Gr 10 Maths – Euclidean Geometry GR ... - The …

1 Gr 10 Maths Euclidean Geometry Copyright The Answer 1 Geometry is a fun topic in which we work with lines angles triangles quadrilaterals polygons while we investigate and discover as much as we can for ourselves; explain what we've found, make conjectures, and then proceed to confirm these various properties by proof. As important as the facts which we discover is the language we use to state these facts. So, be sure to study the vocabulary as well as the facts which you gather. REVISION Lines, Angles & Triangles The Language of Geometry (Vocabulary) Make sure you know the meanings of all the WORDS we use in Geometry .

2 Lines Parallel and perpendicular Parallel lines: AB || CD Perpendicular lines: AB CD Angles their names Angles (single): Angles can be acute, obtuse or reflex; a right angle, a straight angle or a revolution. Angles (pairs): Complementary s add up to 90 40 and 50 ; x and 90 - x Supplementary s add up to 180 135 and 45 ; x and 180 - x Adjacent s have a common vertex and a common arm and lie on opposite sides of the common arm (side) A pair of adjacent A pair of adjacent supplementary s: complementary s: The ANGLE is the amount of rotation about the vertex.

3 40 140 A BC D50 40 acuteanglesobtuseanglesreflexangles0 90 180 270 360 a right angle a straight anglea revolution A CB D ACBD90 NOTE: The plural of vertex is vertices! ABCD135 45 x180 - x x40 50 90 - x armsvertexcommon arm commonvertex REVISION Lines, Angles & Triangles .. 1 The Language of Geometry (Vocabulary) .. 1 || and lines names of angles types of s The Facts .. 4 Lines & Angles Triangles quadrilaterals .. Revision of Properties of quadrilaterals .. Sides Angles Diagonals Defining quadrilaterals .. NB: See 'Pathways of definitions, properties and areas' Theorems and Proofs An assignment.

4 Area of quadrilaterals & Triangles .. A summary of formulae for areas of quadrilaterals Important facts on areas of quadrilaterals & triangles THE MIDPOINT THEOREM .. Investigation, proofs and application POLYGONS .. Definition & types of polygons .. regular/irregular polygons congruent & similar polygons Sum of Interior s of Polygons .. Sum of Exterior s of Polygons .. TOPIC OUTLINEGR 10 Maths Euclidean Geometry Gr 10 Maths Euclidean Geometry Copyright The Answer 2 When 2 lines intersect, we have .. ( adjacent supplementary s 1 and 4 and 4 and 3, etc. ( vertically opposite s, 1 and 3 or 2 and 4 or, when 2 lines are cut by a transversal.))

5 2 'families' of 4 angles are formed Taking one from each family, we have the following pairs: Pairs of corresponding s: 1 2 3 4 & 8 Pairs of alternate s: 3 & 5 and 4 & 6 Pairs of co-interior s: 4 & 5 and 3 & 6 ( Corresponding means their positions correspond 1 2 3 4 & 8 ( Interior s: 3, 4, 5 & 6 on the inside & Exterior s: 1, 2, 7 & 8 on the outside ( We say 1, 4, 5 and 8 are alternate to 2, 3, 6 and 7, the two groups lie on opposite sides of the transversal whereas, 'co-' means: 'on the same side of the transversal'. corresponding s interior alternate s co-interior s Often the transversal doesn't go right across both || lines.)))

6 Types of Triangles In an isosceles triangle: We can classify s (triangles) according to sides and s (angles) simultaneously. This is a/an .. -angled This is a/an .. -angled This is a/an .. -angled The Facts Lines & Angles Now that you know the meanings of the words, let's revise the FACTS. Intersecting Lines FACT 1 When two lines intersect, any pair of adjacent angles is supplementary. FACT 2 When 2 lines intersect, the vertically opposite angles are equal. Special case: perpendicular lines The sizes of the other angles? Parallel Lines 3 types of pairs of angles: corresponding ; alternate ; and co-interior FACT 3 If 2 parallel lines are cut by a transversal, corresponding angles are equal ; alternate angles are equal ; and co-interior angles are 23 4 NOTE: We are just talking about the naming, not the relationships ( whether s are equal or supplementary, etc.)

7 Answers: (1) An isosceles right- d (2) An isosceles acute- d (3) A scalene obtuse- d :1 2 3 45 6 7 82 intersecting linesform four angles 1 2 3 45678 When 2 lines are cut by a transversal, 2 'families' of four angles are formed: whether the lines are parallel or not! 12345678!7124356871243568 BASEthe vertical angle the base angles 1 2 3 4 Classification according to .. SIDES ANGLES Scalene (all 3 sides different in length) Acute d (all 3 s are acute) Isosceles (2 sides equal in length) Right d (one = 90 ) Equilateral (all 3 sides equal in length) Obtuse d (one is obtuse) 12435687the transversal(3)(2)(1) Gr 10 Maths Euclidean Geometry : Exercise_Questions Copyright The Answer 3 Converse Facts 1 The original statement (see FACT 1) is: If ABC is a straight line, then 1 + 2 = 180 Converse statement: If 1 + 2 = 180 , then ABC is a straight line.

8 Note: Given: x = 40 and y = 130 Question: Is PQR a straight line? Answer: x + y = 40 + 130 = 170 180 No, PQR is not a straight line .. 2 If the original statement is FACT 3, then the Converse statement: IF corresponding angles are equal; OR IF alternate angles are equal; OR IF co-interior angles are supplementary, THEN, line AB is parallel to line CD .. whether it looks like it or not! 1. 140 and 40 are adjacent supplementary angles: The angle supplementary to x is: 2. In this figure : x + 90 + y = 180 .. s on a straight line x + y = and so they are called angles.

9 3. Is ADB a straight line? Give a reason for your answer. 4. 5. x + y = 180 .. s on a straight line and z + y = 180 .. s on a straight line 6. Given: reflex AOD= 200 (see figure below). Reasons: Obtuse AOD= BOD= AOC= 7. 1 and 2are 1 and 2are 1 and 2are s (NAME) s (NAME) s (NAME) RELATIONSHIP: RELATIONSHIP: RELATIONSHIP: 8. 1 and are corresponding s and they are 1 and are alternate s and they are 1 and are co-interior s and they are 9. NB: It is ONLY BECAUSE THE LINES ARE , that the corresponding and alternate s ARE EQUAL and the co-interior s are SUPPLEMENTARY !

10 !! 12 ABCABCDy x P QRWhether itlooks likeitor not!40 140 ?x(1)y x( )( )145 45 A DBC( ) ( )x90 - xA DBCE100 40 60 ?( )x?( )The sum of adjacent anglesabout a point is 360 . When two lines intersect, the vertically opposite s are equal. Why?(5)OACBD200 ( )( )( ) ( )( ) ( ) ( )( )( ) ( )( )123 4 56( )( ) ( )( ) ( )( ) (9)( ) 12121 2 EXERCISE Revision: Lines and Angles QUESTIONSxzyGr 10 Maths Euclidean Geometry Copyright The Answer 4 The Facts Triangles FACT 1: Sum of the interior angles of a triangle The sum of the (interior) angles of a triangle is 180 . FACT 2: The exterior angle of a triangle The exterior angle of a triangle equals the sum of the interior opposite angles.