By properties of parallel lines A+B = 180^o, B = 45^o, C = A = 135^o, so 2*|B-C| = 2* |45-135| = 180^o

Since we know opposite angles are equal, it follows that angle \(\displaystyle \angle AFE=10^{\circ}\) and \(\displaystyle \angle BGE=50^{\circ}\).  

Imagine a parallel line passing through point \(\displaystyle E\).  The imaginary line would make opposite angles with \(\displaystyle \angle AFE\) & \(\displaystyle \angle BGE\), the sum of which would equal \(\displaystyle \angle FEG\).  Therefore, \(\displaystyle \angle FEG=60^{\circ}\).

\(\displaystyle \cos (60)=.5=\frac{EG}{EF}\rightarrow EF=\frac{EG}{.5}=20\)

When the measure of an angle is added to the measure of its supplement, the result is always 180 degrees. Put differently, two angles are said to be supplementary if the sum of their measures is 180 degrees. For example, two angles whose measures are 50 degrees and 130 degrees are supplementary, because the sum of 50 and 130 degrees is 180 degrees. We can thus write the following equation:

\(\displaystyle \dpi{100} measure\ of\ \angle A+ measure\ of\ supplement\ of\ \angle A=180\)

\(\displaystyle \dpi{100} 40-10x+ measure\ of\ supplement\ of\ \angle A=180\)

Subtract 40 from both sides.

\(\displaystyle \dpi{100} -10x+ measure\ of\ supplement\ of\ \angle A=140\)

Add \(\displaystyle \dpi{100} 10x\) to both sides.

\(\displaystyle \dpi{100} measure\ of\ supplement\ of\ \angle A=140+10x=10x+140\)

The answer is \(\displaystyle (10x+140)^{\circ}\).

When two parallel lines are intersected by another line, the sum of the measures of the interior angles on the same side of the line is 180°. Therefore, the sum of the angle that is labeled as 100° and angle y is 180°. As a result, angle y is 80°.

Another property of two parallel lines that are intersected by a third line is that the corresponding angles are congruent. So, the measurement of angle x is equal to the measurement of angle y, which is 80°.

Let A represent the measure, in degrees, of angle A. By definition, the sum of the measures of A and its complement is 90 degrees. We can write the following equation to determine an expression for the measure of the complement of angle A.

A + measure of complement of A = 90

Subtract A from both sides.

measure of complement of A = 90 – A

Similarly, because the sum of the measures of angle A and its supplement is 180 degrees, we can represent the measure of the supplement of A as 180 – A.

The problem states that the measure of the supplement of A is 40 degrees larger than twice the measure of the complement of A. We can write this as 2(90-A) + 40.

Next, we must set the two expressions 180 – A and 2(90 – A) + 40 equal to one another and solve for A:

180 – A = 2(90 – A) + 40

Distribute the 2:

180 - A = 180 – 2A + 40

Add 2A to both sides:

180 + A = 180 + 40

Subtract 180 from both sides:

A = 40

Therefore the measure of angle A is 40 degrees. 

The question asks us to find the sum of the measures of the supplement and complement of A. The measure of the supplement of A is 180 – A = 180 – 40 = 140 degrees. Similarly, the measure of the complement of A is 90 – 40 = 50 degrees.

The sum of these two is 140 + 50 = 190 degrees.

\(\displaystyle \dpi{100} \small \angle AEC\) & \(\displaystyle \dpi{100} \small \angle BEC\) must add up to 180 degrees. So, if \(\displaystyle \dpi{100} \small \angle AEC\) is 120, \(\displaystyle \dpi{100} \small \angle BEC\) (the supplementary angle) must equal 60, for a total of 180.

When a line intersects two parallel lines as a transversal, it always passes through both at identical angles (regardless of distance or length of arc).

The question states that \(\displaystyle \overleftrightarrow{FH} \parallel \overleftrightarrow{AD}\). The alternate interior angle theorem states that if two parallel lines are cut by a transversal, then pairs of alternate interior angles are congruent; therefore, we know the following measure:

\(\displaystyle m\angle GCA = 58^\circ\)

The sum of angles of a triangle is equal to 180 degrees. The question states that \(\displaystyle \overline{BE}\perp \overleftrightarrow{GC}\); therefore we know the following measure:

\(\displaystyle m \angle BEC = 90^\circ\)

Use this information to solve for the missing angle: \(\displaystyle \angle EBC\)

\(\displaystyle 180^\circ=m\angle EBC+58^\circ+90^\circ\)

\(\displaystyle m\angle EBC=32^\circ\)

The degree measure of a straight line is 180 degrees; therefore, we can write the following equation:

\(\displaystyle 180^\circ=m\angle ABE+32^\circ\)

\(\displaystyle m\angle ABE=148^\circ\)

The measure of \(\displaystyle \angle ABE\) is 148 degrees. 

Since A and B are parallel, and the triangle is isosceles, we can use the supplementary rule for the two angles, \(\displaystyle 4x\) and \(\displaystyle 2x+30\) which will sum up to \(\displaystyle 180\). Setting up an algebraic equation for this, we get \(\displaystyle 4x+2x+30=180\). Solving for \(\displaystyle x\), we get \(\displaystyle x=25\). With this, we can get either \(\displaystyle 2(25)+30=80\) (for the smaller angle) or \(\displaystyle 4(25)=100\) (for the larger angle - must then use supplementary rule again for inner smaller angle). Either way, we find that the inner angles at the top are 80 degrees each. Since the sum of the angles within a triangle must equal 180, we can set up the equation as \(\displaystyle 80+80+y=180\)

\(\displaystyle y=20\) degrees.