1 / 20

5.5 Inequalities in One Triangle

5.5 Inequalities in One Triangle. Geometry Mrs. Spitz Fall, 2004. Objectives:. Use triangle measurements to decide which side is longest or which angle is largest. Use the Triangle Inequality. Assignment. pp. 298-300 #1-25, 34.

terrel
Download Presentation

5.5 Inequalities in One Triangle

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 5.5 Inequalities in One Triangle Geometry Mrs. Spitz Fall, 2004

  2. Objectives: • Use triangle measurements to decide which side is longest or which angle is largest. • Use the Triangle Inequality

  3. Assignment pp. 298-300 #1-25, 34

  4. In activity 5.5, you may have discovered a relationship between the positions of the longest and shortest sides of a triangle and the position of its angles. Objective 1: Comparing Measurements of a Triangle The diagrams illustrate Thms. 5.10 and 5.11.

  5. If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. Theorem 5.10 mA > mC

  6. If one ANGLE of a triangle is larger than another ANGLE, then the SIDE opposite the larger angle is longer than the side opposite the smaller angle. Theorem 5.11 60° 40° EF > DF You can write the measurements of a triangle in order from least to greatest.

  7. Write the measurements of the triangles from least to greatest. m G < mH < m J JH < JG < GH Ex. 1: Writing Measurements in Order from Least to Greatest 100° 45° 35°

  8. Write the measurements of the triangles from least to greatest. QP < PR < QR m R < mQ < m P Ex. 1: Writing Measurements in Order from Least to Greatest 8 7 5

  9. Paragraph Proof – Theorem 5.10 Given►AC > AB Prove ►mABC > mC Use the Ruler Postulate to locate a point D on AC such that DA = BA. Then draw the segment BD. In the isosceles triangle ∆ABD, 1 ≅ 2. Because mABC = m1+m3, it follows that mABC > m1. Substituting m2 for m1 produces mABC > m2. Because m2 = m3 + mC, m2 > mC. Finally because mABC > m2 and m2 > mC, you can conclude that mABC > mC.

  10. NOTE: The proof of 5.10 in the slide previous uses the fact that 2 is an exterior angle for ∆BDC, so its measure is the sum of the measures of the two nonadjacent interior angles. Then m2 must be greater than the measure of either nonadjacent interior angle. This result is stated in Theorem 5.12

  11. Theorem 5.12-Exterior Angle Inequality • The measure of an exterior angle of a triangle is greater than the measure of either of the two non adjacent interior angles. • m1 > mA and m1 > mB

  12. Ex. 2: Using Theorem 5.10 • DIRECTOR’S CHAIR. In the director’s chair shown, AB ≅ AC and BC > AB. What can you conclude about the angles in ∆ABC?

  13. Because AB ≅ AC, ∆ABC is isosceles, so B ≅ C. Therefore, mB = mC. Because BC>AB, mA > mC by Theorem 5.10. By substitution, mA > mB. In addition, you can conclude that mA >60°, mB< 60°, and mC < 60°. Ex. 2: Using Theorem 5.10Solution

  14. Objective 2: Using the Triangle Inequality • Not every group of three segments can be used to form a triangle. The lengths of the segments must fit a certain relationship.

  15. Ex. 3: Constructing a Triangle • 2 cm, 2 cm, 5 cm • 3 cm, 2 cm, 5 cm • 4 cm, 2 cm, 5 cm Solution: Try drawing triangles with the given side lengths. Only group (c) is possible. The sum of the first and second lengths must be greater than the third length.

  16. 2 cm, 2 cm, 5 cm 3 cm, 2 cm, 5 cm 4 cm, 2 cm, 5 cm Ex. 3: Constructing a Triangle

  17. The sum of the lengths of any two sides of a Triangle is greater than the length of the third side. AB + BC > AC AC + BC > AB AB + AC > BC Theorem 5.13: Triangle Inequality

  18. A triangle has one side of 10 cm and another of 14 cm. Describe the possible lengths of the third side SOLUTION: Let x represent the length of the third side. Using the Triangle Inequality, you can write and solve inequalities. x + 10 > 14 x > 4 10 + 14 > x 24 > x ►So, the length of the third side must be greater than 4 cm and less than 24 cm. Ex. 4: Finding Possible Side Lengths

  19. Solve the inequality: AB + AC > BC. (x + 2) +(x + 3) > 3x – 2 2x + 5 > 3x – 2 5 > x – 2 7 > x #24 - homework

  20. AB + BC > AC MC + CG > MG 99 + 165 > x 264 > x x + 99 < 165 x < 66 66 < x < 264 5. Geography

More Related