Question
Parallelogram ABCD is a rectangle.
Which statements are true?
m∠ABX+m∠CBX=90°
m∠DXC=m∠BXC
2AX=CD
DX=AX
Which statements are true?
m∠ABX+m∠CBX=90°
m∠DXC=m∠BXC
2AX=CD
DX=AX
Answers
GPT-5 mini
Assuming X is the intersection of the diagonals of rectangle ABCD:
1) m∠ABX + m∠CBX = 90° — True.
- At B the two angles together equal ∠ABC, and in a rectangle ∠ABC = 90°.
2) m∠DXC = m∠BXC — False (in general).
- B, X, D are collinear (X is on diagonal BD), so XB and XD are opposite rays. Thus ∠BXC + ∠DXC = 180°, so they are equal only when each is 90° (i.e. only for a square).
3) 2AX = CD — False.
- AX = (1/2)AC, so 2AX = AC, not CD; AC ≠ CD unless degenerate.
4) DX = AX — True.
- In a rectangle diagonals are equal, so their halves are equal: AX = 1/2 AC = 1/2 BD = DX.
So: statements 1 and 4 are true; 2 and 3 are false.
1) m∠ABX + m∠CBX = 90° — True.
- At B the two angles together equal ∠ABC, and in a rectangle ∠ABC = 90°.
2) m∠DXC = m∠BXC — False (in general).
- B, X, D are collinear (X is on diagonal BD), so XB and XD are opposite rays. Thus ∠BXC + ∠DXC = 180°, so they are equal only when each is 90° (i.e. only for a square).
3) 2AX = CD — False.
- AX = (1/2)AC, so 2AX = AC, not CD; AC ≠ CD unless degenerate.
4) DX = AX — True.
- In a rectangle diagonals are equal, so their halves are equal: AX = 1/2 AC = 1/2 BD = DX.
So: statements 1 and 4 are true; 2 and 3 are false.