Given: bisects Given: In rt. ,

diagonals. In such a case, the formula of Theorem 2.5.1 is most convenient to use. Al- though this theorem is given without proof, Exercise 39 of this section provides some insight for the proof. EXAMPLE 1 Use Theorem 2.5.1 to find the number of diagonals for any pentagon. Solution To use the formula of Theorem 2.5.1, we note that n ⫽ 5 in a pentagon. Then . 쮿 D = 55 - 3 2 = 52 2 = 5 Reminder The sum of the interior angles of a triangle is 180°. Exs. 1–5 SUM OF THE INTERIOR ANGLES OF A POLYGON The following theorem provides the formula for the sum of the interior angles of any polygon. The sum S of the measures of the interior angles of a polygon with n sides is given by . Note that for any polygon. n 7 2 S = n - 2 180° THEOREM 2.5.2 Let us consider an informal proof of Theorem 2.5.2 for the special case of a penta- gon. The proof would change for a polygon of a different number of sides but only by the number of triangles into which the polygon can be separated. Although Theorem 2.5.2 is also true for concave polygons, we consider the proof only for the case of the convex polygon. Proof Consider the pentagon ABCDE in Figure 2.33 with auxiliary segments diagonals from one vertex as shown. With angles marked as shown in triangles ABC, ACD, and ADE, m 1 ⫹ m 2 ⫹ m 3 ⫽ 180⬚ m 6 ⫹ m 5 ⫹ m 4 ⫽ 180⬚ m 8 ⫹ m 9 ⫹ m 7 ⫽ 180⬚ m E ⫹ m A ⫹ m D ⫹ m B ⫹ m C ⫽ 540⬚ adding For pentagon ABCDE, in which , the sum of the measures of the interior angles is , which equals 540°. When drawing diagonals from one vertex of a polygon of n sides, we always form triangles. The sum of the measures of the interior angles always equals . n - 2 180° n - 2 5 - 2 180° n = 5 ∠ ∠ ∠ ∠ ∠ ∠ ∠ ∠ ∠ ∠ ∠ ∠ ∠ ∠ E D C B 2

