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].

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].

[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?
- Parallel Postulate
- Given
- If two lines are cut by a transversal and alternate interior angles are supplementary, the lines are parallel
- 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?
- Reflexive Property of Congruence
- Identity Property of Congruence
- Reflection Property of Congruence
- Given
C.
What is the missing statement in step 4?
- [math]Delta BED \ ~ \ Delta ABC[/math]
- [math]Delta DEB \ ~ \ Delta ABC[/math]
- [math]Delta BDE \ ~ \ Delta ABC[/math]
- [math]Delta DBE \ ~ \ Delta ABC[/math]
D.
What is the missing statement in step 5?
- [math](AC)/(DE) = (AB)/(AD)[/math]
- [math](AB)/(AD) = (BC)/(BE)[/math]
- [math](BC)/(CE) = (AC)/(DE)[/math]
- [math](AB)/(BD) = (BC)/(BE)[/math]
E.
What is the missing reason in step 8?
- Ratios of corresponding sides of similar triangles are equal
- Division Property of Equality
- Substitution Property of Equality
- 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].

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].

[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?
- Definition of a midpoint
- Given
- Segment Equality Postulate
- Substitution Property of Equality
B.
What is the missing reason in step 9?
- Definition of a midpoint
- Given
- Segment Addition Postulate
- Substitution Property of Equality
C.
What is the missing reason in step 14?
- Ratios of corresponding sides in similar triangles are equal
- Division Property of Equality
- Segment Division Postulate
- Transitive Property of Equality
D.
What is the missing statement in step 15?
- [math]Delta ABC \ ~ \ Delta ANM[/math]
- [math]Delta ABC \ ~ \ Delta AMN[/math]
- [math]Delta ABC \ ~ \ Delta MAN[/math]
- [math]Delta ABC \ ~ \ Delta MNA[/math]
E.
What is the missing reason in step 17?
- If two lines are cut by a transversal, and vertical angles are congruent, the two lines are parallel
- If two lines are cut by a transversal, and alternate exterior angles are congruent, the two lines are parallel
- If two lines are cut by a transversal, and corresponding angles are congruent, the two lines are parallel
- If two lines are cut by a transversal, and alternate interior angles are congruent, the two lines are parallel
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