Answer:
33° + 112° + (7x - 7)° = 180°
x = 6
Step-by-step explanation:
Refer to the attached diagram.
When two straight lines intersect, the opposite vertical angles are congruent. Therefore, the measure of angle A is 45°.
Angles on a straight line sum to 180°. Therefore, the measure of angle B can be found by subtracting 135° from 180°, so:
\(m\angle B + 135^{\circ} = 180^{\circ}\\\\m\angle B = 180^{\circ} - 135^{\circ}\\\\m\angle B = 45^{\circ}\)
The interior angles of a triangle sum to 180°.
As angles A and B are both 45°, this means that angle C measures 90° and is therefore a right angle.
\(m\angle A + m\angle B + m\angle C = 180^{\circ}\\\\45^{\circ} + 45^{\circ} + m\angle C = 180^{\circ}\\\\90^{\circ} + m\angle C = 180^{\circ}\\\\m\angle C = 180^{\circ} - 90^{\circ}\\\\m\angle C = 90^{\circ}\)
As angles on a straight line sum to 180°, the measure of angle D can be found by subtracting angle C and 57° from 180°:
\(m∠C + 57^{\circ} + m\angle D = 180^{\circ}\\\\90^{\circ} + 57^{\circ} + m\angle D = 180^{\circ}\\\\147^{\circ} + m\angle D = 180^{\circ}\\\\m\angle D = 180^{\circ} - 147^{\circ}\\\\m\angle D = 33^{\circ}\)
As the interior angles of a triangle sum to 180°, then:
\(m\angle D + 112^{\circ} + (7x - 7)^{\circ} = 180^{\circ}\)
Substitute m∠D = 33° and solve for x:
\(33^{\circ} + 112^{\circ} + (7x - 7)^{\circ} = 180^{\circ}\\\\145^{\circ} + (7x - 7)^{\circ} = 180^{\circ}\\\\(7x - 7)^{\circ} = 35^{\circ}\\\\7x - 7 = 35\\\\7x = 42\\\\x = 6\)
Therefore, the equation is:
\(\Large\boxed{33^{\circ} + 112^{\circ} + (7x - 7)^{\circ} = 180^{\circ}}\)
The value of x is:
\(\Large\boxed{x=6}}\)
