Question 46·Hard·Geometry and Trigonometry

You know the measure of one acute angle is $8x - 22$. Use the fact that an acute angle and its adjacent obtuse angle are supplementary to write an expression for the obtuse angle.

Find an expression for the sum of four acute angles and one obtuse angle using your expressions. Then set this equal to $24x + w$ and compare the constant terms.

In Desmos, type the expression for the sum of four acute angles and one obtuse angle in terms of $x$:

S(x) = 4*(8x - 22) + (180 - (8x - 22))

This represents the total of four acute angles plus one obtuse angle.

Choose an easy value for $x$, for example x = 10. In Desmos, either define x = 10 or just type S(10) to see the numeric value of the sum of the angles for that $x$.

For the same $x$-value (for example, $x = 10$), compute 24*10 in Desmos. Subtract this from the value of S(10):

S(10) - 24*10

The result is the constant amount that remains after removing the $24x$ part; that constant is the value of $w$. You will get the same result for any other $x$ you choose.

When a transversal cuts two parallel lines:

• There are 4 congruent acute angles and 4 congruent obtuse angles.
• Each acute angle and its adjacent obtuse angle form a linear pair, so they are supplementary and add up to $180^\circ$.

So, if one acute angle measures $8x - 22$, then the obtuse angle next to it must be $180^\circ$ minus that acute angle.

The measure of one acute angle is given as $(8x - 22)^\circ$.

The adjacent obtuse angle is supplementary to it, so its measure is:

Now find the sum of four acute angles and one obtuse angle:

• Four acute angles: $4(8x - 22) = 32x - 88$
• One obtuse angle: $202 - 8x$

Total sum:

The problem states that this same sum is also $(24x + w)^\circ$.

So the two expressions represent the same angle sum:

Since the $24x$ terms are the same, the constants must be equal:

So, the value of $w$ is $114$.