Question

Please answer all these

Answer 1

`$m\angle 4 = 70^\circ + 50^\circ = 120^\circ$.`

The Exterior Angle Theorem is a useful theorem in geometry, and it can be used to solve a variety of problems.

Proof of the Exterior Angle Theorem

Given:
`$\angle 4$ is an exterior angle of $\triangle ABC$`

Prove:
`$m\angle 1 + m\angle 2 = m\angle 4$`

Proof:

1.
`$\angle 4$ is an exterior angle of $\triangle ABC$`.

2.
`$\angle 3$ and $\angle 4$ form a linear pair.` (Definition of an exterior angle)

3.
`$\angle 3$ is supplementary to $\angle 4$.` (Supplementary angles theorem)

4.
`$m\angle 3 + m\angle 4 = 180^\circ$`. (Definition of supplementary angles)

5.
`$m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ$`. (Triangle Sum Theorem)

6.
`$m\angle 1 + m\angle 2 = 180^\circ - m\angle 3$`. (Subtracting
`$m\angle 3$` from both sides of equation 5)

7.
`$m\angle 1 + m\angle 2 = m\angle 4$`. (Substituting equation 4 into equation 6)

Therefore, the exterior angle of a triangle is equal to the sum of the two remote interior angles.

In the diagram below,
`$\angle 4$` is an exterior angle of
`$\triangle ABC$.`

[Image of a triangle ABC with an exterior angle 4]

Using the Exterior Angle Theorem, we can prove that
`$m\angle 4 = m\angle 1 + m\angle 2$.`

Proof:

1.
`$\angle 4$ is an exterior angle of $\triangle ABC$.`

2.
`$\angle 3$ and $\angle 4$ form a linear pair`

3.
`$\angle 3$ is supplementary to $\angle 4$.`

4.
`$m\angle 3 + m\angle 4 = 180^\circ$.`

5.
`$m\angle 1 + m\angle 2 + m\angle 3 = 180^\circ$.`

6.
`$m\angle 1 + m\angle 2 = 180^\circ - m\angle 3$.`

7.
`$m\angle 1 + m\angle 2 = m\angle 4$.`