It is necessary and sufficient to prove that one of the angles of the parallelogram is a right angle.

Assume Statement 1 alone - that .  and  are supplementary, since they are same-side interior angles of parallel lines. Since ,  is also supplementary to . But as corresponding angles of parallel lines, . Two angles that are conruent and supplementary are both right angles, so  is a right angle.

Assume Statement 2 alone -  that  and  are complementary angles, or, equivalently, . Since the angles of a triangle have measures that add up to , the third angle of , which is , measures , and is a right angle.

Either statement alone proves  a right angle and subsequently proves  a rectangle.

Assume Statement 1 alone. By congruence,  and , making Quadrilateral  a parallelogram. However, no clue is given to whether any angles are right or not, so whether the quadrilateral is a rectangle or not remains open.

Assume Statement 2 alone. By congruence, opposite sides , but no clue is provided as to the lengths of opposite sides  and  . Also, , but no clue is provided as to whether the angles are right. A rectangle would have both characteristics, but so would an isosceles trapezoid with legs  and .

Assume both statements are true. Quadrilateral  is a parallelogram as a consequence of Statement 1. Since   and  are consecutive angles of the parallelogram, they are supplementary, but they are also congruent as a consequence of Statement 2. Therefore, they are right angles, and a parallelogram with right angles is a rectangle.

Statement 1 alone is insufficient to answer the question; A quadrilateral in which   and  are right angles, , and  fits the statement, as well as a rectangle, which by defintion has four right angles. 

Statement 2 alone is insufficient as well, as a parallelogram with acute and obtuse angles, as well as a rectangle, fits the description.

Assume both statements, and construct diagonal  to form two triangles  and . By Statement 1, both triangles are right with congruent legs , and congruent hypotenuses, both being the same segment . By the Hypotenuse Leg Theorem, . By congruence, . The quadrilateral, having two sets of congruent opposite sides, is a parallelogram; a parallelogram with right angles is a rectangle.

Assume Statement 1 alone. A rhombus being a parallelogram, its opposite angles and its adjacent angles are supplementary. From this fact and Statement 1 alone, it follows that

, , , and .

By definition of a rhombus, all of its sides are congruent. By substitution,

 .

All side proportions hold as well as all angle congruences, so the similarity statement holds.

Assume Statement 2 alone. Construct the diagonals of the rhombuses, as follows:

In each rhombus, the diagonals are each other's perpendicular bisector. If 

then 

Since , both angles being right, it follows via the Side-Angle-Side Smiilarity Theorem that 

,

and , by similarity,

.

By a similar argument, 

,

and by angle addition,

.

As with Statement 1 alone, congruence of one set of corresponding angles in two rhombuses leads to the similarity of the two.

Each statement gives the length of one diagonal of the rhombus. Knowing one diagonal is not enough to give the perimeter of the rhombus.

Knowing the lengths of both diagonals, which is the case if both statements are assumed, is enough to determine the perimeter. The rhombus in question, along with its diagonals, is as shown below:

As marked in the diagram, the diagonals are perpendicular, and they are also are each other's bisector. It follows from the given lengths that  and , so  can be calculated using the Pythagorean Theorem. The perimeter is four times , since all sides of a rhombus are congruent.

To prove two figures similar, we must prove that their corresponding angles are congruent, and that their corresponding sides are in proportion.

Assume both statements are true. We show that they provide insufficient information by examining two scenarios.

If Trapezoid  Trapezoid , then  and , so the conditions of both statements are met; also, since the trapezoids are congruent, they are also similar.

Now examine isosceles trapezoid  below, in which , , , and  are positioned on the bases so that  and .

Since  and , Quadrilateral  is a parallelogram, and ; similarly, . Therefore, Trapezoid  is also isosceles, and the conditions of both statements are met. However, corresponding sides are not in proportion, since , but ; consequently, the trapezoids are not similar.

To prove two figures similar, we must prove that their corresponding angles are congruent, and that their corresponding sides are in proportion.

All four sides of a rhombus are congruent, so it easily follows that corresponding sides of two rhombuses are in proportion, regardless of whether they are similar or not; it is therefore necessary and sufficient to prove that corresponding angles are congruent. Also, since a rhombus is a parallelogram, opposite angles are congruent and consecutive angles are supplementary—that is, their angle measures total . Therefore, it is necessary and sufficient to prove just one pair of corresponding angles congruent.

