Summarizing Angles

Angles in a triangle For any triangle, a b c a + b + c = 180°

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. These angles are equal because they are alternate angles. a a b b Call this angle c. 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°.

Sum of the interior angles in a pentagon What is the sum of the interior angles in a pentagon? We can work this out by using lines from one vertex to divide the pentagon into three triangles. a + b + c = 180°and d + e + f = 180° So,( a + b + c ) + ( d + e + f ) + ( g + h + i ) = 540° The sum of the interior angles in a pentagon is 540°. c a b and g + h + i = 180° d f e g i h

Sum of the interior angles in a polygon We’ve seen that a quadrilateral can be divided into two triangles … … and a pentagon can be divided into three triangles. How many triangles can a hexagon be divided into? A hexagon can be divided into four triangles. To find the interior angles of a convex polygon: (n-2) x 180

Angles made with parallel lines

Types of angle Acute angle 0º < a < 90º Acute angle 0º < a < 90º a Right angle a = 90º Right angle a = 90º a Obtuse angle 90º < a < 180º Obtuse angle 90º < a < 180º a Reflex angle 180º < a < 360º a

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

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

Intersecting lines

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

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

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

dd hh a b c e f g Corresponding angles There are four pairs of corresponding angles, or F-angles. a b c e f g d = h because Corresponding angles are equal

ee aa b c d f g h Corresponding angles There are four pairs of corresponding angles, or F-angles. b c d f g h a = e because Corresponding angles are equal

gg cc Corresponding angles There are four pairs of corresponding angles, or F-angles. c = g because a b d e f h Corresponding angles are equal

ff Corresponding angles There are four pairs of corresponding angles, or F-angles. b = f because a b c d e g h b Corresponding angles are equal

ff dd Alternate angles There are two pairs of alternate angles, or Z-angles. d = f because Alternate angles are equal a b c e g h

cc ee Alternate angles There are two pairs of alternate angles, or Z-angles. c = e because a b g h d f Alternate angles are equal