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This printable supports Common Core Mathematics Standard HSG-SRT.B.5

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# Proofs Involving Geometric Figures (Grade 10)

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## Proofs Involving Geometric Figures

1.
For the following proof, choose the correct missing statements and reasons in the questions below.

Given that $Delta AFB ~= Delta EDC$, $bar{BC} " || " bar{FD}$, $m ang BFE = 90°$, and points $A,F,E$ are collinear, prove that quadrilateral $BCDF$ is a rectangle.

 $\ \ \ \ \ \ \ \ \ \ \ \ \mathbf{"Statement"} \ \ \ \ \ \ \ \ \ \ \ \$ $\mathbf{"Reason"}$ $1. m ang BFE = 90°$ $1. "Given"$ $2. A,F,E " are collinear"$ $2. "Given"$ $3. m ang AFB + m ang BFE = 180°$ $3. ""$ $4.$ $4. "Substitution Property of Equality"$ $5. m ang AFB = 180° - 90°$ $5. "Subtraction Property of Equality"$ $6. m ang AFB = 90°$ $6. "Algebra (subtract)"$ $7. Delta AFB ~= Delta EDC$ $7. "Given"$ $8.$ $8. "Corr. angles of congruent triangles congruent"$ $9. m ang AFB = m ang EDC$ $9. "Definition congruent angles"$ $10. 90° = m ang EDC$ $10. "Substitution Property of Equality"$ $11. bar{BC} " || " bar{DF}$ $11. "Given"$ $12. m ang EDC + m ang BCD = 180°$ $12. ""$ $13. 90° + m ang BCD = 180°$ $13. "Substitution Property of Equality"$ $14. m ang BCD = 180° - 90°$ $14. "Subtraction Property of Equality"$ $15. m ang BCD = 90°$ $15. "Algebra (subtract)"$ $16. m ang BFE + m ang EDC + m ang BCD +$ $\ \ \ \ m ang CBF = 360°$ $16. ""$ $17. 90° + 90° + 90° + m ang CBF = 360°$ $17. "Substitution Property of Equality"$ $18. 270° + m ang CFB = 360°$ $18. "Algebra (add)"$ $19. m ang CFB = 360° - 270°$ $19. "Subtraction Property of Equality"$ $20. m ang CFB = 90°$ $20. "Algebra (subtract)"$ $21. ang BFE, \ ang EDC, \ ang BCD, \ ang CBF$ $\ \ \ " are right angles"$ $21. "Definition of right angles"$ $22. "Quad. " BCDF " is a rectangle"$ $22. "Quad. with 4 right angles is a rectangle"$
A.
What is the missing reason in step 3?
1. The sum of the angles on a straight line is 180°
2. $ang ABF$ and $ang BFE$ are complimentary
3. Right angles are congruent
4. $ang AFB ~= ang BFE$
B.
What is the missing statement in step 4?
1. $m ang AFB + m ang BFE = m ang AFE$
2. $90° + 90° = 180°$
3. $m ang AFB + 90° = 180°$
4. $90° + m ang BFE = 180°$
C.
What is the missing statement in step 8?
1. $ang AFB ~= ang BFE$
2. $ang AFB ~= ang EDC$
3. $ang BAF ~= ang EDC$
4. $ang BAF ~= ang ECD$
D.
What is the missing reason in step 12?
1. If 2 parallel lines are cut by a transversal, alternate angles are supplementary
2. If 2 parallel lines are cut by a transversal, consecutive exterior angles are supplementary
3. If 2 parallel lines are cut by a transversal, alternate interior angles are congruent
4. If 2 parallel lines are cut by a transversal, consecutive interior angles are supplementary
E.
What is the missing reason in step 16?
1. The sum of four non-obtuse angles is 360°
2. The sum of the interior angles of any two dimensional figure is 360°
3. Four angles make a full rotation, which is 360°
4. The sum of the interior angles in a quadrilateral is 360°
2.
For the following proof, choose the correct missing statement and reasons in the questions below.

