The length of an altitude of an acute triangle is less than

In algebra, it is shown that ; not by coincidence, the set ⭋, which has 0 elements, has 1 subset. Just as 2 1 = 2, the set {a} which has 1 element, has 2 subsets. The pattern continues so that a set with 2 elements has 2 2 ⫽ 4 subsets and a set with 3 elements has 2 3 ⫽ 8 subsets. A quick examination suggests this fact: 2 = 1 Looking back at Example 1, we notice that the number of sub- sets of the four-element set {a, b, c, d} is , which equals 16, or . The preceding principle can be restated in the following equivalent form: 2 4 1 + 4 + 6 + 4 + 1 EXAMPLE 3 The sixth row of Pascal’s Triangle is 1 5 10 10 5 1. Use the principle above to find the sum of the entries of this row. Solution With , it follows that . Then , or 32. 쮿 1 + 5 + 10 + 10 + 5 + 1 = 2 5 n - 1 = 5 n = 6 EXAMPLE 1 List all 16 subsets of the set {a, b, c, d} by considering the fifth row of Pascal’s Triangle, namely 1 4 6 4 1. Notice also that 1 + 4 + 6 + 4 + 1 must equal 16. Solution ⭋, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d}. 쮿 EXAMPLE 2 Find the number of subsets for a set with six elements. Solution The number of subsets is 2 6 , or 64. 쮿 NOTE: There are 32 subsets for a set containing five elements; consider {a, b, c, d, e}. In closing, we note that only a few of the principles based upon Pascal’s Triangle have been explored in this Perspective on Application The total number of subsets for a set with n elements is . 2 n The sum of the entries in row n of Pascal’s Triangle is . 2 n - 1 The entries of the fifth row of Pascal’s Triangle correspond to the numbers of subsets of the four-element set {a, b, c, d}; of course, the subsets of {a, b, c, d} must have 0 elements, 1 element each, 2 elements each, 3 elements each, or 4 elements each. Based upon the preceding principle, there will be a total of subsets for {a, b, c, d}. 2 4 = 16 Pascal’s Triangle Set Number of Elements Subsets of the Set Number of Subsets 1 ⭋ ⭋ 1 1 1 {a} 1 ⭋, {a} subsets 1 + 1 2 1 2 1 {a, b} 2 ⭋, {a}, {b}, {a, b} subsets 1 + 2 + 1

4 1

3 3 1 {a, b, c} 3 ⭋, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} subsets 1 + 3 + 3 + 1 8 1 subset of 0 elements, 3 subsets of 1 element each, 3 subsets of 2 elements each, 1 subset of 3 elements Summary A LOOK BACK AT CHAPTER 3 In this chapter, we considered several methods for prov- ing triangles congruent. We explored properties of isosceles triangles and justified construction methods of earlier chapters. Inequality relationships for the sides and angles of a triangle were also investigated. A LOOK AHEAD TO CHAPTER 4 In the next chapter, we use properties of triangles to de- velop the properties of quadrilaterals. We consider several special types of quadrilaterals, including the parallelo- gram, kite, rhombus, and trapezoid. KEY CONCEPTS

3.1

Congruent Triangles • SSS, SAS, ASA, AAS • Included Angle, Included Side • Reflexive Property of Congruence Identity • Symmetric and Transitive Properties of Congruence 3.2 CPCTC • Hypotenuse and Legs of a Right Triangle • HL • Pythagorean Theorem • Square Roots Property 3.3 Isosceles Triangle • Vertex, Legs, and Base of an Isosceles Triangle • Base Angles • Vertex Angle • Angle Bisector • Median • Altitude • Perpendicular Bisector • Auxiliary Line • Determined, Underdetermined, Overdetermined • Equilateral and Equiangular Triangles • Perimeter

