Illustrating Theorems on Triangle Inequalities.pptx

School purposes

1. MATH 8- 4 th Quarter ILLUSTRATING THEOREMS ON TRIANGLE INEQUALITIES

2. At the end of the lesson, students are expected to: Illustrate Triangle Inequality Theorems; Showing the importance of Triangle Inequality Theorems in real-life situation; and Evaluate illustrations on Triangle Inequality Theorems. Lesson Objectives:

3. An inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. Common Symbol are , , , , .   WHAT IS INEQUALITY?

4. These theorems can be illustrated in ONE TRIANGLE and in TWO TRIANGLES. VARIOUS THEOREMS ON TRIANGLE INEQUALITIES

5. Most commonly used… INEQUALITIES IN ONE TRIANGLE Angle-Side Relationship Theorem, Triangle Inequality Theorem, And, Exterior Angle Inequality Theorem.

6. Most commonly used… INEQUALITIES IN TWO TRIANGLES The Hinge Theorem (SAS Inequality Theorem) AND The Converse of Hinge Theorem (SSS Inequality Theorem)

7. INEQUALITIES IN ONE TRIANGLE

8. If two angles of a triangle are not congruent, then the larger side is opposite the larger angle. If two sides of a triangle are not congruent, then the larger angle is opposite the larger side. ANGLE-SIDE RELATIONSHIP

9. LET’S SEE SOME EXAMPLES!

10. If two angles of a triangle are not congruent, then the larger side is opposite the larger angle. A B C    

11. If two angles of a triangle are not congruent, then the larger side is opposite the larger angle. A B C     SOLUTION: Step 1: Find the measure of the third angle. The sum of all the angles in any triangle is 180º.

12. If two angles of a triangle are not congruent, then the larger side is opposite the larger angle. A B C     SOLUTION: Step 2: Look at the relative sizes of the angles and compare.

13. If two angles of a triangle are not congruent, then the larger side is opposite the larger angle. A B C     SOLUTION: Step 3: Following the angle-side relationship we can order the sides accordingly. Remember it is the side opposite the angle.

14. If two sides of a triangle are not congruent, then the larger angle is opposite the larger side. A B C      

15. If two sides of a triangle are not congruent, then the larger angle is opposite the larger side. A B C       SOLUTION: Step 1: Since the length of the sides were given, we can easily compare the lengths from shortest to longest.

16. If two sides of a triangle are not congruent, then the larger angle is opposite the larger side. A B C       SOLUTION: Step 2: Following the angle-side relationship we can order the angles opposite to these sides accordingly.

17. The sum of the lengths of any two sides of a triangle is greater than the length of the remaining side. Let a, b, and c be the lengths of a triangle. These lengths may only form a triangle if the three conditions are satisfied: ; ; .   TRIANGLE INEQUALITY THEOREM Let b an unknown side of a triangle. To find the range of possible measure of side b, the inequality below may be used: (a- c) < b < (a + c)

18. LET’S SEE SOME EXAMPLES!

19. In , find the range of the possible lengths of   𝒂+𝒃 >𝒄 𝒂+𝒄 >𝒃 𝒄+𝒃 >𝒂

20. Which length/s can form a triangle? 𝒂+𝒃 >𝒄 𝒂+𝒄 >𝒃 𝒄+𝒃 >𝒂

21. The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle. EXTERIOR ANGLE INEQUALITY THEOREM

22. LET’S SEE SOME EXAMPLES!

23. If and , then by Exterior Angle Inequality Theorem:   The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.

24. INEQUALITIES IN TWO TRIANGLES

25. If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is greater than the included angle of the second, then the third side of the first triangle is longer than the third side of the second. HINGE THEOREM OR SAS INEQUALITY

26. LET’S SEE SOME EXAMPLES!

27. Compare the lengths of the third side of and .   Compare these:

28. Compare the lengths of the third side of and .   Based on the angle-side theorem, the opposite side of the greater angle is longer side.

29. If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second, then the included angle of the first triangle is larger than the included angle of the second. CONVERSE OF HINGE THEOREM OR SSS INEQUALITY THEOREM

30. LET’S SEE SOME EXAMPLES!

31. Given that and have two congruent sides as shown in the figure, which angle is greater, or ?   Compare these:

32. Given that and have two congruent sides as shown in the figure, which angle is greater, or ?   With these, Given: Solution:

33. In summary,

34. NOW IT’S YOUR TURN! Let’s have an activity to know how far you learned for today’s lesson.

35. RESOURCES