Question

Triangle angle sum theorem and exterior angle theorem

Answer 1

the angle sum of triangle is 180 degrees. it can be explained because all the angle sum is just the number of sides - 2 and multiply them by 180.
and the exterior angle sum of any sides of polygon, including triangle is always 360 degrees

Answer 2

5.
`m\angle 1=47^(\circ)`

`m\angle 2=137^(\circ)`

`m\angle 3=31^(\circ)`

7.
`m\angle 1=87^(\circ)`

`m\angle 2=45^(\circ)`

`m\angle 3=45^(\circ)`

`m\angle 4=52^(\circ)`

Explanation:

5.
`\angle 1+43^(\circ)+90^(\circ)=180^(\circ)`

By triangle angle sum theorem

`\angle +133^(\circ)=180^(\circ)`

Addition property of integers

`\angle 1=180-133=47^(\circ)`

By using subtraction property of equality

`m\angle 1=47^(\circ)`

`m\angle 2=90+47=137^(\circ)`

By using exterior angle theorem

`90+47+12+m\angle 3=180^(\circ)`

By using triangle angle sum theorem

`149+m\angle 3=180`

Addition property of integers

`m\angle 3=180-149=31^(\circ)`

Using subtraction property of equality

`m\angle 3=31^(\circ)`

7.We are given that

`m\angle ACD=90^(\circ)`

BC bisects angle ACD

`m\angle 2=m\angle 3`

Therefore,
`m\angle 2=(90)/(2)=45^(\circ)`

`m\angle 2=m\angle 3=45^(\circ)`

`m\angle 1+m\angle 2+48=180^(\circ)`

By using triangle angle sum theorem

Substitute the value

`45+m\angle 1+48=180`

`m\angle 1+93=180`

By addition property of integers

`m\angle 1=180-93=87^(\circ)`

By using subtraction property of equality

`m\angle 3+m\angle 4+83=180^(\circ)`

By using triangle angle sum property

Substitute the values

`45+m\angle 4+83=180`

`128+m\angle 4=180`

Using addition property of integers

`m\angle 4=180-128=52^(\circ)`

Using subtraction property of equality