Notes on Triangle Theorems | Grade 8 > Compulsory Maths > Geometry | KULLABS.COM

Notes, Exercises, Videos, Tests and Things to Remember on Triangle Theorems

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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:

 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.
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#### Click on the questions below to reveal the answers

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 using protactor