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TRIANGLE INEQUALITY, UNEQUAL SIDE THEOREM, AND UNEQUAL ANGLE THEOREM
TRIANGLE INEQUALITY, UNEQUAL SIDE THEOREM, AND UNEQUAL ANGLE THEOREM
WHAT IS TRIANGLE INEQUALITY?
WHAT IS TRIANGLE INEQUALITY?
The triangle inequality theorem describes the relationship between the three sides of a triangle. According to this theorem, for any triangle, the sum of lengths of two sides is always greater than the third side.
EXAMPLE:
EXAMPLE:
TRIANGLE INEQUALITY THEOREM
TRIANGLE INEQUALITY THEOREM
We know that a triangle has three sides. But, have you ever thought what is necessary for the three line segments to form a triangle. Is it possible to make a triangle with any three line segments?
In the given figure, line segments 6, 8, and 10 units form a triangle. What about if the line segments change to 6, 8, and 17 units?
We would not be able to create the triangle as shown by an incomplete triangle. This proves that we cannot create a triangle from any combination of three line segments. This relationship is explained using the triangle inequality theorem.
The above theorem describes the relationship between the three sides of a triangle. It tells us that for 3 line segments to form a triangle, it is always true that none of the 3 line segments is greater than the lengths of the other two line segments combined.
Let us take our initial example. We could make a triangle with line segments having lengths 6, 8, and 10 units. This is because those line segments satisfy the triangle inequality theorem.
6 + 8 = 14 and 10 < 14
8 + 10 = 18 and 6 < 18
6 + 10 = 16 and 8 < 16
Here, we see that none of the line segments are longer that the sum of the other two line segments.
In contrast, if we consider the line segments of lengths 6, 8, and 17 units, we find that the line segment measuring 17 units is longer than the length of the other two line segments combined.
This proves that we cannot make a triangle with these three line segments. Thus, they don’t satisfy the triangle inequality theorem.
The inequality theorem is applicable to all types of triangles such as scalene, isosceles, and equilateral.
Hence proved that, the sum of the lengths of any two sides of a triangle is greater than the third side.
Let us solve some problems involving the above theorem to understand the concept better.
TRIANGLE INEQUALITY PROOF
TRIANGLE INEQUALITY PROOF
To prove: |YZ| <|XY| + |XZ|
Proof:
The side XZ is extended to a point W such that XW=XY as shown in the given figure.
TRIANGLE INEQUALITIES: SIDES & ANGLES
TRIANGLE INEQUALITIES: SIDES & ANGLES
WHAT IS TRIANGLE INEQUALITIES: SIDES & ANGLES?
WHAT IS TRIANGLE INEQUALITIES: SIDES & ANGLES?
You have just seen that if a triangle has equal sides, the angles opposite these sides are equal, and if a triangle has equal angles, the sides opposite these angles are equal. There are two important theorems involving unequal sides and unequal angles in triangles. They are:
Theorem 36: If two sides of a triangle are unequal, then the measures of the angles opposite these sides are unequal, and the greater angle is opposite the greater side.
Theorem 37: If two angles of a triangle are unequal, then the measures of the sides opposite these angles are also unequal, and the longer side is opposite the greater angle.
EXAMPLE #1:
EXAMPLE #1:
Figure 1 shows a triangle with angles of different measures. List the sides of this triangle in order from least to greatest.
Figure 1 List the sides of this triangle in increasing order.
Figure 1 List the sides of this triangle in increasing order.
Because 30° < 50° < 100°, then RS < QR < QS.
EXAMPLE # 2:
EXAMPLE # 2:
Figure 2 shows a triangle with sides of different measures. List the angles of this triangle in order from least to greatest.
Figure 2 List the angles of this triangle in increasing order.
Figure 2 List the angles of this triangle in increasing order.
Because 6 < 8 < 11, then m ∠ N < m ∠ M < m ∠ P.
EXAMPLE #3:
EXAMPLE #3:
Figure 3 shows right Δ ABC. Which side must be the longest?
Figure 3 Identify the longest side of this right triangle.
Figure 3 Identify the longest side of this right triangle.
Because ∠ A + m ∠ B + m ∠ C = 180 ° (by Theorem 25) and m ∠ = 90°, we have m ∠ A + m ∠ C = 90°. Thus, each of m ∠ A and m ∠ C is less than 90°. Thus ∠ B is the angle of greatest measure in the triangle, so its opposite side is the longest. Therefore, the hypotenuse, AC , is the longest side in a right triangle.
SUBJECT TEACHER
SUBJECT TEACHER
Sir Leo Esteban
Sir Leo Esteban
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