q4-ppt-w2-applying-theorems-on-triangle-inequalities_compress (1).pptx

This PowerPoint presentation provides an in-depth exploration of the Triangle Inequality Theorem and its applications in geometry. It covers the fundamental principles of triangle inequalities, including Triangle Inequality Theorem 1, 2, and 3, as well as the Exterior Angle Inequality Theorem and the Hinge Theorem with its Converse.

The presentation includes clear explanations, step-by-step proofs, real-world applications, and problem-solving examples to help learners understand how triangle inequalities determine the possible side lengths and angle relationships in a triangle. Visual aids such as diagrams, solved examples, and practice exercises are incorporated to enhance comprehension and engagement.

By the end of the presentation, learners will be able to apply the Triangle Inequality Theorem to solve problems, determine whether given side lengths form a triangle, and analyze angle-side relationships within one or more triangles

1. APPLYING THEOREMS ON TRIANGLE INEQUALITIES

2. INEQUALITIES IN TRIANGLES INEQUALITIES IN ONE TRIANGLE INEQUALITIES IN TWO TRIANGLES TRIANGLE INEQUALITY 1 TRIANGLE INEQUALITY 2 TRIANGLE INEQUALITY 3 EXTERIOR ANGLE INEQUALITY THEOREM HINGE THEOREM CONVERSE OF HINGE THEOREM

3. INEQUALITIES IN TRIANGLES INEQUALITIES IN ONE TRIANGLE TRIANGLE INEQUALITY 1 If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side. SIDE – ANGLE THEOREM

4. INEQUALITIES IN TRIANGLES INEQUALITIES IN ONE TRIANGLE TRIANGLE INEQUALITY 1 If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side. SIDE – ANGLE THEOREM O G T 8 in ∠𝐎 , ∠𝐆, ∠𝐓 𝐄𝐗𝐀𝐌𝐏𝐋𝐄 𝟏 : List down the angles of ∆ 𝐺𝑂𝑇 from greatest to least measure.

5. INEQUALITIES IN TRIANGLES INEQUALITIES IN ONE TRIANGLE TRIANGLE INEQUALITY 2 If one angle of a triangle is larger than a second angle, then the side opposite the first angle is longer than the side opposite the second angle. ANGLE – SIDE THEOREM

6. INEQUALITIES IN TRIANGLES INEQUALITIES IN ONE TRIANGLE TRIANGLE INEQUALITY 2 If one angle of a triangle is larger than a second angle, then the side opposite the first angle is longer than the side opposite the second angle. ANGLE – SIDE THEOREM 𝑳𝑫, 𝑶𝑫, 𝑳𝑶 O L D 30º 40⁰ 𝐄𝐗𝐀𝐌𝐏𝐋𝐄 𝟏 : List down the sides of ∆ 𝐿𝑂𝐷 from longest to shortest.

7. EXAMPLE 2 C A 63º 43⁰ List down the sides of ∆BAC from longest to shortest. Solve for the measure of the missing angle, m∠B: 𝑚∠𝐵 + 𝑚∠𝐶 + 𝑚∠𝐴 = 180° 𝑚∠𝐵 + 43° + 63° = 180° 𝑚∠𝐵 + 106° = 180° 𝑚∠𝐵 + 106° − 106° = 180° − 106° 𝑚∠𝐵 + = 180° − 106° 𝑚 ∠𝐵 = 74° B 74º 𝑨𝑪, 𝑩𝑪, 𝑨𝑩

8. TRY THIS Arrange the angles of ∆ 𝐴𝐵𝐶 from greatest to least measure given the lengths of its sides. 𝐴𝐵 = 5 𝑐𝑚 , 𝐵𝐶 = 10 𝑐𝑚 , 𝐴𝐶 = 12 𝑐𝑚 𝐵𝐴 = 10 𝑐𝑚 , 𝐶𝐵 = 19 𝑐𝑚 , 𝐶𝐴 = 11 𝑐𝑚 𝐴𝐶 = 7 𝑐𝑚 , 𝐴𝐵 = 3 𝑐𝑚 , 𝐵𝐶 = 5 𝑐𝑚

9. INEQUALITIES IN TRIANGLES INEQUALITIES IN ONE TRIANGLE TRIANGLE INEQUALITY 3 The sum of the lengths of any two sides of a triangle is greater than the length of the third side. (𝑺 𝟏 +𝑺 𝟐 > 𝑺 𝟑 ) SIDE – SIDE THEOREM

