Proofs Involving Triangles, #2 (Grade 10)

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Proofs Involving Triangles, #2

1. 
In the questions below, choose the correct missing statements and reasons from the following proof.

Let line [math]l[/math] (not pictured) be parallel to side [math]bar{AC}[/math] in [math]Delta ABC[/math]. Line [math]l[/math] intersects [math]bar{AB}[/math] at point [math]D[/math] (not pictured) and [math]bar{BC}[/math] at point [math]E[/math] (not pictured). Prove that [math](AD)/(BD) = (CE)/(BE)[/math].

Equilateral Triangle ABC v1

[math] \ \ \ \ \ \ \ \ \ \ \ \ \mathbf{"Statement"} \ \ \ \ \ \ \ \ \ \ \ \ [/math][math]
\mathbf{"Reason"}[/math]
[math]1. l " || " bar{AC} [/math][math]1. ""[/math]
[math]2. ang BDE ~= ang A[/math][math]2. "If two || lines cut by a transversal, corr. angles congruent"[/math]
[math]3. ang B ~= ang B[/math][math]3. ""[/math]
[math]4. [/math][math]4. "AA Similarity Theorem"[/math]
[math]5. [/math][math]5. "Ratios corr. sides of similar triangles are equal"[/math]
[math]6. BD + AD = AB[/math][math]6. "Segment Addition Postulate"[/math]
[math]7. BE + CE = BC [/math][math]7. "Segment Addition Postulate"[/math]
[math]8. (BD + AD)/(BD) = (BE + CE)/(BE)[/math][math]8. "" [/math]
[math]9. (BD)/(BD) + (AD)/(BD) = (BE)/(BE) + (CE)/(BE) [/math][math]9. "Algebra (addition of fractions)"[/math]
[math]10. 1 + (AD)/(BD) = 1 + (CE)/(BE) [/math][math]10. "Any non-zero quantity divided by itself equals 1"[/math]
[math]11. (AD)/(BD) = (CE)/(BE) [/math][math]11. "Subtraction Property of Equality"[/math]
A. 
What is the missing reason in step 1?
  1. Parallel Postulate
  2. Given
  3. If two lines are cut by a transversal and alternate interior angles are supplementary, the lines are parallel
  4. If two lines are cut by a transversal and corresponding angles are congruent, the lines are parallel
B. 
What is the missing reason in step 3?
  1. Reflexive Property of Congruence
  2. Identity Property of Congruence
  3. Reflection Property of Congruence
  4. Given
C. 
What is the missing statement in step 4?
  1. [math]Delta BED \ ~ \ Delta ABC[/math]
  2. [math]Delta DEB \ ~ \ Delta ABC[/math]
  3. [math]Delta BDE \ ~ \ Delta ABC[/math]
  4. [math]Delta DBE \ ~ \ Delta ABC[/math]
D. 
What is the missing statement in step 5?
  1. [math](AC)/(DE) = (AB)/(AD)[/math]
  2. [math](AB)/(AD) = (BC)/(BE)[/math]
  3. [math](BC)/(CE) = (AC)/(DE)[/math]
  4. [math](AB)/(BD) = (BC)/(BE)[/math]
E. 
What is the missing reason in step 8?
  1. Ratios of corresponding sides of similar triangles are equal
  2. Division Property of Equality
  3. Substitution Property of Equality
  4. Corresponding sides of congruent triangles are congruent
2. 
In the questions below, choose the correct missing statement and reasons from the following proof.

In [math]Delta ABC[/math], let [math]M[/math] (not pictured) be the midpoint of [math]bar{AB}[/math] and [math]N[/math] (not pictured) be the midpoint of [math]bar{AC}[/math]. Prove that [math]bar{MN} " || " bar{BC}[/math].

Equilateral Triangle ABC v2

[math] \ \ \ \ \ \ \ \ \ \ \ \ \mathbf{"Statement"} \ \ \ \ \ \ \ \ \ \ \ \ [/math][math]
\mathbf{"Reason"}[/math]
[math]1. ang A ~= ang A [/math][math]1. "Reflexive Property of Congruence"[/math]
[math]2. M " is the midpoint of " bar{AB} \ \ \ \ \ \ \ [/math][math]2. "Given"[/math]
[math]3. AM = BM[/math][math]3. ""[/math]
[math]4. AM + BM = AB[/math][math]4. "Segment Addition Postulate"[/math]
[math]5. AM + AM = AB[/math][math]5. "Substitution Property of Equality"[/math]
[math]6. 2 AM = AB[/math][math]6. "Algebra (addition)"[/math]
[math]7. (AM)/(AB) = 1/2 [/math][math]7. "Division Property of Equality"[/math]
[math]8. N " is the midpoint of " bar{AC}[/math][math]8. "Given" [/math]
[math]9. AN = CN [/math][math]9. ""[/math]
[math]10.AN + CN = AC [/math][math]10. "Segment Addition Postulate"[/math]
[math]11. AN + AN = AC [/math][math]11. "Substitution Property of Equality"[/math]
[math]12. 2 AN = AC [/math][math]12. "Algebra (addition)"[/math]
[math]13. (AN)/(AC) = 1/2 [/math][math]13. "Division Property of Equality"[/math]
[math]14. (AN)/(AC) = (AM)/(AB) [/math][math]14. ""[/math]
[math]15. [/math][math]15. "SAS Similarity Theorem"[/math]
[math]16. ang B ~= ang AMN [/math][math]16. "Coor. angles in similar triangles congruent"[/math]
[math]17. bar{BC} " || " bar{MN} [/math][math]17. ""[/math]
A. 
What is the missing reason in step 3?
  1. Definition of a midpoint
  2. Given
  3. Segment Equality Postulate
  4. Substitution Property of Equality
B. 
What is the missing reason in step 9?
  1. Definition of a midpoint
  2. Given
  3. Segment Addition Postulate
  4. Substitution Property of Equality
C. 
What is the missing reason in step 14?
  1. Ratios of corresponding sides in similar triangles are equal
  2. Division Property of Equality
  3. Segment Division Postulate
  4. Transitive Property of Equality
D. 
What is the missing statement in step 15?
  1. [math]Delta ABC \ ~ \ Delta ANM[/math]
  2. [math]Delta ABC \ ~ \ Delta AMN[/math]
  3. [math]Delta ABC \ ~ \ Delta MAN[/math]
  4. [math]Delta ABC \ ~ \ Delta MNA[/math]
E. 
What is the missing reason in step 17?
  1. If two lines are cut by a transversal, and vertical angles are congruent, the two lines are parallel
  2. If two lines are cut by a transversal, and alternate exterior angles are congruent, the two lines are parallel
  3. If two lines are cut by a transversal, and corresponding angles are congruent, the two lines are parallel
  4. If two lines are cut by a transversal, and alternate interior angles are congruent, the two lines are parallel

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