Solve for a in the diagram below: In the diagram below, AB //CD, determine the value of x:

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ScanSolve AI Tutor · Accepted Answer

here are the solutions for the geometry problems:

Identify that the angles $85^\circ$ , $110^\circ$ , and $a$ are angles around a point.
The sum of angles around a point is $360^\circ$ .
Set up the equation and solve for $a$ .
$85^\circ + 110^\circ + a = 360^\circ$ $195^\circ + a = 360^\circ$ $a = 360^\circ - 195^\circ$ $a = 165^\circ$
The value of $a$ is: $\boxed{165^\circ}$

In the diagram below, AB // CD, determine the value of $x$ :
Identify the relationship between $\angle BAC$ and $\angle ACD$ .
Since AB // CD and AC is a transversal, $\angle BAC$ and $\angle ACD$ are consecutive interior angles.
Consecutive interior angles are supplementary, meaning their sum is $180^\circ$ .
$\angle BAC + \angle ACD = 180^\circ$ $80^\circ + \angle ACD = 180^\circ$
Solve for $\angle ACD$ .
$\angle ACD = 180^\circ - 80^\circ$ $\angle ACD = 100^\circ$
Identify the relationship between $\angle ACD$ and $\angle DCE$ .
The line ACE is a straight line. Angles on a straight line sum to $180^\circ$ .
$\angle ACD + \angle DCE = 180^\circ$
Substitute the known values and solve for $x$ .
$100^\circ + (2x + 20^\circ) = 180^\circ$ $120^\circ + 2x = 180^\circ$ $2x = 180^\circ - 120^\circ$ $2x = 60^\circ$ $x = \frac{60^\circ}{2}$ $x = 30^\circ$
The value of $x$ is: $\boxed{30^\circ}$

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