3.4

Justifying Constructions 3.5 Lemma • Inequality of Sides and Angles of a Triangle • The Triangle Inequality Methods of Proving Triangles Congruent: 䉭ABC ⬵ 䉭DEF FIGURE NOTE MARKS METHOD STEPS NEEDED IN PROOF SSS , and BC ⬵ EF AC ⬵ DF, AB ⬵ DE SAS , , and AC ⬵ DF ∠A ⬵ ∠D AB ⬵ DE ASA , and ∠C ⬵ ∠F AC ⬵ DF, ∠A ⬵ ∠D AAS , , and BC ⬵ EF ∠C ⬵ ∠F ∠A ⬵ ∠D HL and are rt. , and BC ⬵ EF AC ⬵ DF, ∠s ∠D ∠A TABLE 3.2 An Overview of Chapter 3 B C A E F D B C A E F D B C A E F D B C A E F D B C A E F D Special Relationships FIGURE RELATIONSHIP CONCLUSION Pythagorean Theorem c 2 = a 2 + b 2 two sides ⬵ DF ⬵ EF opposite ∠s ⬵ ∠E ⬵ ∠D two angles ⬵ ∠D ⬵ ∠E opposite sides ⬵ EF ⬵ DF A C c b a B F D E F D E Inequality Relationships in a Triangle FIGURE RELATIONSHIP CONCLUSION ST 7 RS opposite angles m ∠R 7 m∠T m ∠Y 7 m∠X opposite sides XZ 7 YZ S T R X Y Z TABLE 3.2 continued Chapter 3 REVIEW EXERCISES

1. Given:

Prove: 䉭AEB ⬵ 䉭DEC AE ⬵ ED ∠AEB ⬵ ∠DEC

6. Given: B is the midpoint of

Prove: is isosceles 䉭ADC BD ⬜ AC AC A D B C E H G A B C D E F 1 3 2

2. Given:

Prove: ∠B ⬵ ∠E ∠1 ⬵ ∠2 AC ⬵ DF AB ⬵ EF

3. Given: bisects

Prove: AE ⬵ ED DC ⬜ BC AB ⬜ BC BC AD A B E C D A B C O

4. Given:

is the median to Prove: OC ⬜ AB AB OC OA ⬵ OB

5. Given:

Prove: BC 7 FE AC ⬵ DF AB 7 DE AB ⬵ DE A D F C B E A B C D

7. Given:

and Prove: GM ⬵ JK GH ⬵ HJ GK ⬜ KJ JM ⬜ GM M K G J H N O P T R S

8. Given:

Prove:

9. Given: is the base of an isosceles triangle;

Prove: ∠1 ⬵ ∠2 XA 7 YZ YZ ∠N ⬵ ∠R TO ⬵ TS TS ⬜ PR TO ⬜ NP TN ⬵ TR Y Z X A B 1 2

10. Given:

C is the midpoint of Prove: AC 7 DE BE AB ⬵ DC AB 7 DC B E C A D

11. Given:

Prove: HINT: Prove first. 䉭 BAD ⬵ 䉭CDA AE ⬵ ED AB ⬵ CD ∠BAD ⬵ ∠CDA

17. In not shown,

, , and . List the angles in order of size, starting with the smallest angle.

18. Name the longest line segment shown in quadrilateral

ABCD. QR = 2.5 PR = 2 PQ = 1.5 䉭PQR A D B C E 1 2 A E D B C A D C B E

12. Given: is the altitude to

is the altitude to Prove: HINT: Prove . 䉭 CBE ⬵ 䉭CDA BE ⬵ AD BC ⬵ CD CE AD AC BE

13. Given:

Prove: is isosceles HINT: Prove by CPCTC. ∠CAD ⬵ ∠BDA 䉭AED ∠BAD ⬵ ∠CDA AB ⬵ CD

14. Given: bisects

Prove: AD 7 CD ∠BAD AC B D C A 1 2

15. In not shown,

and . a Name the shortest side. b Name the longest side.

16. In not shown,

and . List the sides in order of their lengths, starting with the smallest side. m ∠B = 65° m ∠A = 40° 䉭ABC m ∠Q = 23° m ∠P = 67° 䉭PQR D A B C 80° 55° 30°

19. Which of the following can be the lengths of the sides of a

triangle? a 3, 6, 9 b 4, 5, 8 c 2, 3, 8 20. Two sides of a triangle have lengths 15 and 20. The length of the third side can be any number between ? and ? .

21. Given:

Find: m ∠ADB m ∠C = 70° AD ⬵ DC DB ⬜ AC C B A D A C B D

22. Given:

Find: m ∠ADC m ∠B = 50° ∠DAC ⬵ ∠BCD AB ⬵ BC

23. Given: is isosceles with base

Find: m ∠C m ∠4 = 5 2 x + 18 m ∠2 = 3x + 10 AB 䉭ABC E F C B A D 1 2 4 3 Exercises 23, 24

24. Given: with perimeter 40

Find: Whether is scalene, isosceles, or equilateral 䉭ABC AC = 2x - 3 BC = x + 6 AB = 10 䉭ABC E F C B A D 1 2 4 3 Exercises 25, 26 c A 25. Given: is isosceles with base