Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium]
235. The Parallelogram's Mirror · Opposite angles are always equal
Opposite angles of a parallelogram are always equal.
In a parallelogram, opposite angles are equal (∠A = ∠C and ∠B = ∠D). It follows from the equal, parallel sides: a diagonal forms congruent triangles whose matching angles are equal.
What this lesson covers
Try to break it
Drag A, B, or C to reshape the parallelogram. ∠A always equals ∠C and ∠B always equals ∠D, and any two adjacent angles always sum to 180°. Try to drag a vertex so two opposite angles drift apart — impossible. The parallel sides force the mirror match.
How you build it
Construct a parallelogram.
- Point tool: mark point A.
- Point tool: mark point B.
- Segment tool: join A to B.
- Point tool: mark point D, off the line AB.
- Segment tool: join A to D.
- Parallel tool: click B, then line AD — a line through B parallel to AD.
- Parallel tool: click D, then line AB — a line through D parallel to AB.
- Point tool: mark C where the two parallel lines cross — ABCD is the parallelogram.
The proof, step by step
Prove that the opposite angles of a parallelogram are equal.
- Draw diagonal BD.
- AB // DC and AD // BC (Given).
- ∠ABD = ∠CDB and ∠ADB = ∠CBD (Alt. Int. Angles).
- ΔABD ≅ ΔCDB (ASA Congruence).
- ∠A = ∠C (CPCT).
- Draw diagonal AC.
- ΔABC ≅ ΔCDA (ASA Congruence).
- ∠B = ∠D (CPCT).
- Opposite angles are equal.
Worked example
In a parallelogram ABCD, if ∠A = 70°, find ∠C.
Opposite angles of a parallelogram are equal. Therefore, ∠C = ∠A = 70°.
- 70° — correct
- 110°
- 90°
- 55°