The two statements together give no information about the measures of any of the angles of the rhombus. Therefore, together, they do not answer the question of whether they are similar or not.

To prove two figures similar, we must prove that their corresponding angles are congruent, and that their corresponding sides are in proportion.

All four sides of a rhombus are congruent, so it easily follows that corresponding sides of two rhombuses are in proportion, regardless of whether they are similar or not; it is therefore necessary and sufficient to prove that coresponding angles are congruent. Also, since a rhombus is a parallelogram, opposite angles are congruent and consecutive angles are supplementary - that is, their angle measures total . Therefore, it is neccessary and sufficient to prove just one pair of corresponding angles congruent.

Assume Statement 1 alone.  is a  angle, so any angle consecutive to it, which includes , is supplementary to it—that is, the angle measures total . This makes  a  angle. Its corresponding angle in Rhombus  is , which is a  angle. Since  and  are noncongruent, it follows that Rhombus  Rhombus . (Note that it can be demonstrated that the rhombuses are similar, but the correct statement is Rhombus  Rhombus .)

Statement 2 alone provides no useful information; the relationship between the areas of the rhombuses is irrelevant.

To prove two figures similar, we must prove that their corresponding angles are congruent, and that their corresponding sides are in proportion.

All four sides of a rhombus are congruent, so it easily follows that corresponding sides of two rhombuses are in proportion, regardless of whether they are similar or not; it is therefore necessary and sufficient to prove that coresponding angles are congruent. Also, since a rhombus is a parallelogram, opposite angles are congruent and consecutive angles are supplementary—that is, their angle measures total . Therefore, it is neccessary and sufficient to prove just one pair of corresponding angles congruent.

Assume Statement 1 alone.  and  are not corresponding angles, so their congruence does not prove the rhombuses are similar; however it does not prove they are not similar either; for example, two squares are both rhombuses with four right angles, so they are similar and fit this condition.

Assume Statement 2 alone.  and  are two angles of the same rhombus, so similarity cannot be proved or disproved without information about the other rhombus.

Now assume both statements.   and  are consecutive angles of Rhombus ; a rhombus being a parallelogram, the degree measures of the angles total . From Statement 2, they are congruent, so each measures . Since  from Statement 1,  also measures . Since each parallelogram has at least one right angle, each has four right angles. Therefore, corresponding angles are congruent, so the rhombuses are similar.

To prove two figures similar, we must prove that their corresponding angles are congruent and that their corresponding sides are in proportion.

We show that Statement 1 alone provides insufficient information by examining two cases.

Case 1: Trapezoid  Trapezoid , with  a  angle.

A pair of base angles of an isosceles trapezoid are congruent, so, since  measures , so does . The leg angles of a trapezoid have measures whose sum is , so  and  measure . By congruence of corresponding angles, , so  measures , and the conditions of Statement 1 are met. Furthermore, two congruent triangles are also similar, so Trapezoid  Trapezoid .

Case 2: Examine the figure below, in which . Note that these congruent upper bases have been superimposed upon each other:

The trapezoids are isosceles since their base angles are congruent; also, the conditions of Statement 1 are met. However, , but . The sides are not in proportion, so the trapezoids are not similar.

Assume Statement 2 alone. Two congruent trapezoids whose side lengths fit the conditions of the statement are also similar. But examine this diagram, in which , , and . Note that the congruent upper bases have been superimposed upon each other:

The conditions of Statement 2 are met, but , so the trapezoids are not similar.

Now assume both statements to be true. As stated before, it follows from Statement 1 that all corresponding angles are congruent.  and , so it follows from the Division Property of Equality that

It remains to be demonstrated that  is equal to the above ratios as well. If the diagonals  and  are constructed,  and  are formed; since  and , by the Side-Angle-Side Similarity Theorem, . It follows that , and by angle addition, since , it follows that . From the Angle-Angle Postulate, the other two triangles formed are similar—that is, —and it follows that . Therefore, all four sides of the trapezoids are in proportion, and the trapezoids are similar.