Question

Please help with both answers

Answer 1

`m\angle PTR = 70`

`m\angle 1 = 34`

Explanation:

Solving (1):

Given: Trapezium APTR

Find
`m\angle PTR`

In the trapezium:

`AR = PT`

And AP is parallel to RT.

This implies that:

`m\angle APT = m\angle RAP =110` and
`m\angle PTR = m\angle TRA`

The sum of all angles is:

`m\angle APT + m\angle RAP + m\angle PTR + m\angle TRA = 360`

`110 + 110 + m\angle PTR + m\angle TRA = 360`

Recall that:

`m\angle PTR = m\angle TRA`

`110 + 110 + m\angle PTR + m\angle PTR = 360`

`220 + 2m\angle PTR = 360`

`2m\angle PTR = 360 - 220`

`2m\angle PTR = 140`

Divide by 2

`m\angle PTR = 70`

Solving (2):

Given

`m\angle 1 = 3x + 4`

`m\angle 2 = x + 24`

Required

Find
`m\angle 1`

Both angles are vertically opposie:

So:

`m\angle 1 =m\angle 2`

`3x + 4= x + 24`

Collect Like Terms

`3x -x= 24 - 4`

`2x = 20`

`x = 10`

So:

`m\angle 1 = 3x + 4`

`m\angle 1 = 3 * 10 + 4`

`m\angle 1 = 30 + 4`

`m\angle 1 = 34`