Parallelograms - quadrilateral

Question

1- Given a parallelogram MNPQ such that MN = 2a-3b and NO = 20-3b. Find the value of a and all possible value of b that make MNPQ a rhombus.

 

2- Given a quadrilateral ABCd such that ∠A and ∠B are supplementary and ∠A = ∠C. Show that ABCD is a parallelogram.

 

3- Prove that every rhombus is a parallelogram.

Answer 1

Note: I think you meant NP instead of NO. Your finger must have hit the wrong key.

 

 

1)

 

The rhombus will have 4 consecutive sides:

 

MN, NP, PQ, and MQ

  

 

A rhombus has 4 equal sides.  So we set the given sides equal to each other.

 

2a - 3b = 20 - 3b

 

Since -3b is on both sides of the equation, b can be an infinite number of values as long as b is constant. All that is left is to equate 2a and 20.

 

2a = 20

 

a = 10

 

 

2)

 

The sides of the quadrilateral will be arrange as:

 

AB, BC, CD, and AD

 

∠A and ∠B  are adjacent to each other.  ∠C and ∠D are also adjacent to each other. ∠A and ∠C are opposite to each other.  These angles are always equal in measure.

 

You can draw a picture of this to get a better visual.

 

Parallelograms are 4 sided closed figures that have two pairs of parallel sides that are equal to each other, 2 pairs of equal and opposite angles, and 2 pairs of supplementary angles.  If ∠A is an angle less than 90 degrees and ∠B is an angle greater than 90 degrees, these two angles together must form a straight line.  These angles are supplement to each other.

 

 

3)

 

A parallelogram is a 4 sided figure with 2 pairs of parallel sides, 2 pairs of equal side lengths, and 2 pairs of opposite and equal angles.  A rhombus fits this description.