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MrRoyalidn

Answer:

[tex]m\angle PTR = 70[/tex]

[tex]m\angle 1 = 34[/tex]

Step-by-step explanation:

Solving (1):

Given: Trapezium APTR

Find [tex]m\angle PTR[/tex]

In the trapezium:

[tex]AR = PT[/tex]

And AP is parallel to RT.

This implies that:

[tex]m\angle APT = m\angle RAP =110[/tex] and [tex]m\angle PTR = m\angle TRA[/tex]

The sum of all angles is:

[tex]m\angle APT + m\angle RAP + m\angle PTR + m\angle TRA = 360[/tex]

[tex]110 + 110 + m\angle PTR + m\angle TRA = 360[/tex]

Recall that:

[tex]m\angle PTR = m\angle TRA[/tex]

[tex]110 + 110 + m\angle PTR + m\angle PTR = 360[/tex]

[tex]220 + 2m\angle PTR = 360[/tex]

[tex]2m\angle PTR = 360 - 220[/tex]

[tex]2m\angle PTR = 140[/tex]

Divide by 2

[tex]m\angle PTR = 70[/tex]

Solving (2):

Given

[tex]m\angle 1 = 3x + 4[/tex]

[tex]m\angle 2 = x + 24[/tex]

Required

Find [tex]m\angle 1[/tex]

Both angles are vertically opposie:

So:

[tex]m\angle 1 =m\angle 2[/tex]

[tex]3x + 4= x + 24[/tex]

Collect Like Terms

[tex]3x -x= 24 - 4[/tex]

[tex]2x = 20[/tex]

[tex]x = 10[/tex]

So:

[tex]m\angle 1 = 3x + 4[/tex]

[tex]m\angle 1 = 3 * 10 + 4[/tex]

[tex]m\angle 1 = 30 + 4[/tex]

[tex]m\angle 1 = 34[/tex]

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