Question

given: ( \frac{ad}{dc} = \frac{be}{ec} )
prove: ( overline{ab} parallel overline{de} )
diagram with points a, b, d, e, c; segments labeled 1, 2, 3
complete the steps of the proof.
statements | reasons
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1. ( \frac{ad}{dc} + 1 = \frac{be}{ec} + 1 ) | 2. addition property
2. ( \frac{ad}{dc} + \frac{dc}{dc} = \frac{be}{ec} + \frac{ec}{ec} ) | 3. property of proportion
3. ( \frac{ad + dc}{dc} = \frac{be + ec}{ec} ) | 4. addition of fractions
4. ( ac = ad + dc ); ( bc = be + ec ) | 5. segment addition postulate
5. ( \frac{ac}{dc} = \frac{bc}{ec} ) | 6. substitution property
6. ( angle 3 cong angle 3 ) | 7. reflexive property
7. ( \triangle abc sim \triangle dec ) | 8. dropdown
8. ( angle 1 cong angle 2 ) | 9. dropdown
9. ( overline{ab} parallel overline{de} ) | 10. corresponding angles theorem

dropdown options: definition of congruent angles, definition of corresponding angles, definition of similar triangles

Explanation:

Step-by-Step Explanation for Step 8 (Reason for $\triangle ABC \sim \triangle DEC$):

Step 1: Recall Similar Triangle Criteria

To prove two triangles are similar, we can use the SAS (Side - Angle - Side) Similarity Criterion, which states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar.

Step 2: Identify Proportional Sides and Included Angle

• From step 6, we have $\frac{AC}{DC}=\frac{BC}{EC}$.
• From step 7, we know that $\angle 3\cong\angle 3$ (the included angle between the sides $AC, BC$ of $\triangle ABC$ and $DC, EC$ of $\triangle DEC$).

So, by the SAS Similarity Criterion, $\triangle ABC \sim \triangle DEC$. The reason for step 8 is the SAS (Side - Angle - Side) Similarity Criterion (or we can also say "SAS similarity" or "Side - Angle - Side similarity").

Step-by-Step Explanation for Step 9 (Reason for $\angle 1\cong\angle 2$):

Step 1: Recall Properties of Similar Triangles

When two triangles are similar, their corresponding angles are congruent. This is a fundamental property of similar triangles (corresponding angles of similar triangles are congruent).

Step 2: Apply the Property to $\triangle ABC$ and $\triangle DEC$

Since $\triangle ABC \sim \triangle DEC$ (from step 8), $\angle 1$ (in $\triangle ABC$) and $\angle 2$ (in $\triangle DEC$) are corresponding angles. Therefore, by the property of similar triangles (corresponding angles of similar triangles are congruent), we have $\angle 1\cong\angle 2$.

Answer:

s:

• For step 8 (reason for $\triangle ABC \sim \triangle DEC$): SAS (Side - Angle - Side) Similarity Criterion (or "SAS similarity")
• For step 9 (reason for $\angle 1\cong\angle 2$): Corresponding angles of similar triangles are congruent