ANGLES KHM

These figures are not polygonsThese figures are polygons Definition:A closed figure formed by a finite number of coplanar segments so that each segment intersects exactly two others, but only at their endpoints. Polygons

Interior Angle of a Polygon For example, ∆ ABC has interior angles:  ABC,  BAC,  BCA The interior angles of a polygon are the angles inside the polygon, formed by two adjacent sides.

Exterior Angle of a Polygon For example, ∆ ABC has exterior angle:  ACD. It forms a linear pair with  ACB. An exterior angle of a polygon is an angle that forms a linear pair with an interior angle. It is an angle outside the polygon formed by one side and one extended side of the polygon. A B C D Exterior Angle Interior Angles

What is the sum of the measures of the interior angles of a convex n-gon? Polygon Number of Sides Sum of Measures of Interior Angles Triangle3180° Quadrilateral4360° Pentagon5540° Hexagon6720° n-gonn(n - 2)180°

What is the sum of the measures of the exterior angles of a 3-gon? (a triangle) Sum of Measures of Exterior Angles = 360 

What is the measure of each exterior angle of a regular 3-gon? ? ? ?

What is the measure of each exterior angle of a regular 4-gon? ? ? ? ?

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

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 Corresponding angles are equal a b d e f h

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

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 Alternate angles are equal a b g h d f

Angles in a triangle For any triangle, ab c a + b + c = 180° The angles in a triangle add up to 180°.

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

Interior and exterior angles in a triangle a b c Any exterior angle in a triangle is equal to the sum of the two opposite interior angles. a = b + c We can prove this by constructing a line parallel to this side. These alternate angles are equal. These corresponding angles are equal. b c

Interior and exterior angles in a triangle

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

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