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In Delta DEF and Delta PQR, it is give...

In `Delta DEF and Delta PQR`, it is given that `angle D=angle Q and angle R= angleE`, then which of the following is not true?

A

`(EF)/(PR)=(DF)/(PQ)`

B

`(DE)/(PQ)=(EF)/(RP)`

C

`(DE)/(QR)=(DF)/(PQ)`

D

`(EF)/(RP)=(DE)/(QR)`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the given information about triangles DEF and PQR. ### Step-by-Step Solution: 1. **Identify the Given Angles:** We are given that: - \( \angle D = \angle Q \) - \( \angle R = \angle E \) 2. **Apply the Angle-Angle (AA) Similarity Criterion:** Since two angles of triangle DEF are equal to two angles of triangle PQR, we can conclude that: - Triangle DEF is similar to triangle PQR by the AA similarity criterion. 3. **List the Corresponding Angles:** From the similarity, we can state: - \( \angle D = \angle Q \) - \( \angle E = \angle R \) - Let’s denote the third angle in triangle DEF as \( \angle F \) and the third angle in triangle PQR as \( \angle P \). Therefore, \( \angle F = \angle P \) (since the sum of angles in a triangle is 180 degrees). 4. **Write the Proportionality of Sides:** Since the triangles are similar, the corresponding sides are proportional: \[ \frac{DE}{QP} = \frac{EF}{PR} = \frac{DF}{QR} \] 5. **Evaluate the Options:** We need to determine which statement is not true based on the proportionality of the sides. The options typically involve ratios of the sides of the triangles. 6. **Identify the Incorrect Statement:** Given the proportionality, if an option states something like \( \frac{DE}{PQ} \), it would not be true because \( PQ \) is not a corresponding side to \( DE \) in the similarity relation. The correct corresponding side to \( DE \) is \( QP \). 7. **Conclusion:** Therefore, the option that states \( \frac{DE}{PQ} \) is not true, and we can conclude that option B is the incorrect statement. ### Final Answer: The option that is not true is option B.

To solve the problem, we need to analyze the given information about triangles DEF and PQR. ### Step-by-Step Solution: 1. **Identify the Given Angles:** We are given that: - \( \angle D = \angle Q \) - \( \angle R = \angle E \) ...
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