CONSTRUCTIONS (Using ruler and compasses only)
176. The Parallel Promise · equal alternate angles, forever parallel
Alternate interior angles ∠PQB and ∠QPC remain equal, keeping CD parallel to AB.
To construct a parallel line through external point P: draw a transversal from P to AB. Copy the angle at the AB intersection at P on the same side, creating equal alternate interior angles. The new line through P is then parallel to AB.
What this lesson covers
Try to break it
Drag P off line AB. The transversal PQ shifts and the alternate angles ∠PQB and ∠QPC change size together — but they always stay equal. Because of that, CD stays parallel to AB. Try to drag P so CD eventually crosses AB; impossible.
How you build it
Construct a line parallel to AB through P.
- Draw line AB — a long horizontal line across the canvas.
- Mark point P somewhere above line AB. The parallel line will pass through P.
- Select the Parallel tool. Click on point P, then click on line AB. The new line through P is parallel to AB — alternate interior angles are equal.
The proof, step by step
Prove that the constructed line CD is parallel to AB.
- Draw transversal PQ and construct ∠QPC equal to ∠PQB.
- Identify the alternate interior angles: ∠PQB and ∠QPC.
- Since ∠PQB = ∠QPC, the lines AB and CD are parallel by the Alternate Angle Converse Theorem.
Worked example
In the figure, line AB || line CD. Transversal PQ intersects them at Q and P respectively. If ∠PQB = 55°, find ∠QPC.
When a transversal intersects two parallel lines, alternate interior angles are equal. Therefore, ∠QPC = ∠PQB = 55°.
- 35°
- 55° — correct
- 125°
- 135°