Question 174472
Prove: If a diagonal of a parallelogram bisects an angle of the parallelogram, the parallelogramis a rhombus. State your plan and give a proof. Proof must include statements and reasons. 
Given: ABCD is a parallelogram with <font face = "symbol">Ð</font>1 <font face = "symbol">@</font> <font face = "symbol">Ð</font>2.
To Prove: ABCD is a rhombus.
<pre><font size = 3 color = "indigo"><b>
{{{drawing(200,120,-1,9,-1,5, triangle(0,0,5,0,8,4),
triangle(0,0,3,4,8,4),locate(0,0,A), locate(5,0,B),
locate(3,4.7,D), locate(8,4.7,C), locate(1.4,.7,1),
locate(1.2,1.5,2), locate(6.3,4,3),locate(6.7,3.5,4)
 )}}} 

 1. ABCD is a parallelogram           1. Given
 2. AB <font face = "arial" size = 2>||</font> CD                           2. Opposite sides of a
                                         parallelogram are parallel
 3. <font face = "symbol">Ð</font>1 = <font face = "symbol">Ð</font>2                          3. Given

 4. <font face = "symbol">Ð</font>1 = <font face = "symbol">Ð</font>3                          4. Alternate interior angles
                                         formed when transversal AC
                                         cuts parallel lines AB and
                                         CD.
 5. <font face = "symbol">Ð</font>2 = <font face = "symbol">Ð</font>3                          5. Things equat to the same 
                                                 thing are equal to each 
                                                 other
                                        
 6. <font face = "symbol">Ð</font>2 <font face = "symbol">@</font> <font face = "symbol">Ð</font>3                          6. Angles congruent to the
                                         same angle are congruent.
 7. <font face = "symbol">D</font>ACD is isosceles                 7. Base angles 2 and 3 are
                                         congruent
 8. AD <font face = "symbol">@</font> CD                           8. Legs of an isosceles
                                         triangle are congruent
 9. AD <font face = "symbol">@</font> BC                         9. Opposite sides of a
                                         parallelogram are congruent 
10. CD <font face = "symbol">@</font> AB                          10. Opposite sides of a
                                         parallelogram are congruent
11. AB <font face = "symbol">@</font> BC <font face = "symbol">@</font> CD <font face = "symbol">@</font> AD                11. By 8,9, and 10 and "things
                                         congruent to the same thing
                                         are congruent to each other".
12. ABCD is a rhombus                12. All 4 sides are congruent. 

Edwin</pre>