The following two-column proof with missing statements and reasons proves that if a line parallel to one side of a triangle also intersects the other two sides, the line divides the sides proportionally:
Statement Reason
1. Line segment DE is parallel to line segment AC 1. Given
2. Line segment AB is a transversal that intersects two parallel lines. 2. Conclusion from Statement 1.
3. 3.
4. ∠B ≅ ∠B 4. Reflexive Property of Equality
5. 5.
6. BD over BA equals BE over BC 6. Converse of the Side-Side-Side Similarity Theorem
Which statement and reason accurately completes the proof?
3. ∠BDE ≅ ∠BAC; Corresponding Angles Postulate
5. ΔBDE ~ ΔBAC; Angle-Angle (AA) Similarity Postulate
3. ΔBDE ~ ΔBAC; Corresponding Angles Postulate
5. ∠BDE ~ ∠BAC; Angle-Angle (AA) Similarity Postulate
3. ∠BDE ≅ ∠BAC; Congruent Angles Postulate
5. ΔBDE ~ ΔBAC; Angle-Angle (AA) Similarity Postulate
3. ∠BDE ≅ ∠BAC; Congruent Angles Postulate
5. ΔBDE ~ ΔBAC; Side-Angle-Side (SAS) Similarity Postulate
I’m still confused even though I read all this.
To complete the two-column proof, we need to provide statements and reasons that demonstrate the similarity between triangles ΔBDE and ΔBAC.
Since line segment DE is parallel to line segment AC, and AB is a transversal that intersects two parallel lines, we can conclude that ∠BDE and ∠BAC are corresponding angles. Therefore, we can use the Corresponding Angles Postulate to state that ∠BDE ≅ ∠BAC as our missing statement (Statement 3) with the reason as the Corresponding Angles Postulate.
Now, in order to show the similarity between ΔBDE and ΔBAC, we need to show the presence of another pair of congruent angles. We can use the fact that ∠B ≅ ∠B (Reflexive Property of Equality, Statement 4) as our missing statement (Statement 5) with the reason as the Reflexive Property of Equality.
Therefore, the correct completion of the proof should be:
3. ∠BDE ≅ ∠BAC; Corresponding Angles Postulate
5. ΔBDE ~ ΔBAC; Angle-Angle (AA) Similarity Postulate
For the statement/reason, the ones that have the same number are the statement and its reason.
Statement - 2. Line segment AB is a transversal that intersects two parallel lines. Reason - 2. Conclusion from Statement 1.