Thus, we have shown that if a quadrilateral has one pair of parallel, congruent opposite sides, then it is a parallelogram.

Thus, we have shown that if a quadrilateral has two pairs of opposite congruent angles, it is a parallelogram.

Statement                                                                            Reasoning

 is a parallelogram.                                           This is given in the problem.

We can add the line  across the diagonal                   Connecting any two points make a line, so this is a valid line we can add.

 is parallel to 

 is parallel to   We know this to be true according to the definition of a parallelogram.

 Line  is a transversal line intersection two parallel lines. We could extend lines  and  or lines  and  to make this relationship more clear. Alternate interior angles are formed by this transversal line.

 is a common side between  and 

 We are able to use the Angle-Side-Angle Theorem because we have one congruent side between these two triangles, , and two pairs of congruent angles, , 

 Congruent triangles have congruent corresponding parts by definition

 Because these angles are congruent they are also equal

 Since  we can add one of these angles to each side and still keep the equation balanced and equal

 The Angle Addition Postulate says that two side by side angles create a new angle whose measure is equal to their sum

  We are simply substituting these equalities into the equation 

 Equal angles are congruent

Therefore  and 

Thus, we have proven that this parallelogram has two pairs of opposite congruent angles.

Given only a line segment there are numerous steps that need to be taken to construct a diamond. Recall a diamond is a parallelogram with four equilateral sides where the diagonals are vertical and horizontal.

The first step in constructing a diamond would be to calculate the length of the line segment . After the length is known then the slope can be found and from there, each of the connecting lines can be drawn. 

Therefore, the correct answer is,

"Calculate the length of ."

To determine which image illustrates a triangle that is inscribed in a circle, first understand what the term "inscribed" means.

"Inscribed" means to draw inside of. Therefore, a triangle inscribed in a circle means the triangle will be drawn inside of a circle.

The image that illustrates this is as followed.

To determine which image illustrates a rectangle that is inscribed in a circle, first understand what the term "inscribed" means.

"Inscribed" means to draw inside of. Therefore, a rectangle inscribed in a circle means the rectangle will be drawn inside of a circle.

The image that illustrates this is as followed.

To determine which image illustrates a circle that is inscribed in a rectangle, first understand what the term "inscribed" means.

"Inscribed" means to draw inside of. Therefore, a circle inscribed in a rectangle means the circle will be drawn inside of a rectangle.

The image that illustrates this is as followed.

To determine which image illustrates a circle that is inscribed in a triangle, first understand what the term "inscribed" means.

"Inscribed" means to draw inside of. Therefore, a circle inscribed in a triangle means the circle will be drawn inside of a triangle.

The image that illustrates this is as followed.

To determine which image illustrates a pentagon that is inscribed in a circle, first understand what the term "inscribed" means.

"Inscribed" means to draw inside of. Therefore, a pentagon inscribed in a circle means the pentagon will be drawn inside of a circle.

The image that illustrates this is as followed.