Given: Isosceles trapezoid CPRQ with base
R CXQ
Prove: QCX is isosceles
it needs to be a two column proof
To prove that triangle QCX is isosceles, we must demonstrate that QC and CX are congruent. In a two-column proof, we provide statements on the left column and the reasons or justifications for each statement on the right column.
Here is the two-column proof:
Statement Reason
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1. Isosceles trapezoid CPRQ with base R CXQ Given
2. CP ≅ RQ Definition of isosceles trapezoid
3. ∠CPQ ≅ ∠RQC Base angles of an isosceles trapezoid are congruent
4. ∠CQR ≅ ∠CPR Alternate interior angles are congruent
5. ∠CQR ≅ ∠QCR CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
6. QC ≅ CX Definition of congruent angles
7. Triangle QCX is isosceles Definition of isosceles triangle
In this proof, we initially state that we have an isosceles trapezoid CPRQ with base R CXQ. From this given information, we conclude that CP is congruent to RQ (statement 2) since an isosceles trapezoid has congruent bases.
Next, we observe that the base angles ∠CPQ and ∠RQC are congruent (statement 3) because the opposite angles of an isosceles trapezoid are equal.
Then, by examining the alternate interior angles, we assert that the angles ∠CQR and ∠CPR are congruent (statement 4).
Using the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) theorem, we establish that ∠CQR is congruent to ∠QCR (statement 5).
Finally, since we have congruent angles ∠CQR and ∠QCR, we conclude that side QC is equal to side CX (statement 6). Therefore, triangle QCX is isosceles (statement 7).
This completes the two-column proof showing that triangle QCX is indeed an isosceles triangle.