3 4

5 7 8 1 6 9 A Figure 2.33 The total number of diagonals D in a polygon of n sides is given by the formula . D = n n - 3 2 THEOREM 2.5.1 Theorem 2.5.1 reaffirms the fact that a triangle has no diagonals; when , D = 33 - 3 2 = 0. n = 3 쮿 REGULAR POLYGONS Figure 2.34 shows polygons that are, respectively, a equilateral, b equiangular, and c regular both sides and angles are congruent. Note the dashes that indicate congru- ent sides and the arcs that indicate congruent angles. EXAMPLE 2 Find the sum of the measures of the interior angles of a hexagon. Then find the measure of each interior angle of an equiangular hexagon. Solution For the hexagon, , so the sum of the measures of the interior angles is or 4180° or 720°. In an equiangular hexagon, each of the six interior angles measures , or 120°. 쮿 720 ° 6 S = 6 - 2 180° n = 6 EXAMPLE 3 Find the number of sides in a polygon whose sum of interior angles is 2160°. Solution Here in the formula of Theorem 2.5.2. Because , we have . Then The polygon has 14 sides. 쮿 n = 14 180n = 2520 180n - 360 = 2160 n - 2 180 = 2160 S = 2160 Exs. 6–9 a Figure 2.34 b c A regular polygon is a polygon that is both equilateral and equiangular. DEFINITION The polygon in Figure 2.34c is a regular pentagon. Other examples of regular polygons include the equilateral triangle and the square. Based upon the formula from Theorem 2.5.2, there is also a formula for the measure of each interior angle of a regular polygon having n sides. It applies to equiangular polygons as well. S = n - 2 180° The measure I of each interior angle of a regular polygon or equiangular polygon of n sides is . I = n - 2 180° n COROLLARY 2.5.3 A second corollary to Theorem 2.5.2 concerns the sum of the interior angles of any quadrilateral. For the proof, we simply let in the formula . Then . Also, see the Discover at the left. S = 4 - 2 180° = 2 180° = 360° S = n - 2 180° n = 4 EXAMPLE 4 Find the measure of each interior angle of a ceramic floor tile in the shape of an equiangular octagon Figure 2.35. Solution For an octagon, . Then Each interior angle of the tile measures 135°. NOTE: For the octagonal tiles of Example 4, small squares are used as “fillers” to cover the floor. The pattern, known as a tessellation, is found in Section 8.3. 쮿 = 1080 8 , so I = 135° = 6 180 8 I = 8 - 2 180 8 n = 8 EXAMPLE 5 Each interior angle of a certain regular polygon has a measure of 144°. Find its number of sides, and identify the type of polygon it is. Solution Let n be the number of sides the polygon has. All n of the interior angles are equal in measure. The measure of each interior angle is given by Then multiplying by n With 10 sides, the polygon is a regular decagon. 쮿 n = 10 36n = 360 180n - 360 = 144n n - 2 180 = 144n n - 2 180 n = 144 I = n - 2 180 n where I = 144 Discover From a paper quadrilateral, cut the angles from the “corners.” Now place the angles so that they have the same vertex and do not overlap. What is the sum of measures of the four angles? ANSWER 360° Exs. 10–12 Figure 2.35 On the basis of Corollary 2.5.4, it is clearly the case that each interior angle of a square or rectangle measures 90°. The following interesting corollary to Theorem 2.5.2 can be established through algebra. The sum of the four interior angles of a quadrilateral is 360°. COROLLARY 2.5.4 We now consider an algebraic proof for Corollary 2.5.5. Proof A polygon of n sides has n interior angles and n exterior angles, if one is considered at each vertex. As shown in Figure 2.36, these interior and exterior angles may be grouped into pairs of supplementary angles. Because there are n pairs of angles, the sum of the measures of all pairs is degrees. Of course, the sum of the measures of the interior angles is . In words, we have Let S represent the sum of the measures of the exterior angles. The next corollary follows from Corollary 2.5.5. The claim made in Corollary 2.5.6 is applied in Example 6. ‹ S = 360 - 360 + S = 0 180n - 360 + S = 180n n - 2 180 + S = 180n Sum of Measures Sum of Measures Sum of Measures of All of Interior Angles of Exterior Angles Supplementary Pairs n - 2 180° 180 n The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°. COROLLARY 2.5.5 EXAMPLE 6 Use Corollary 2.5.6 to find the number of sides of a regular polygon if each interior angle measures 144°. Note that we are repeating Example 5. Solution If each interior angle measures 144°, then each exterior angle measures 36° they are supplementary, because exterior sides of these adjacent angles form a straight line. Now each of the n exterior angles has the measure In this case, , and it follows that , so . The polygon a decagon has 10 sides. 쮿 n = 10 36n = 360 360 n = 36 360° n The measure E of each exterior angle of a regular polygon or equiangular polygon of n sides is . E = 360° n COROLLARY 2.5.6 POLYGRAMS A polygram is the star-shaped figure that results when the sides of convex polygons with five or more sides are extended. When the polygon is regular, the resulting poly- gram is also regular—that is, the interior acute angles are congruent, the interior reflex 1 n 1 2 2 3

3 4

4 n Figure 2.36 Exs. 13, 14 ⫹ ⫽ 쮿 angles are congruent, and all sides are congruent. The names of polygrams come from the names of the polygons whose sides were extended. Figure 2.37 shows a pentagram, a hexagram, and an octagram. With congruent angles and sides indicated, these figures are regular polygrams. Exs. 15, 16 Pentagram Figure 2.37 Hexagram Octagram Geometry in Nature The starfish has the shape of a pentagram. Exercises 2.5

1. As the number of sides of a regular polygon increases,

does each interior angle increase or decrease in measure?

2. As the number of sides of a regular polygon increases,

does each exterior angle increase or decrease in measure?

3. Given:

, , , with angle measures as indicated Find: x , y, and z

4. In pentagon ABCDE with

, find the measure of interior angle D.

