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If in ∆DEF and ∆PQR, ∠D ≅ ∠Q, ∠R ≅ ∠E then which of the following statements is false?

Problem Set 1 | Q 1.2 | Page 26
If in ∆DEF and ∆PQR, ∠D ≅ ∠Q, ∠R ≅ ∠E then which of the following statements is false?

\[\frac{EF}{PR} = \frac{DF}{PQ}\]

\[\frac{DE}{PQ} = \frac{EF}{RP}\]

\[\frac{DE}{QR} = \frac{DF}{PQ}\]

\[\frac{EF}{RP} = \frac{DE}{QR}\]

Solution

In ∆DEF and ∆PQR
∠D ≅ ∠Q
∠R ≅ ∠E
By AA test of similarity
∆DEF~ ∆PQR

\[\therefore \frac{DE}{PQ} = \frac{EF}{QR} = \frac{DF}{PR} \left( \text{ Corresponding sides of similar triangles are proportional } \right)\] \[\therefore \frac{DE}{PQ} \neq \frac{EF}{RP}\]

Hence, the correct option is \[\frac{DE}{PQ} = \frac{EF}{RP}\]

Answer:-

In ∆DEF and ∆PQR, ∠D ≅ ∠Q, ∠R ≅ ∠E, we can say that ∆DEF ~ ∆PQR by AA similarity test.

So, we can use the ratios of corresponding sides to find the false statement.

Let’s compare each statement:

\[\frac{EF}{PR} = \frac{DF}{PQ}\]
This statement is true since it corresponds to the ratio of sides EF and PQ with sides PR and DF respectively.
\[\frac{DE}{PQ} = \frac{EF}{RP}\]
This statement is true as well since it corresponds to the ratio of sides DE and EF with sides PQ and RP respectively.
\[\frac{DE}{QR} = \frac{DF}{PQ}\]
This statement is true also since it corresponds to the ratio of sides DE and DF with sides QR and PQ respectively.
\[\frac{EF}{RP} = \frac{DE}{QR}\]
This statement is false since it corresponds to the ratio of sides EF and DE with sides RP and QR respectively. However, in the similarity we established earlier, the ratio of sides EF and DE is not equivalent to the ratio of sides RP and QR.

Therefore, the false statement is:

\[\frac{EF}{RP} = \frac{DE}{QR}\]

Problem Set 1 | Q 1.2 | Page 26

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