Question

Please help with both answers

Answer 1

Answer:

$m\angle PTR = 70$

$m\angle 1 = 34$

Step-by-step 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:

$m\angle 1 = 3x + 4$

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

$m\angle 1 = 30 + 4$

$m\angle 1 = 34$