Pearson Unit 1 Topic 5: Relationships Within Triangles 5-7: Inequalities in One Triangle Pearson Texas Geometry ©2016 Holt Geometry Texas ©2007

TEKS Focus: (5)(D) Verify the Triangle Inequality theorem using constructions and apply the theorem to solve problems. (1)(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate. (1)(G) Display, explain, or justify mathematical ideas and arguments using precise mathematical language in written or oral communication. (6)(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems.

Recall this previous theorem:

The positions of the longest and shortest sides of a triangle are related to the positions of the largest & smallest angles.

A triangle is formed by three segments, but not every set of three segments can form a triangle.

Shortcut: The sum of the lengths of the two shortest sides of a triangle is greater than the length of the longest side.

Example: 1 A town park is triangular. A landscape architect wants to place a bench at the corner with the largest angle. Which two streets form the corner with the largest angle? Hollingsworth Road is the longest Street, so it is opposite the largest angle. MLK Boulevard and Valley Road form the largest angle.

Example: 2 Your family is currently in Oklahoma driving north on I-35 with a final destination of Topeka, Kansas. You can continue north to Salina and then turn east to Topeka. Or you can go directly from Wichita to Topeka. Which is the shortest choice? Why? The cities of Wichita, Salina, and Topeka roughly form a triangle. The shortest route is directly from Wichita to Topeka. The sum of 2 lengths of sides in a triangle is longer than the third side of the triangle.

Example: 3 Write the angles in order from smallest to largest. The shortest side is , so the smallest angle is F. The longest side is , so the largest angle is G. The angles from smallest to largest are F, H and G.

Example: 4 Write the sides in order from shortest to longest. mR = 180° – (60° + 72°) = 48° The smallest angle is R, so the shortest side is . The largest angle is Q, so the longest side is . The sides from shortest to longest are

Example: 5 Tell whether a triangle can have sides with the given lengths. Explain. 2.3, 3.1, 4.6    Yes—the sum of each pair of lengths is greater than the third length.

Example: 6 Do you need to check out the other two Tell whether a triangle can have sides with the given lengths. Explain. 7, 10, 19 Do you need to check out the other two possibilities? No, it is good enough to just sum the two shortest lengths to compare to the longest length. No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths.

Example: 7 Tell whether a triangle can have sides with the given lengths. Explain. n + 6, n2 – 1, 3n, when n = 4. Step 1 Evaluate each expression when n = 4. n + 6 n2 – 1 3n 4 + 6 (4)2 – 1 3(4) 10 15 12 Step 2 Compare the lengths. Yes—the sum of each pair of lengths is greater than the third length.

Example: 8 Can a side length be a negative number? No! The lengths of two sides of a triangle are 8 inches and 13 inches. Find the range of possible lengths for the third side. Let x represent the length of the third side. Then apply the Triangle Inequality Theorem. x + 8 > 13 x + 13 > 8 8 + 13 > x x > 5 x > –5 21 > x Can a side length be a negative number? No! Write a compound inequality: 5 < x < 21. The length of the third side is greater than 5 inches and less than 21 inches.

Example: 9 The figure shows the approximate distances between cities in California. What is the range of distances from San Francisco to Oakland? Did you notice a shortcut in the last example to get the two numbers in the inequality? To get the smaller number you subtract the two side lengths of the triangle. To get the larger number you sum the two side lengths of the triangle. 5 < x < 97 The distance from San Francisco to Oakland is greater than 5 miles and less than 97 miles.

Example: 10 The distance from San Marcos to Johnson City is 50 miles, and the distance from Seguin to San Marcos is 22 miles. What is the range of distances from Seguin to Johnson City? 28 < x < 72 The distance from Seguin to Johnson City is greater than 28 miles and less than 72 miles.