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PART 3 Angle Relationships

PART 3 Angle Relationships. Angle Relationships. Lesson Objectives - Students will: Discover angle properties such as: Complementary, Supplementary, vertically opposite, Corresponding, Alternate, interior. Investigate properties of triangles and quadrilaterals.

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PART 3 Angle Relationships

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  1. PART 3 Angle Relationships

  2. Angle Relationships • Lesson Objectives - Students will: • Discover angle properties such as: Complementary, Supplementary, vertically opposite, Corresponding, Alternate, interior. • Investigate properties of triangles and quadrilaterals

  3. Angles on a straight line and at a point a Angles at a point add up to 360 Angles on a line add up to 180 b b d c a c a + b + c = 180° a + b + c + d = 360 because there are 180° in a half turn. because there are 360 in a full turn.

  4. Angles on a straight line Angles on a line add up to 180. a b a + b = 180° because there are 180° in a half turn.

  5. Complementary and supplementary angles Two supplementary angles add up to 180°. Two complementary angles add up to 90°. a b a b a + b = 90° a + b = 180°

  6. Angles on a straight line

  7. Angles around a point

  8. Vertically opposite angles a d b c When two lines intersect, two pairs of vertically opposite angles are formed. and a = c b = d Vertically opposite angles are equal.

  9. Angles made with parallel lines When a straight line crosses two parallel lines eight angles are formed. a b d e c f h This line is called a traversal. g Which angles are equal to each other?

  10. Corresponding, alternate and interior angles Corresponding angles are equal Alternate angles are equal Interior angles add up to 180° a a a b b b a = b a = b a + b = 180° Look for an F-shape Look for a Z-shape Look for a C- or U-shape

  11. Angles and parallel lines

  12. Angles made with parallel lines

  13. Calculating angles Calculate the size of angle a. 28º Hint: Add another parallel line. a 45º 73º a = 28º + 45º =

  14. Interior and exterior angles in a triangle

  15. Calculating angles Calculate the size of the lettered angles in each of the following triangles. 116° b 33° a 82° 64° 34° 31° 43° c 25° d 131° 152° 127° 272°

  16. Interior angles in polygons b c a The angles inside a polygon are called interior angles. The sum of the interior angles of a triangle is 180°.

  17. Angles in a triangle We can prove that the sum of the angles in a triangle is 180° by drawing a line parallel to one of the sides through the opposite vertex. a b c a b These angles are equal because they are alternate angles. Call this angle c. a + b + c = 180° because they lie on a straight line. The angles a, b and c in the triangle also add up to 180°.

  18. Calculating angles in a triangle Calculate the size of the missing angles in each of the following triangles. 64° b 116° 33° a 326° 31° 82° 49° 43° 25° d 88° c 28° 233°

  19. Calculating angles Calculate the size of the lettered angles in each of the following triangles. 116° b 33° a 82° 64° 34° 31° 43° c 25° d 131° 152° 127° 272°

  20. Calculating angles Calculate the size of the lettered angles in this diagram. 38º 38º 56° 73° 86° a b 69° 104° Base angles in the isosceles triangle = (180º – 104º) ÷ 2 = 76º ÷ 2 = 38º = 86º Angle a = 180º – 56º – 38º = 69º Angle b = 180º – 73º – 38º

  21. Sum of the interior angles in a quadrilateral What is the sum of the interior angles in a quadrilateral? d c f a e b We can work this out by dividing the quadrilateral into two triangles. a + b + c = 180° and d + e + f = 180° So, (a + b + c)+ (d + e + f )= 360° The sum of the interior angles in a quadrilateral is 360°.

  22. Sum of interior angles in a polygon We have just shown that the sum of the interior angles in any quadrilateral is 360°. d a c b We already know that the sum of the interior angles in any triangle is 180°. c a + b + c = 180 ° a b a + b + c + d = 360 ° Do you know the sum of the interior angles for any other polygons?

  23. Sum of the interior angles in a polygon We’ve seen that a quadrilateral can be divided into two triangles … … a pentagon can be divided into three triangles … … and a hexagon can be divided into four triangles. How many triangles can a hexagon be divided into?

  24. Interior angles in regular polygons A regular polygon has equal sides and equal angles. We can work out the size of the interior angles in a regular polygon as follows: Equilateral triangle 180° 180° ÷ 3 = 60° Square 2 × 180° = 360° 360° ÷ 4 = 90° Regular pentagon 3 × 180° = 540° 540° ÷ 5 = 108° Regular hexagon 4 × 180° = 720° 720° ÷ 6 = 120°

  25. Sum of the interior angles in a polygon The number of triangles that a polygon can be divided into is always two less than the number of sides. We can say that: A polygon with n sides can be divided into (n– 2) triangles. The sum of the interior angles in a triangle is 180°. So, The sum of the interior angles in an n-sided polygon is (n– 2) × 180°.

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