Problem

Corresponding and alternate angles are equal

Two parallel lines p (top) and q (bottom) are crossed by two slanted lines that meet in an X between them. One angle where a transversal meets the top line p is 55 deg; one angle where a transversal meets the bottom line q is 40 deg. At the crossing point between the lines, the angle that opens downward toward q is marked with a circle. I need that circle angle.

Geometry

Your answer

How to solve

Strategy Draw a Diagram — The two transversals and line q bound a triangle. Its apex is the crossing point (the circle angle), and its base sits on q. The 40 deg is one base angle directly. The 55 deg at line p moves down to the other base angle on q by alternate interior angles (p is parallel to q). Then the apex angle is what is left to reach 180 deg.

1STEP 1

Spot the triangle

The two transversals and line q cut off a triangle, and the circle is the apex angle.

∠(circle) = 180° - (base angle_1) - (base angle_2)

2STEP 2

Move the 55 deg down to line q

The 55 deg at line p equals its alternate interior angle down at line q, so one base angle of the triangle is 55 deg.

55° at p = 55° base angle at q

3STEP 3

Use the two base angles

The triangle's two base angles on q are 55 deg and 40 deg. The apex (circle) angle is 180 deg minus their sum.

∠(circle) = 180° - 55° - 40° = 85°

Answer

85 °

180° - 55° - 40° = 85°

The base angles 55 deg and 40 deg add to 95 deg, leaving 85 deg for the apex, which is just under a right angle, matching the crossing shown. All three angles 55 + 40 + 85 = 180 deg.

Takeaway

Find the hidden triangle, slide the 55 deg down between the parallel lines, and the last angle is just what is left of 180 deg!

Spot the triangle
Move the 55 deg down to line q
Use the two base angles

Where next?

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