Question

(60 POINTS) Please explain this in detail

Answer 1

H=D

O=C

B=M

R=A

M=B

First we will prove that the diagonals are bisectors of each other.

Angle DBA is congruent to angle BDC.

Angle CMD is congruent to angle AMB.

Triangle CMD is congruent to triangle AMB.

Segment AM is congruent to segment MC.

M is the midpoint of segment AC.

Segment BD bisects segment AC.

Segment BM is congruent to segment MD.

M is the midpoint of segment BD.

Segment AC bisects segment BD.

Then we will prove that the diagonals are perpendicular to each other.

10. Segment CD is congruent to segment CB.

11. Segment BM is congruent to segment MB.

12. Segment CM is congruent to segment CM.

13. Triangle CDM is congruent to triangle CBM.

14. Angle CMD is congruent to triangle BMC.

15. Angle CMD + angle BMC = 180

16. Angle CMD = 90 and angle BMC = 90

17. Angle AMB = 90

18. Angle CMD is congruent to angle DMA.

19. Angle DMA = 90

20. Segment AC is perpendicular to segment BD.

Answer 2

In Quadrilateral A B C D

Point of intersection of Diagonals is point T

If, AB ║CD

then BD being a transversal and using Alternate Interior Angles Theorem gives

∠ADB≅∠DBC

∠DBA≅∠BDC

To prove, AB = DC,

If this quadrilateral is a parallelogram or rhombus or Square or rectangle

AB= CD →[Opposite sides of Quadrilateral]

The Quadrilateral can be any of the four i.e a parallelogram or rhombus or Square or rectangle .