Question

A triangle is shown with its exterior angles. The interior angles of the triangle are angles 2, 3, 5. The exterior angle at angle 2 is angle 1. The exterior angle at angle 3 is angle 4. The exterior angle at angle 5 is angle 6.
Which statements are always true regarding the diagram? Select three options.

m∠5 + m∠3 = m∠4
m∠3 + m∠4 + m∠5 = 180°
m∠5 + m∠6 =180°
m∠2 + m∠3 = m∠6
m∠2 + m∠3 + m∠5 = 180°

Answer 1

The three right options are:

m∠5 ₊ m∠6 = 180° which represents a linear pair.

∠2 ₊ ∠3 = ∠6 showing the exterior angle property of triangle.

m∠2  m∠3  m∠5 = 180° as the sum-triangle

sum-triangleproperty.

Given,

the interior angles of the triangle are 2,3,5

the exterior angle at ∠2 is ∠1.

the exterior angle at ∠3 is ∠4.

the exterior angle at ∠5 is ∠6.

Now, take into account a triangle with external angles as ∠1 , ∠4 and ∠6.

Apply the exterior angle property of a triangle.

The sum of the opposing internal angles determines the outer angle of a triangle.

For exterior ∠1 we get

∠1 = ∠5 ₊ ∠3

Similarly, for exterior ∠4 we get

∠4 = ∠5 ₊ ∠2

Then for exterior ∠6 we get

∠6 = ∠2 ₊ ∠3

∴ through the exterior angle property we get the correct option as

m∠6 = m∠2 ₊ m∠3

Now we know the sum-triangle property that sum of the angles in triangle is equal to 180°

∴ m∠2 ₊ m∠3 ₊ m∠5 = 180°

We know the linear pair property i.e, If the angles follow the point where the two lines cross, they are considered to be linear. A linear pair's angles add up to 180° in every case.,

∴ m∠5 ₊ m∠6 = 180°

Hence we get the statements which match the diagram are:

m∠5 ₊ m∠6 = 180°

m∠2 ₊ m∠3 ₊ m∠5 = 180°

m∠6 = m∠2 ₊ m∠3

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Answer 2

Answer:

• m∠5 + m∠6 =180°
• m∠2 + m∠3 = m∠6
• m∠2 + m∠3 + m∠5 = 180°

Step-by-step explanation:

m∠5 + m∠6 = 180° - This is true because the exterior angle and interior angle form a linear pair, and are thus supplementary.

m∠2 + m∠3 = m∠6 - This is true by the exterior angle theorem.

m∠2 + m∠3 + m∠5 = 180° - This is true because the interior angles of a triangle add to 180 degrees.