In the following figure, $bar{AB} " || " bar{CE}$, $ang BFE$ and $ang CDE$ are supplementary, and points $A,F,E$ are collinear. Also, $ang BCD ~= ang CDE$ and $bar{AF} ~= bar{ED}$. If $BC = 2CD$ and $CD = 2 ED$, prove that $BD*BF = AB*BC$ ($bar{BD}$ is not drawn).

 $\ \ \ \ \ \ \ \ \ \ \ \ \mathbf{"Statement"} \ \ \ \ \ \ \ \ \ \ \ \$ $\mathbf{"Reason"}$ $1. ang BCD ~= ang CDE$ $1. "Given"$ $2. BC = 2CD$ $2. "Given"$ $3. (BC)/(CD) = 2$ $3. "Division Property of Equality"$ $4. CD = 2ED$ $4. "Given"$ $5. (CD)/(ED) = 2$ $5. "Division Property of Equality"$ $6. (BC)/(CD) = (CD)/(ED)$ $6. "Transitive Property of Equality"$ $7.$ $7. "SAS Similarity Theorem"$ $8. (BD)/(CE) = (BC)/(CD)$ $8. "Ratios corr. sides of similar triangles are equal"$ $9. bar{AF} ~= bar{ED}$ $9. "Given"$ $10. bar{AB} " || " bar{CE}$ $10. "Given"$ $11. ang BAF ~= ang CED$ $11. ""$ $12. ang BFE " and " ang CDE " are supplementary"$ $12. "Given"$ $13. m ang BFE + m ang CDE = 180°$ $13. ""$ $14. m ang BFE = 180° - m ang CDE$ $14. "Subtraction Property of Equality"$ $15. A, F, E " are collinear"$ $15. "Given"$ $16. m ang AFB + m ang BFE = 180°$ $16. ""$ $17. m ang BFE = 180° - m ang AFB$ $17. "Subtraction Property of Equality"$ $18. 180° - m ang CDE = 180° - m ang ABF$ $18. "Transitive Property of Equality"$ $19. - m ang CDE = - m ang AFB$ $19. "Subtraction Property of Equality"$ $20. m ang CDE = m ang AFB$ $20. "Multiplication Property of Equality"$ $21. ang CDE ~= ang AFB$ $21. "Definition of Congruent Angles"$ $22. Delta AFB ~= Delta EDC$ $22. ""$ $23. bar{AB} ~= bar{CE}$ $23. "Corr. sides of congruent triangles congruent"$ $24. AB = CE$ $24. "Definition of Congruent Segments"$ $25. bar{BF} ~= bar{CD}$ $25. "Corr. sides of congruent triangles congruent"$ $26. BF = CD$ $26. "Definition of Congruent Segments"$ $27. (BD)/(CE) = (BC)/(CD)$ $27. "Earlier result"$ $28. (BD)/(AB) = (BC)/(BF)$ $28. "Substitution Property of Equality"$ $29. BD*BF = AB*BC$ $29. "Multiplication Property of Equality"$
A.
What is the missing statement in step 7?
1. $Delta BCD \ ~ \ Delta CDE$
2. $Delta BCD \ ~ \ Delta CED$
3. $Delta BCD \ ~ \ Delta DEC$
4. $Delta BCD \ ~ \ Delta DCE$
B.
What is the missing reason in step 11?
1. If two parallel lines are cut by a transversal, alternate exterior angles are congruent
2. If two parallel lines are cut by a transversal, alternate interior angles are congruent
3. If two parallel lines are cut by a transversal, alternate angles are congruent
4. If two parallel lines are cut by a transversal, corresponding angles are congruent
C.
What is the missing reason in step 13?
1. Given
2. Definition of supplementary angles
3. The sum of two right angles is 180°
4. The sum of any two angles of a rectangle is 180°
D.
What is the missing reason in step 16?
1. The sum of the exterior angle and its corresponding interior angle is 180°
2. The sum of adjacent angles which are also equal is 180°
3. The sum of the angles on a straight line is 180°
4. The sum of two right angles is 180°
E.
What is the missing reason in step 22?
1. SAS Congruence Theorem
2. AAS Congruence Theorem
3. HL Congruence Theorem
4. ASA Congruence Theorem
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