5. Find the total number of diagonals for a polygon of n

sides if: a b

6. Find the total number of diagonals for a polygon of n

sides if: a b

7. Find the sum of the measures of the interior angles of a

polygon of n sides if: a b

8. Find the sum of the measures of the interior angles of a

polygon of n sides if: a b

9. Find the measure of each interior angle of a regular

polygon of n sides if: a b n = 12 n = 4 n = 8 n = 6 n = 10 n = 5 n = 8 n = 6 n = 10 n = 5 ∠B ⬵ ∠D ⬵ ∠E AE 7 FC AD 7 BC AB 7 DC

10. Find the measure of each interior angle of a regular

polygon of n sides if: a b

11. Find the measure of each exterior angle of a regular

polygon of n sides if: a b

12. Find the measure of each exterior angle of a regular

polygon of n sides if: a b

13. Find the number of sides that a polygon has if the sum of

the measures of its interior angles is: a 900° b 1260°

14. Find the number of sides that a polygon has if the sum of

the measures of its interior angles is: a 1980° b 2340°

15. Find the number of sides that a regular polygon has if the

measure of each interior angle is: a 108° b 144°

16. Find the number of sides that a regular polygon has if the

measure of each interior angle is: a 150° b 168°

17. Find the number of sides in a regular polygon whose

exterior angles each measure: a 24° b 18°

18. Find the number of sides in a regular

polygon whose exterior angles each measure: a 45° b 9°

19. What is the measure of each interior angle

of a stop sign? n = 10 n = 6 n = 12 n = 4 n = 10 n = 6 z 36 ° x A F B D E C 77 ° y A C D E B 93 93

20. Lug bolts are equally

spaced about the wheel to form the equal angles shown in the figure. What is the measure of each of the equal acute angles? In Exercises 21 to 26, with P ⫽ {all polygons} as the universe, draw a Venn Diagram to represent the relationship between these sets. Describe a subset relationship, if one exists. Are the sets described disjoint or equivalent? Do the sets intersect?

21. T

⫽ {triangles}; I ⫽ {isosceles triangles}

22. R

⫽ {right triangles}; S ⫽ {scalene triangles}

23. A

⫽ {acute triangles}; S ⫽ {scalene triangles}

24. Q

⫽ {quadrilaterals}; L ⫽ {equilateral polygons}

25. H

⫽ {hexagons}; O ⫽ {octagons}

26. T

⫽ {triangles}; Q ⫽ {quadrilaterals}

27. Given: Quadrilateral RSTQ with exterior

at R and T Prove: m ∠1 + m∠2 = m∠3 + m∠4 ∠s

31. A father wishes to make a home plate for his son to use in

practicing baseball. Find the size of each of the equal angles if the home plate is modeled on the one in a and if it is modeled on the one in b. 1 3 2 4 S R T Q a b ? ? ?

28. Given: Regular hexagon

ABCDEF with diagonal and exterior Prove:

29. Given: Quadrilateral RSTV with diagonals

and intersecting at W Prove: m ∠1 + m∠2 = m∠3 + m∠4 SV RT m ∠2 + m∠3 = m∠1 ∠1 AC A F C D E B 2 3 1 1 2

3 4

W S R V T

30. Given:

Quadrilateral ABCD with and Prove: B and D are supplementary ∠s BC ⬜ DC BA ⬜ AD A B C D

32. The adjacent interior and exterior angles of a certain

polygon are supplementary, as indicated in the drawing. Assume that you know that the measure of each interior angle of a regular polygon is a Express the measure of each exterior angle as the supplement of the interior angle. b Simplify the expression in part a to show that each exterior angle has a measure of . 360 n n - 2180 n . 1 2

33. Find the measure of each acute interior angle of a regular

pentagram.

34. Find the measure of each acute interior angle of a regular

octagram.

35. Consider any regular polygon; find and join in order the

midpoints of the sides. What does intuition tell you about the resulting polygon?

36. Consider a regular hexagon RSTUVW. What does intuition

tell you about 䉭 , the result of drawing diagonals , , and ?

37. The face of a clock has the shape of a

regular polygon with 12 sides. What is the measure of the angle formed by two consecutive sides?

38. The top surface of a picnic table is in

the shape of a regular hexagon. What is the measure of the angle formed by two consecutive sides?

39. Consider a polygon of n sides

determined by the n noncollinear vertics A, B, C, D, and so on. a Choose any vertex of the polygon. To how many of the remaining vertices of the polygon can the selected vertex be joined to form a diagonal? b Considering that each of the n vertices in a can be joined to any one of the remaining vertices to form diagonals, the product appears to represent the total number of diagonals possible. However, this number n n - 3 n - 3 VR TV RT RTV ? A B C D E ? 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.