10. INEQUALITIES IN TRIANGLES INEQUALITIES IN ONE TRIANGLE TRIANGLE INEQUALITY 3 The sum of the lengths of any two sides of a triangle is greater than the length of the third side. (𝑺 𝟏 +𝑺 𝟐 > 𝑺 𝟑 ) SIDE – SIDE THEOREM Illustration Consider ∆ 𝐸𝐹𝐺 as shown below, with 𝑒 , 𝑓 , and 𝑔 as the side lengths. The triangle inequality theorem 3 states that : 𝑓 + 𝑔 > 𝑒 𝑒 + 𝑔 > 𝑓 𝑒 + 𝑓 > 𝑔

11. INEQUALITIES IN TRIANGLES INEQUALITIES IN ONE TRIANGLE TRIANGLE INEQUALITY 3 The sum of the lengths of any two sides of a triangle is greater than the length of the third side. (𝑺 𝟏 +𝑺 𝟐 > 𝑺 𝟑 ) SIDE – SIDE THEOREM Is it possible to form a triangle with the side lengths 13, 14, and 22? 13 + 14 > 22 27 > 22 ✓ 13 + 22 > 14 35 > 14 ✓ 14 + 22 > 13 36 > 13 ✓ Since all the sum of the lengths of any two sides of a triangle is greater than the length of the third side, therefore the answer is YES, it is possible to form a triangle . 𝐄𝐗𝐀𝐌𝐏𝐋𝐄 𝟏 :

12. INEQUALITIES IN TRIANGLES INEQUALITIES IN ONE TRIANGLE TRIANGLE INEQUALITY 3 The sum of the lengths of any two sides of a triangle is greater than the length of the third side. (𝑺 𝟏 +𝑺 𝟐 > 𝑺 𝟑 ) SIDE – SIDE THEOREM Tell whether it is possible to form a triangle with lengths 2cm,8cm, and 12cm. 2 + 8 > 12 10 > 12 X 2 + 12 > 8 14 > 8 ✓ 8 + 12 > 2 20 > 2 ✓ One of the inequalities is false. Therefore, it is NOT POSSIBLE to form a triangle 𝐄𝐗𝐀𝐌𝐏𝐋𝐄 𝟐 :

13. 8 + 15 > 10 23 > 10 ✓ 8 + 10 > 15 18 > 15 ✓ 15 + 10 > 8 25 > 8 ✓ Answer: Therefore, the length of the third side can be any value between 7 in. and 23 in. 𝐄𝐗𝐀𝐌𝐏𝐋𝐄 𝟑 : If two of the sides of a triangular frame measure 8 inches and 15 inches, find the possible lengths of the third side x . 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ𝑠 𝑜𝑓 𝑡𝑤𝑜 𝑠𝑖𝑑𝑒𝑠 < 𝑈𝑛𝑘𝑛𝑜𝑤𝑛 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑠𝑖𝑑𝑒 < 𝑆𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑤𝑜 𝑠𝑖𝑑𝑒𝑠 . 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ𝑠 𝑜𝑓 𝑡𝑤𝑜 𝑠𝑖𝑑𝑒𝑠 < x < 𝑆𝑢𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑤𝑜 𝑠𝑖𝑑𝑒𝑠 . 15 − 8 < 𝑥 < 15 + 8 7 < 𝑥 < 23 For Checking: use a value between 7 and 23; let’s use 10.

14. TRY THIS A. Use the Triangle Inequality Theorem 3 ( 𝑆 1 + 𝑆 2 > 𝑆 3) in determining whether the following numbers in cm can be side lengths of a triangle. Put a check mark (√) to indicate your answer. B. Find the possible values for the length of the third side 𝑥 using the Triangle Inequality Theorem 3 ( 𝑆 1 + 𝑆 2 > 𝑆 3). 1)14, 36 2) 8, 21 3) 13, 40

15. INEQUALITIES IN TRIANGLES INEQUALITIES IN ONE TRIANGLE EXTERIOR ANGLE INEQUALITY THEOREM The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle. TERMS TO REMEMBER! Exterior Angle – an angle that forms a linear pair with one of the interior angles of a triangle. Remote Interior Angle - an angle of a triangle that is NOT adjacent to a specified exterior angle. Linear Pair Theorem – if two angles form a linear pair, then the two angles are supplementary and adjacent.

