Theorem 1 :
If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side.
Theorem 2 :
If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.
The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles.
In the triangle above, according to theorem 3, we have
m∠1 > m∠A
m∠1 > m∠B
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
In the triangle ABC above, according to theorem 4, we have
AB + BC > AC
AC + BC > AB
AB + AC > BC
Example 1 :
Write the sides of the triangle shown below in order from least to greatest.
Solution :
In the triangle GHJ above, we have
m∠G < m∠H < m∠J
So, the smallest angle is m∠G and the largest angle m∠J.
In any triangle, the smallest angle is always across from the shortest side and the largest angle is always across from the longest side.
Hence, the order of sides of the triangle from least to greatest is
JH < JG < GH
Example 2 :
Write the angles of the triangle shown below in order from smallest to largest.
Solution :
In the triangle PQR above, we have
PQ < PR < QR
So, the shortest side is PQ and the longest side is QR.
In any triangle, the smallest angle is always across from the shortest side and the largest angle is always across from the longest side.
Hence, the order of angles of the triangle from smallest to largest is
m∠R < m∠Q < m∠P
Example 3 :
In the triangle shown below AB ≅ AC and BC > AB. What can we conclude about the angles in triangle ABC ?
Because AB ≅ AC, triangle ABC is isosceles.
So, we have
m∠B ≅ m∠C
Therefore, m∠B = m∠C.
Because BC > AB, m∠A > m∠C by Theorem 1.
By substitution, m∠A > m∠B.
In addition, you can conclude that m∠A > 60°. m∠B < 60° and m∠C < 60°.
Example 4 :
Construct a triangle with the given group of side lengths, if possible.
2 cm, 2 cm, 5 cm
Solution :
We can not construct a triangle with the given side lengths.
Because, sum of the lengths of any two sides of a triangle must be greater than the third side by Theorem 4.
In the given side lengths, we have
2 + 2 < 5
(Does not satisfy the theorem)
The diagram given below illustrates that a triangle can not be constructed with the given side lengths.
Example 5 :
Construct a triangle with the given group of side lengths, if possible.
3 cm, 2 cm, 5 cm
Solution :
We can not construct a triangle with the given side lengths.
Because, sum of the lengths of any two sides of a triangle must be greater than the third side by Theorem 4.
In the given side lengths, we have
3 + 2 = 5
(Does not satisfy the theorem)
The diagram given below illustrates that a triangle can not be constructed with the given side lengths.
Example 6 :
Construct a triangle with the given group of side lengths, if possible.
4 cm, 2 cm, 5 cm
Solution :
In the given side lengths, we have
4 + 2 > 5
4 + 5 > 2
2 + 5 > 4
In the given side lengths, it is clear that the sum of any two sides is greater than the third side.
Because the given side lengths satisfy Theorem 4, we can construct a triangle with the given side lengths.
The diagram given below illustrates that a triangle can be constructed with the given side lengths.
Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.
If you have any feedback about our math content, please mail us :
v4formath@gmail.com
We always appreciate your feedback.
You can also visit the following web pages on different stuff in math.
WORD PROBLEMS
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Trigonometry word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits