Write a two column proof proving figure ABCD is a rhombus. Given - diagonal BC.
To write a two-column proof proving that figure ABCD is a rhombus, we need to use the properties of a rhombus and the given information. The given information states that diagonal BC is given, but to prove that ABCD is a rhombus, we need to show that all the sides of ABCD are equal.
Here's a step-by-step explanation of how to write the two-column proof:
Step 1: Write "Given" in the first row of the left column and "To prove" in the second row. In the first row of the right column, write the given information: diagonal BC.
Step 2: In the second row of the right column, write that we want to prove figure ABCD is a rhombus.
Step 3: Write "Properties/Reasons" in the third row of the left column. In this column, we will list the properties or reasons that support our proof.
Step 4: Start with the first property or reason, which is that a rhombus has all sides equal. Write this in the first row of the right column.
Step 5: In the second row of the left column, write "Diagonals of a rhombus bisect each other." This property will help us prove that the sides are equal.
Step 6: In the second row of the right column, write that diagonals of a rhombus bisect each other.
Step 7: In the third row of the left column, write "BC is a diagonal of ABCD." This is a property of a rhombus since we are given diagonal BC.
Step 8: In the third row of the right column, write that BC is a diagonal of ABCD.
Step 9: In the fourth row of the left column, write "Diagonals bisect each other at right angles." This is a well-known property of a quadrilateral.
Step 10: In the fourth row of the right column, write that diagonals of ABCD bisect each other at right angles.
Step 11: In the fifth row of the left column, write "Right angles imply congruent triangles." With the right angles formed between the diagonals, we can prove congruence.
Step 12: In the fifth row of the right column, write that right angles imply congruent triangles.
Step 13: In the sixth row of the left column, write "By SAS (Side-Angle-Side) congruence, triangles ABC and CBD are congruent." This property is based on the fact that the sides and the included angle of the triangles are congruent.
Step 14: In the sixth row of the right column, write that triangles ABC and CBD are congruent by SAS congruence.
Step 15: In the seventh row of the left column, write "Since triangles ABC and CBD are congruent, their corresponding sides are congruent."
Step 16: In the seventh row of the right column, write that corresponding sides of triangles ABC and CBD are congruent.
Step 17: In the eighth row of the left column, write "Therefore, AB = CB." This follows from the congruence of triangles ABC and CBD.
Step 18: In the eighth row of the right column, write that AB = CB.
Step 19: In the ninth row of the left column, write "Similarly, AD = DC."
Step 20: In the ninth row of the right column, write that AD = DC.
Step 21: In the tenth row of the left column, write "Therefore, all sides of ABCD are equal."
Step 22: In the tenth row of the right column, write that all sides of ABCD are equal.
Step 23: In the eleventh row of the left column, write "Hence, figure ABCD is a rhombus."
Step 24: In the eleventh row of the right column, write that figure ABCD is a rhombus.
By following these steps, you can construct a two-column proof proving that figure ABCD is a rhombus using the given information of diagonal BC.