16. INEQUALITIES IN TRIANGLES INEQUALITIES IN ONE TRIANGLE EXTERIOR ANGLE INEQUALITY THEOREM The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle. Given the figure below, name the following: linear pair ∠𝐴𝐶𝐷 and ∠𝐴𝐶𝐵 form a linear pair. exterior angle ∠𝐴𝐶𝐷 is an exterior angle. remote interior angles of ∠ACD ∠𝐴 𝑎𝑛𝑑 ∠𝐵 are the remote interior angles of ∠𝐴𝐶𝐷 𝐄𝐗𝐀𝐌𝐏𝐋𝐄 𝟏 :

17. INEQUALITIES IN TRIANGLES INEQUALITIES IN ONE TRIANGLE EXTERIOR ANGLE INEQUALITY THEOREM The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle. Given the figure below, name the following: Find the measure of ∠𝑆𝑀𝑅 . 𝑚∠𝑆𝑀𝑅 = 105° Compare the measure of ∠𝑆𝑀𝑅 to the measure of each remote interior angle of ∆ 𝐴𝑀𝑅 . 𝑚∠𝑆𝑀𝑅 > 𝑚∠𝐴 and 𝑚∠𝑆𝑀𝑅 > 𝑚∠𝑅 𝐄𝐗𝐀𝐌𝐏𝐋𝐄 𝟐 : 75⁰ 45⁰ S M A R

18. INEQUALITIES IN TRIANGLES INEQUALITIES IN ONE TRIANGLE EXTERIOR ANGLE INEQUALITY THEOREM The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle. Four billiard balls are left on the table as shown below. Use the expressions to determine the measure of the exterior angle. 𝐄𝐗𝐀𝐌𝐏𝐋𝐄 𝟑 : B A C D Answer: Exterior angle D, m ∠𝑫 = 130°

19. TRY THIS Study the figure on the left and use >, <, or = to compare the measures of angles and sides. 1. 𝑚∠𝑅 𝑚∠𝐴 𝑚∠𝑅𝐶𝐴 𝑚∠𝐸𝐶𝐴 𝑚∠𝑅𝐴𝐶 𝑚∠𝐶𝑅𝐴 𝑚∠𝐶𝐴𝑅 𝑚∠𝐸𝐶𝐴 𝑚∠𝑅 + 𝑚∠𝐴 𝑚∠𝐴𝐶𝐸 𝑅𝐴 𝐶𝑅 7. 𝐴𝐶 𝐶𝑅 0⁰ 4 75⁰ R A C E

20. INEQUALITIES IN TRIANGLES INEQUALITIES IN TWO TRIANGLES 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. SAS INEQUALITY Since 𝐴𝐵 ≅ 𝑌𝑍 𝑎𝑛𝑑 𝐵𝐶 ≅ 𝑋𝑍 but ∠B (45⁰) < ∠X (60⁰), then 𝑨𝑪 < 𝒀𝒁 Consider ∆ 𝐴𝐵𝐶 and ∆ 𝑋𝑌𝑍 . Describe the lengths of sides 𝐴𝐶 and 𝑌𝑍.

21. INEQUALITIES IN TRIANGLES INEQUALITIES IN TWO TRIANGLES 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. SAS INEQUALITY The distance between the tips of the two blades in Figure 1 is greater since the angle opposite this side has a greater measure which is 50° as compared to that in Figure 2 which is only 26°. The figures below show two pairs of scissors of the same size in two different positions. In which figure is the distance between the tips of the two blades greater? Use the Hinge Theorem to justify your answer.

22. INEQUALITIES IN TRIANGLES INEQUALITIES IN TWO TRIANGLES CONVERSE OF HINGE THEOREM 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. SSS INEQUALITY Since 𝑂𝑀 ≅ 𝑄𝑆 𝑎𝑛𝑑 𝑀𝑃 ≅ 𝑅𝑆 but 𝑂𝑃 (16 𝑐𝑚) > 𝑄𝑅 (9 𝑐𝑚) , then ∠𝐌 > ∠S Consider 𝛥𝑀𝑂𝑃 and 𝛥𝑄𝑅𝑆 . Which angle is larger, ∠𝑀 or ∠𝑆 ?

