Question

(SAT Prep) In the given figure, find x for which a ∥ b?

A. 50° B. 65° C. 75° D. 95°

Please explain your answer, and thanks in advance :)

Answer 1

`\huge\boxed{x = 65 \ \text{degrees}}`

Explanation:

We can use angle relationships to find the measure of
`x` here.

Let's assume that the triangle has the points ABC, where m∠a is the leftmost angle, m∠b is the right angle, and m∠c is the rightmost angle (I've labeled it in the attached picture).

We have to realize here that a and b are parallel lines. This means that the angles on one side will be equivalent to the other as long as the triangle in the center is symmetrical along both.

It's not. However, we can make it symmetrical by realizing that the 85° angle is the exact same as Line A AC. These are alternate interior angles, meaning their angle measurements will be the same. So Line A A C is also 85°.

Line A A C covers the area even in the triangle. Since we know the total measure and the measure outside the triangle, we can find the measure of m∠a.

`25 + a = 85\\\\a = 85-25\\\\a = 60`

Now we know that m∠a is 60°. We also know that m∠b is a 90° angle. All triangle angles add up to 180°. Since we know two out of the three, let's find the measure of m∠c by adding up what we know and subtracting from 180.

`60+90+c=180\\\\150+c=180\\\\c=180-150\\\\c=30`

Now that we know m∠c, we can note that the 85° angle, m∠a, and x are supplementary. This means their angle lengths add up to 180°. Since we know two out of the three, we can do a process similar to finding the missing triangle angle.

`85+30+x=180\\\\115+x=180\\\\x=180-115\\\\x = 65`

Therefore, x is 65°.

Hope this helped!

Answer 2

x= 65 degrees

Explanation:

If the triangles vertices are marked CDE (d as the right angle and c as the leftmost vertex) then:

Alternate interior angles

⇒m∠ACE=85 degrees

m∠ACD=25 and m∠ACD+m∠DCE=m∠ACE=85 degrees

⇒ 25 + m∠DCE= 85 ⇒ m∠DCE=60 degrees

Since the sum of the angles in a triangle is 180 degrees

⇒ 60 + 90 + m∠DEC = 180 ⇒ m∠DEC= 30 degrees

Finally, since ∠BEC is supplementary to the given 85 degree angle

⇒ 30 + 85 + x = 180 ⇒ x=65