Proving Parallelogram Theorems
Aligned to G-CO.C.11 — Common Core State Standards for Mathematics.
What this lesson teaches
A parallelogram has both pairs of opposite sides parallel. From this, opposite sides are congruent, opposite angles are congruent, and the diagonals bisect each other. A rectangle is a parallelogram with a right angle, and its diagonals are congruent.
Worked example
Prove the diagonals of a parallelogram bisect each other. In parallelogram ABCD, draw diagonals AC and BD meeting at P. Because AB is parallel to DC, alternate interior angles give angle BAP = angle DCP and angle ABP = angle CDP. Also AB = DC (opposite sides). By ASA, triangle ABP = triangle CDP, so AP = CP and BP = DP. Thus each diagonal is cut in half at P.
Practice questions
- In parallelogram ABCD, angle A = 65°. Find angle C and angle B.
- A parallelogram has one side 9 cm. What is the length of the opposite side?
- Explain why the diagonals of a rectangle must be congruent, using the fact that a rectangle is a parallelogram with right angles.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Demostrar teoremas de paralelogramos
Un paralelogramo tiene los dos pares de lados opuestos paralelos. De esto, los lados opuestos son congruentes, los ángulos opuestos son congruentes y las diagonales se bisecan mutuamente. Un rectángulo es un paralelogramo con un ángulo recto, y sus diagonales son congruentes.
Ejemplo: Demuestra que las diagonales de un paralelogramo se bisecan. En el paralelogramo ABCD, traza las diagonales AC y BD que se cortan en P. Como AB es paralelo a DC, los ángulos alternos internos dan ángulo BAP = ángulo DCP y ángulo ABP = ángulo CDP. Además AB = DC (lados opuestos). Por ALA, triángulo ABP = triángulo CDP, así que AP = CP y BP = DP. Por tanto cada diagonal queda dividida a la mitad en P.
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