Exercise: Rectangle and Diagonals
Given:
ABCD is a rectangle
AB || CD (opposite sides are parallel)
AB = CD (opposite sides are equal)
E = intersection point of the diagonals
Part A
Prove that:
△ABE ≅ △CDE
By which congruence theorem?
Think: we have information about alternate angles between parallel lines, and an equal side.
Which theorem uses two angles and the side between them?
🎉 Correct! 🎉
✓ Proof of Part A
△ABE ≅ △CDE by A.S.A
A
∠ABE = ∠CDE
(Alternate angles between parallel lines AB || CD with transversal BD)
S
AB = CD
(Opposite sides of a rectangle are equal - given)
A
∠BAE = ∠DCE
(Alternate angles between parallel lines AB || CD with transversal AC)
Therefore:
△ABE ≅ △CDE
(By A.S.A theorem - Angle, Side, Angle)
Part B - Conclusion from the Congruence
Already proven:
△ABE ≅ △CDE
Now that the triangles are congruent, what can we conclude?
AE = ? and BE = ?
Corresponding sides are sides in the same "position" in both triangles.
In â–³ABE: the sides are AE, BE, AB
In â–³CDE: the sides are CE, DE, CD
AE corresponds to CE, and BE corresponds to DE
In â–³ABE: the sides are AE, BE, AB
In â–³CDE: the sides are CE, DE, CD
AE corresponds to CE, and BE corresponds to DE
🎉 Excellent! 🎉
✓ Conclusion of Part B
Corresponding sides of congruent triangles are equal
1
AE = CE
(Corresponding sides of congruent triangles △ABE ≅ △CDE)
2
BE = DE
(Corresponding sides of congruent triangles △ABE ≅ △CDE)
Conclusion:
The diagonals of a rectangle bisect each other!
E is the midpoint of AC and also the midpoint of BD
Part C - Auxiliary Construction
Already proven:
AE = CE (from Part B)
Auxiliary Construction:
Drop a perpendicular from E to side AB - the intersection point is F
Drop a perpendicular from E to side CD - the intersection point is G
Prove that:
△AEF ≅ △CEG
By which congruence theorem?
Think: we have a right angle in each triangle (the perpendicular!), we have an equal side AE=CE (proven earlier),
and we have alternate angles. That's A.S.A!
🎉 Excellent! 🎉
✓ Proof of Part C
△AEF ≅ △CEG by A.S.A
A
∠AFE = ∠CGE = 90°
(Two perpendiculars - auxiliary construction)
S
AE = CE
(Proven in Part B - corresponding sides of congruent triangles)
A
∠FAE = ∠GCE
(Alternate angles between parallel lines AB || CD with transversal AC)
Therefore:
△AEF ≅ △CEG
(By A.S.A theorem - Angle, Side, Angle)
Hence: EF = EG (corresponding sides) - the distance from E to both sides is equal!
🌟 Great job! 🌟
You completed the entire exercise successfully!
You proved three things:
A. △ABE ≅ △CDE (by A.S.A)
B. The diagonals bisect each other
C. △AEF ≅ △CEG (with auxiliary construction!)
© Developed by Shir Sivroni