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This printable supports Common Core Mathematics Standard HSG-CO.C.11

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# Parallelogram w/ Congruent Diagonals is Rectangle (Grade 10)

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Fill in the reasons for the following proof.

Given that $ABCD$ is a parallelogram and $bar{BD}~=bar{AC}$ (line segments not pictured) prove that $ABCD$ is a rectangle.
 $\ \ \ \ \ \ \ \ \ \ \ \ " Statement " \ \ \ \ \ \ \ \ \ \ \ \$ $" Reason "$ $1. bar{AC} ~= bar{BD}$ $1. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \$ $2. ABCD " is a parallelogram"$ $2. ""$ $3. bar{AB} ~= bar{DC}$ $3. ""$ $4. bar{AD} ~= bar{AD}$ $4. ""$ $5. Delta ABD ~= Delta DCA$ $5. ""$ $6. ang BAD ~= ang CDA$ $6.$ $7. bar{AB} " || " bar{CD}$ $7. ""$ $8. m ang BAD + m ang CDA = 180°$ $8. ""$ $9. m ang BAD = m ang CDA$ $9. ""$ $10. m ang BAD + m ang BAD = 180°$ $10. ""$ $11. 2m ang BAD = 180°$ $11. ""$ $12. m ang BAD = 90°$ $12. ""$ $13. m ang CDA = 90°$ $13.$ $14. ang BAD and ang CDA " are right angles"$ $14.$ $15. bar{BC} " || " bar{AD}$ $15. ""$ $16. m ang BAD + m ang ABC = 180°,$ $\ \ \ \ \ m ang ADC + m ang DCB = 180°$ $16. ""$ $17. 90° + m ang ABC = 180°,$ $\ \ \ \ \ 90° + m ang DCB = 180°$ $17. ""$ $18. m ang ABC = 90°,$ $\ \ \ \ \ m ang DCB = 90°$ $18. ""$ $19. ang ABC, ang DCB " are right angles"$ $19. ""$ $20. ABCD " is a rectangle"$ $20. ""$

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