MATH 8_Angle-side, Hinge and Converse of Hinge Theorem.pptx

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1. Good day, students! 8th Grade

2. Angle-side Relationship Theorem 8th Grade

3. What is angle-side relationship theorem? In a triangle; If two angles of a triangle are not congruent , then the larger side is opposite the larger angle. This basically means the opposite of the largest angle is always the longest side or the opposite of the smallest angle will always be the shortest side. RS

4. What is angle-side relationship theorem? In a triangle; If two sides of a triangle are not congruent , then the larger angle is opposite the larger side. This basically means “the opposite of the longest side will always be the largest angle and the opposite of the shortest side will always be the smallest angle. ∠ C

5. Let’s take a look at the given examples. Remember: “ If two sides of a triangle are not congruent, then the larger angle is opposite the larger side.” Example 1. Which angle is the largest? Which angle is the smallest? Largest angle: ∠A Smallest angle: ∠C

6. Let’s take a look at the given examples. Remember: “ If two angles of a triangle are not congruent, then the larger side is opposite the larger angle. ” Example 2. Arrange the order of the sides from longest to shortest. Longest side: BC Longer side: AC Shortest side: AB BC ˃ AC ˃ AD

7. 8th Grade To fully understand angle-side relationship theorem, let’s watch this video: Link: https://www.youtube.com/watch?v=4ME9ms1nPtU

8. Hinge Theorem 8th Grade (SAS Inequality Theorem)

9. What is hinge theorem? 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 . So in short,  the triangle having a larger interior angle will also have a longer third side.

10. Example 3: Consider the example of  △ABC and  △XYZ. Let AB = XY and AC = XZ while the length of the side BC and YZ will depend upon the interior angle. Given the interior angle of A = 30 ° while the interior angle of X = 60 ° and t he two sides of the triangles are the same ( AB = XY and AC = XZ ) , the length of the third side varies . Using the hinge theore m, YZ is longer than BC YZ ˃ BC

11. LET’S TRY! YZ is longer than BC YZ ˃ BC 55   ° 78  °

12. 8th Grade To fully understand hinge and converse of hinge theorem, let’s watch this video: Link: https://www.youtube.com/watch?v=-mGDz9tZP2g

13. Converse of Hinge Theorem 8th Grade (SSS Inequality Theorem)

14. What is converse hinge theorem? If two sides of one triangle are congruent to two sides of another triangle , but the t hird 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. So in short,  the triangle having the longer third side will also have a larger included angle .

15. Example 4: Consider the example of  △ B AC and  △ EDF . Let B A = ED and AC = DF and BC = 10 and EF = 8, the m∠A and m∠ D depends on the third side. As you can see,  AB = DE, AC = DF,   and BC = 10 > EF = 8.  Thus, by the Converse of Hinge Theorem or SSS Inequality Theorem, we know that ; m∠A is greater than m∠D

16. LET’S TRY! m∠ D  is greater than m∠ A 15 17

17. Let’s take a look at the given examples. Given: 1st side: SR = NL; 2nd side: ST = ML; included angles: ∠ S = 54°; and ∠ L = 71° Find: Which third side is longer? Is it RT or NM? Example 5. Complete the statement with >, ∠ or =. Determine what theorem is used. RT ____ NM _____________ > Hinge Theorem Since m ∠ L > m ∠ S, therefore by Hinge Theorem, RT > NM

18. Let’s take a look at the given examples. Given: 1st sides: TS = TO; 2nd sides: TP = TP; third sides: PS = 16 cm and PO = 11 cm Find: Which included angle is largest? Example 6. Complete the statement with >, ∠ or =. Determine what theorem is used. m∠STP _____ m∠OTP _____________________ > Converse of Hinge Theorem Since PS > PO, therefore by Converse of Hinge Theorem, ∠ STP > ∠OTP

19. 8th Grade To fully understand converse of hinge theorem, let’s watch this video: Link: https://www.youtube.com/watch?v=OG0QWelNBPw

20. 8th Grade To fully understand converse of hinge theorem, let’s watch this video: Link: https://www.youtube.com/watch?v=eWotAj7wvpo

21. “Mathematics is not just about numbers, equations, computations or algorithms: it is about UNDERSTANDING.” — William Paul Thurston

22. Riza Mae Bayo PREPARED BY: MATH 8- Practice Teacher

23. Thanks! Do you have any questions? You can contact me through: [email protected] + 639 611 925 220 RIZA MAE BAYO