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TRIANGLE INEQUALITY THEOREM

The document discusses several theorems related to triangles: 1) The triangle inequality theorem states that the length of any side of a triangle must be less than the sum of the lengths of the other two sides and greater than their difference. 2) The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the remote interior angles. 3) The hinge theorem states that if two triangles have two congruent sides but not congruent included angles, then the third sides will be unequal, with the longer side opposite the larger angle.

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Inequalities in One Triangle and in Two Triangles
Reaching Goals ❑ State the theorem on triangle inequalities, exterior angle inequality, and hinge theorem. ❑Illustrate the theorem on triangle inequalities, exterior angle inequality, and hinge theorem. ❑Develop positive attitude towards work.
LOOP-A-WORD Find the listed words in the puzzle. They are arranged horizontally, vertically, diagonally, upward or backward.
LOOP-A-WORD Interior Theorem Inequality Exterior Hinge Angle Remote Triangle Adjacent SAS
f f f
MATH-ionary Define all the words in the said Word Hunt.
MATH-ionary ❑ Exterior – Angle outside the triangle ❑ Hinge- If two sides of a triangle are congruent to the sides of another triangle, but their included angles are not, then the remaining sides are unequal. The longer side is opposite the larger angle. ❑Angle- part of the triangle ❑SAS – Side-Angle-Side (with included angle)
MATH-ionary ❑Inequality- not equal ❑Theorem- statement that must be proven ❑Interior- Angles inside the triangle ❑Triangle- closed figure which has three sides ❑Remote- angles of a triangle that are not adjacent to a given angle. ❑Adjacent- angles having a common vertex and a common side.
Observe the figure A B C 1 5 6 3 4 2
Question 1. What are the exterior angles of the given triangle? 2. How will you describe the exterior angle?
Exterior Angle of a Triangle An exterior angle of a triangle is an angle that forms a linear pair with an interior angle of a triangle is extended.
Observe the figure A B C 4 3 5
1. What is the sum of the lengths of AB and AC ? 2. Compare the sum of the lengths of AB and AC to BC 3. What is the sum of the lengths of AB and BC? Question
4. Compare the sum of the lengths of AB and BC to AB 5. What is the sum of the lengths of AC and BC? Question
6. Compare the sum of the lengths of AC and BC to AB. 7. What can you say about the sum of the two sides of a triangle compared to its third side? Question
Triangle Inequality The length of a side of a triangle is less than the sum of the lengths of the other two sides. The length of one side is also greater than the positive difference of the lengths of the other two sides.
Observe the figure 100° 25° 55° A B C 7 4 10 B
1. What is the largest angle? 2. What is the smallest angle? 3. What is the length of the side opposite the largest angle? Question
4. What is the length of the side opposite the smallest angle? 5. What is the relationship between the length of the side and the angle opposite to the given side? Question
Theorem If the length of the two sides of a triangle is unequal, the measures of the angles are also unequal. The longer side is opposite the angle with a greater measure.
Triangle Inequality Theorem 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. In other words, opposite the largest angle is the longest side.
Observe the figure 97° 20° 63° A B C 9 7 15 B
Triangle Inequality Theorem 3 The sum of the lengths of any two sides of a triangle is greater than the length of the third side. In symbol, a + b > c; a + c > b or b + c > a
Observe the figure 97° 20° 63° A B C 9 7 15 B
Observe the figure A B D C E F 56° 56° 50° 9 9 10 10
1. What are the congruent sides? 2. What have you notice on the included angles of the two triangles? Question
What conclusion can you make about the opposite sides of the included angles of the two triangles? Are they congruent? Which side is longer? Question
Hinge Theorem If two sides of a triangle are congruent to the sides of another triangle, but their included angles are not, then the remaining sides are unequal. The longer side is opposite the larger angle.
LET ’S DO THIS! Fill in the blanks with the correct relation symbol ( >, < ) to show the relationship.
O D G 25 56 40 1. m ∠D _____ m ∠O 2. m ∠D _____ m ∠G 3. m ∠O _____ m ∠G 4. m ∠G _____ m ∠O 5. m ∠O _____m ∠D O G
B O K 65° 55° 60°
LET ’S DO THIS! Is it possible for a triangle to have sides with the length indicated? 1. 3, 4, 5 2. 8, 7, 10 3. 2, 5, 6
F U N 1 2 3 1. m ∠2 _____ m ∠4 2. m ∠1 _____ m ∠4 3. m ∠4 _____ m ∠2 4
Group Activity Name the largest and the smallest angle of the triangle.
R T S 22 24 23 Largest Angle- ∠S Smallest Angle- ∠T
W V U 18 16 17 Largest Angle- ∠V Smallest Angle- ∠U
Group Activity Name the longest side and the shortest side of the triangle. .
A D N 75 ° 70° 35 °
A F T 87 ° 43° 50°
G H T 123 ° 32° 25°
Things to Remember! Exterior Angle of a Triangle- An exterior angle of a triangle is an angle that forms a linear pair with an interior angle of a triangle is extended.
Things to Remember! Triangle Inequality Postulate- The length of a side of a triangle is less than the sum of the lengths of the other two sides. The length of one side is also greater than the positive difference of the lengths of the other two sides.
Things to Remember! Theorem- If the length of the two sides of a triangle is unequal, the measures of the angles are also unequal. The longer side is opposite the angle with a greater measure.
Things to Remember! Hinge Theorem- If two sides of a triangle are congruent to the sides of another triangle, but their included angles are not, then the remaining sides are unequal. The longer side is opposite the larger angle.