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In rectangle \(ABCD\), \(E\) is the point of intersection of diagonals. If angle \(ABD\) is twice angle \(EAD\), what is the value of angle \(CED\)?

A. 30 degrees B. 45 degrees C. 60 degrees D. 90 degrees E. 120 degrees

Look at the diagram below:

Notice that since \(\angle{BAD}=90\) degrees then \(\angle{EAB}=90-x\) degrees.

Next, since diagonals of a rectangle are equal and bisect each other then \(AE=BE\). So, \(\angle{EAB}=\angle{EBA}\): \(90-x=2x\), which gives \(x=30\). Thus, \(\angle{EAB}=\angle{EBA}=60\) and \(\angle {AEB}=60\) degrees.

Finally as \(\angle{AEB}=\angle{CED}\) then \(\angle{CED}=60\) degrees.

In rectangle \(ABCD\), \(E\) is the point of intersection of diagonals. If angle \(ABD\) is twice angle \(EAD\), what is the value of angle \(CED\)?

A. 30 degrees B. 45 degrees C. 60 degrees D. 90 degrees E. 120 degrees

Look at the diagram below:

Notice that since \(\angle{BAD}=90\) degrees then \(\angle{EAB}=90-x\) degrees.

Next, since diagonals of a rectangle are equal and bisect each other then \(AE=BE\). So, \(\angle{EAB}=\angle{EBA}\): \(90-x=2x\), which gives \(x=30\). Thus, \(\angle{EAB}=\angle{EBA}=60\) and \(\angle {AEB}=60\) degrees.

Finally as \(\angle{AEB}=\angle{CED}\) then \(\angle{CED}=60\) degrees.

Answer: C

Hi, Aren't the diagonals of a rectangle an angular bisector ? AS, WE KNOW THAT DE = BE. So, angle a should have each angle = 45 degree ? angle DAE = angle BAE = 45 ? thanks

In rectangle \(ABCD\), \(E\) is the point of intersection of diagonals. If angle \(ABD\) is twice angle \(EAD\), what is the value of angle \(CED\)?

A. 30 degrees B. 45 degrees C. 60 degrees D. 90 degrees E. 120 degrees

Look at the diagram below:

Notice that since \(\angle{BAD}=90\) degrees then \(\angle{EAB}=90-x\) degrees.

Next, since diagonals of a rectangle are equal and bisect each other then \(AE=BE\). So, \(\angle{EAB}=\angle{EBA}\): \(90-x=2x\), which gives \(x=30\). Thus, \(\angle{EAB}=\angle{EBA}=60\) and \(\angle {AEB}=60\) degrees.

Finally as \(\angle{AEB}=\angle{CED}\) then \(\angle{CED}=60\) degrees.

Answer: C

Hi, Aren't the diagonals of a rectangle an angular bisector ? AS, WE KNOW THAT DE = BE. So, angle a should have each angle = 45 degree ? angle DAE = angle BAE = 45 ? thanks

The diagonal of a rectangle is a bisector only if the rectangle is a square.
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I think this is a poor-quality question and the explanation isn't clear enough, please elaborate. Hi, can you explain more? Basically, total 3 angles of a triangle is 180. If <AEB = 60, total value of 3 angles in triangle AEB is 150 degree (Because <AEB = 60, <ABE= 60, and <BAE = 30). Moreover, 4 angles AEB + BEC + CED + DEA = 360. As a result, <AEB must be at least 90 degree. Please explain. Thank you so much!

I think this is a poor-quality question and the explanation isn't clear enough, please elaborate. Hi, can you explain more? Basically, total 3 angles of a triangle is 180. If <AEB = 60, total value of 3 angles in triangle AEB is 150 degree (Because <AEB = 60, <ABE= 60, and <BAE = 30). Moreover, 4 angles AEB + BEC + CED + DEA = 360. As a result, <AEB must be at least 90 degree. Please explain. Thank you so much!

The question is fine. You just made mistake in calculations. The measure of angle BAE is 60, not 30. Please re-read the discussion more above carefully this time.
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Can you please explain the angle measure 2x for EBA

We denoted \(\angle EAD\) as x. The stem says that angle \(ABD\) is twice angle \(EAD\), so it's 2x (notice that EBA and ABD are the same angle).
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