What is the measure of each interior angle

LINE SYMMETRY In the figure below, rectangle ABCD is said to have symmetry with respect to line 艎 be- cause each point to the left of the line of symmetry or axis of symmetry has a correspon- ding point to the right; for instance, X and Y are corresponding points. 2.6 includes duplications, such as and . What expression actually represents D, the total number of diagonals in a polygon of n sides?

40. For the concave quadrilateral ABCD,

explain why the sum of the interior angles is 360°. HINT: Draw .

41. If , , and

, find the measure of the reflex angle at vertex D. HINT: See Exercise 40. m ∠C = 31° m ∠B = 88° m ∠A = 20° BD CA AC

42. Is it possible for a polygon to have the following sum of

measures for its interior angles? a 600° b 720°

43. Is it possible for a regular polygon to have the following

measures for each interior angle? a 96° b 140° A B C D Exercises 40, 41 Symmetry Line of Symmetry Axis of Symmetry Point Symmetry Transformations Slides Translations Reflections Rotations Symmetry and Transformations 2.6 KEY CONCEPTS A X D C B Y Figure 2.38 A D C B X Y A figure has symmetry with respect to a line 艎 if for every point A on the figure, there is a second point B on the figure for which 艎 is the perpendicular bisector of AB. DEFINITION In particular, ABCD of Figure 2.38 has horizontal symmetry with respect to line 艎. That is, a vertical axis of symmetry leads to a pairing of corresponding points on a horizon- tal line. In Example 1 on page 108, we see that a horizontal axis leads to vertical symmetry for points. In Chapter 4, we will discover formal definitions of the types of quadrilaterals known as the parallelogram, square, rectangle, kite, rhombus, and rectangle. Some of these are included in Examples 3 and 5. EXAMPLE 1 Rectangle ABCD in Figure 2.38 on page 107 has a second line of symmetry. Draw this line or axis for which there is vertical symmetry. Solution Line m determined by the midpoints of and is the desired line of symmetry. As shown in Figure 2.39b, R and S are located symmetrically with respect to line m. 쮿 BC AD Geometry in Nature Like many of nature’s creations, the butterfly displays line symmetry. © Photo Researchers,Inc. EXAMPLE 2 a Which letters shown below has have a line of symmetry? b Which letters has have more than one line of symmetry? B D F G H Solution a B, D, and H as shown b H as shown Discover The uppercase block form of the letter A is shown below. Does it have symmetry with respect to a line? ANSWER Yes, line 艎 as shown is a line of symmetry. This vertical line 艎 is the only line of symmetry for the uppercase A. A D C B m Figure 2.39 A D C B m R S 쮿 a b