ABCD is a quadrilateral in which AB parallel DC and AD=BC.prove that (1)angle A + angle C=180 degree= angle + angle D (2)AC=BD
To prove the given statements, we can use the properties of the quadrilateral and the given conditions. Here's how you can prove each statement:
1) Proving angle A + angle C = 180 degrees and angle B + angle D = 180 degrees:
Since AB is parallel to DC, we can use the property of alternate angles.
Alternate angles are equal when a transversal intersects two parallel lines.
In this case, the transversal is AD.
So, angle DAB is equal to angle CDA (Alternate angles).
Similarly, angle ABD is equal to angle CAD (Alternate angles).
Now, we know that AD = BC from the given condition.
By using the property of a parallelogram, opposite sides are equal.
Therefore, angle CBD is equal to angle BDA.
Now let's consider the sum of angles in quadrilateral ABCD:
angle A + angle B + angle C + angle D = 360 degrees (Sum of angles in a quadrilateral)
Substituting the equal angles we found:
angle A + angle B + angle A + angle C + angle C + angle D = 360 degrees
Simplifying the equation:
2*(angle A + angle C) + 2*(angle B + angle D) = 360 degrees
Dividing both sides by 2:
angle A + angle C + angle B + angle D = 180 degrees
Therefore, angle A + angle C = 180 degrees and angle B + angle D = 180 degrees.
2) Proving AC = BD:
Since AB is parallel to DC, the opposite sides of the quadrilateral are equal.
So, AB = DC (Opposite sides of a parallelogram)
Since AD = BC (Given condition), we can subtract AD from both sides:
AB - AD = DC - AD
By rearranging the equation, we have:
AB - AD = DC - AD
Substituting the equal sides:
AB - AD = DC - BC
Using the property of subtraction:
AB = DC - BC + AD
Since angle CBD is equal to angle BDA (Proven in statement 1), we can substitute:
AB = DC - angle CDA
Now, considering the quadrilateral ADCB:
AC + AB = DC (Sum of opposite sides of a quadrilateral)
Substituting AB:
AC + (DC - angle CDA) = DC
Rearranging the equation:
AC = DC - DC + angle CDA
Simplifying the equation:
AC = angle CDA
Since angle CDA is equal to angle ACB, we have:
AC = BD
Therefore, we have proved that AC = BD.