23. INEQUALITIES IN TRIANGLES INEQUALITIES IN TWO TRIANGLES CONVERSE OF HINGE THEOREM 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. SSS INEQUALITY Applying the Converse of Hinge Theorem or SSS Inequality Theorem makes the angle opposite the 8-meter side length larger than the angle opposite the 5 - meter side length of a triangle. Thus, the included angle in Figure 1 has a greater measure than the included angle in Figure 2. Apply the determine SSS Inequality Theorem to which figure has a greater measure of the included angle.

24. Summative Test: Triangle Inequalities I. Multiple Choice (Choose the best answer)

25. According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be: Less than the third side b) Equal to the third side c) Greater than the third side d) Any value 2. If one side of a triangle is longer than another side, then the angle opposite the longer side is: Smaller b) Equal c) Larger d) Unchanged 3. The Exterior Angle Inequality Theorem states that an exterior angle of a triangle is always: Less than either remote interior angle b) Equal to the sum of the remote interior angles c) Greater than either remote interior angle d) Equal to one remote interior angle

26. 4. If two sides of a triangle measure 7 cm and 10 cm, what is a possible length for the third side? 2 cm b) 18 cm c) 12 cm d) 20 cm 5. According to the Hinge Theorem, if two sides of a triangle are congruent to two sides of another triangle, but the included angle of one triangle is greater, then: The third side of that triangle is shorter b) The third side of that triangle is longer c) The third side is equal d) The triangles are congruent 6. The Converse of the Hinge Theorem states that if the third side of one triangle is longer than the third side of another, then: The included angle of that triangle is smaller b) The included angle of that triangle is equal c) The included angle of that triangle is larger d) The triangles are congruent

27. 7. Which of the following sets of side lengths CANNOT form a triangle? 5 cm, 6 cm, 7 cm b) 3 cm, 8 cm, 4 cm c) 10 cm, 10 cm, 19 cm d) 4 cm, 4 cm, 7 cm 8. What is the relationship between the angles of a triangle and the lengths of its sides? The largest angle is opposite the shortest side b) The smallest angle is opposite the longest side c) The largest angle is opposite the longest side d) There is no relationship 9. The sum of the measures of the interior angles of a triangle is always: 90° b) 180° c) 270° d) 360° 10. If one triangle has sides of 5 cm and 8 cm and another triangle has sides of 5 cm and 8 cm, but the included angle of one is larger, what can be said about the third side of that triangle? It is shorter b) It is longer c) It is the same length d) It is impossible to determine

28. 11. A builder is constructing a triangular frame with side lengths 4 m, 5 m, and 9 m. Can this form a triangle? Justify your answer. a) Yes, because the sum of two sides is greater than the third b) No, because the sum of two sides is less than or equal to the third c) Yes, because the three sides are nonzero d) No, because one side is longer than both combined 12. If a triangle has two angles measuring 50° and 60°, what is the measure of the third angle? a) 70° b) 60° c) 50° d) 80°

29. 13. carpenter is designing two triangular wooden panels. Both have two sides of equal length, but one panel has a larger included angle. What can be said about the third side of that panel? a) It is longer b) It is shorter c) It is equal d) It is unknown 14. Which of the following sets of side lengths forms a triangle? a) 2 cm, 2 cm, 5 cm b) 6 cm, 8 cm, 14 cm c) 5 cm, 10 cm, 12 cm d) 4 cm, 4 cm, 10 cm

30. 15. The Triangle Inequality Theorem ensures that: The sum of two sides is greater than the third b) Any three lengths form a triangle c) The longest side is always opposite the smallest angle d) The sum of two sides is equal to the third

31. II. True or False

32. 16. The longest side of a triangle is always opposite the smallest angle. 17. The sum of any two sides of a triangle must be greater than the third side. 18. . The Exterior Angle Inequality Theorem states that an exterior angle is greater than either remote interior angle. 19. In an isosceles triangle, the base angles are always different. 20. The Hinge Theorem applies to two triangles with two pairs of congruent sides.

33. 21. If two triangles have three pairs of congruent sides, then they must be congruent. 22. It is possible for a triangle to have sides measuring 2 cm, 4 cm, and 7 cm. 23. The Triangle Inequality Theorem helps determine whether given lengths can form a triangle. 24. The larger the included angle in a triangle, the longer the opposite side. 25. The Converse of the Hinge Theorem states that if the third side is longer, the included angle is larger.

34. GRADE 8