Question

Quadrilateral ABCD has sides AB parallel to CD and AD is equal to AB, and BC is equal to CD. If diagonal BD is an angle bisector to angle B and angle D is 10 degrees, prove that ABCD is a rhombus.

Answer 1

To prove that quadrilateral ABCD is a rhombus, you need to show that all four sides are congruent. Given the information about parallel lines, angle bisectors, and congruent triangles, you can use a step-by-step proof to demonstrate that ABCD is a rhombus.

Step-by-step explanation:

To prove that quadrilateral ABCD is a rhombus, we need to show that all four sides are congruent.

Given that AB is parallel to CD and AD is equal to AB, we can conclude that AD is parallel to BC.

Since BD is an angle bisector of angle B and angle D is 10 degrees, we know that angle ABD is equal to angle CBD.

We can use the properties of parallel lines and congruent triangles to prove that all four sides are congruent.

Here is a step-by-step proof:

1. AB is parallel to CD (given)
2. AD is equal to AB (given)
3. AD is parallel to BC (converse of the alternate interior angles theorem)
4. angle ABD is equal to angle CBD (given)
5. triangle ABD is congruent to triangle CBD (ASA congruence criterion)
6. AB is congruent to CD (corresponding parts of congruent triangles are congruent)
7. AD is congruent to BC (corresponding parts of congruent triangles are congruent)
8. Therefore, all four sides of quadrilateral ABCD are congruent, making it a rhombus.