 2022 Paperzz