Note on Triangle Theorems

  • Note
  • Things to remember
  • Exercise

Triangles are governed by two important inequalities. The first is often referred to as the triangle inequality. It states that the length of a side of a triangle is always less than the sum of the lengths of the other two sides.

The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides.

Theorem 1

The sum of interior angles of a triangle is 180°

Draw three different triangles in your notebook. Measure ∠X, ∠Y and ∠Z using a protector and fill in the table.

Verification:

.

 

Figure ∠X ∠Y ∠Z ∠X +∠Y+∠Z
(i)        
(ii)        
(iii)        


Look at the figure and complete the table given below.

.
Statements Reasons
a+b+c = 180° Sum of adjacent angles on a straight line
a = m, c = n Corresponding angles
m+b+n = 180° ?

Conclusion: The sum of interior angles of a triangle is 180°

Theorem 2

Base angles of an isosceles triangle are equal.

Draw three different triangles making AB = AC, ∠B and ∠C opposite to AC and AB respectively are the base angles. Measure ∠ABC and ∠ACB using a protector and fill in the table.

Verification:

.
Figure ∠ABC ∠ACB Result
(i)     ∠ABC =∠ACB
(ii)      
(iii)      

Conclusion: Base angles of an isosceles triangle are equal.

Theorem 3

Each of the base angles of an isosceles right triangle is 45°.

Draw three triangles making ∠B = 90° and AC = BC. Measure ∠BAC and ∠ACB and fill in the table.

Verification:

 

1
Figure ∠BAC ∠ACB Result
(i)     ∠BAC =∠ACB = 45°
(ii)      
(iii)      

Conclusion: Each of the base angles of an isosceles right triangle is 45°

Theorem 4

The line joining the vertex and midpoint of the base of an isosceles triangle is perpendicular to the base.

Draw three triangles making AB = AC. Join the midpoint P of BC and A, in each figure. Measure the angles APB and APC and fill in the blanks.

Verification:

.
Figure ∠APB ∠APC Result
(i)     ∠APB =∠APC =90°
(ii)      
(iii)      

Conclusion: The line joining the vertex and mid-point of the base of an isosceles triangle is perpendicular to the base.

Theorem 5

All the angles of an equilateral triangle are equal.

Draw three triangles making AB = BC =CA in each figure. Measure∠ABC, ∠BCA and ∠CAB and fill in the table given below.

Verification:

.
Figure ∠ABC ∠BCA ∠CAB Result
(i)       ∠ABC =∠BCA =∠CAB
(ii)        
(iii)        

Conclusion: All angles of an equilateral triangle are equal.

  • Triangles are governed by two important inequalities. 
  • A triangle cannot be constructed from three line segments if any of them is longer than the sum of the other two.
  • The triangle inequality theorem states that any side of a triangle is always shorter than the sum of the other two sides.
.

Very Short Questions

Solution:

\(\angle\)P + \(\angle\)Q + \(\angle\)R = 180o (sum of angles of a triangle is 180o)

or, 3x+ 3x+ 3x=180o

or, 9x=180o

\(\therefore\) x=20o

Also,

3x+y=180o (straight angle)

or, 3\(\times\)20o+y=180o

or, 60o+y=180o

or, y=180o- 60o

\(\therefore\) y=120o

Solution:

Here, \(\angle\)FEG=20o+20o=40o

\(\angle\)EFG= \(\angle\)EGF=x=y [ \(\therefore\)

Now, \(\angle\)FEG+ \(\angle\)EFG+ \(\angle\)EFG=180o [\(\therefore\)sum of angle FEG is 180o]

or, 40o + x + x=180o

or, 2x=180o-40o=140o

or, x= \(\frac{140^o}{2}\)

\(\therefore\)x =y=70o

Solution,

Here, \(\angle\)P= \(\angle\)Q= \(\angle\)R [\(\therefore\) ]

or, xo=yo=zo

or, x=y=z

Now, \(\angle\)P+ \(\angle\)Q+ \(\angle\)R=180o [\(\therefore\)sum of angle FEG is 180o]

or, x+x+x=180o

or, 3x=180o

or, x=\(\frac{180^o}{3}\)=60o

or, x=60o

\(\therefore\) x=y=z=60o

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how to measure traingle

how to measure traingle using protactor


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Adarsha

If the triangle is equilateral triangle and the angles are only given how to find the sides? Please I hope a reply. Thank you


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