in the given figure,PQ=PS and QR=SR.Prove that triangle PQR is congruent to triangle PSR.
Sorry - no figures here. Describe the triangles, if possible.
If they're right triangles, it's possible that SAS will justify the assertion.
Describe the triangles. If PR is a common side to each triangle, then SSS congruency will work.
To prove that triangle PQR is congruent to triangle PSR, we can use the Side-Side-Side (SSS) congruence criterion.
Given: PQ = PS and QR = SR.
To prove: Triangle PQR ≅ Triangle PSR.
Proof:
1. Draw segments PR and SR.
2. Since PQ = PS and QR = SR, we have two pairs of congruent sides.
3. By the definition of congruent segments, we can say that segment PQ ≅ segment PS and segment QR ≅ segment SR.
4. Draw segment PS.
5. Based on steps 2 and 4, we have segment PQ ≅ segment PS, segment QR ≅ segment SR, and segment PR = segment PR. These satisfy the Side-Side-Side (SSS) congruence criterion.
6. Therefore, by the SSS congruence criterion, we can conclude that triangle PQR ≅ triangle PSR.
Hence, triangle PQR is congruent to triangle PSR.
To prove that triangle PQR is congruent to triangle PSR, we need to show that their corresponding sides and angles are equal.
Given: PQ = PS and QR = SR
To start the proof, we can analyze the given information and find the additional information we can deduce from it.
1. Using the given information, we can conclude that the two sides PQ and PS are equal. This establishes the first pair of corresponding sides that are equal.
- Therefore, we have PQ = PS (given)
2. Similarly, according to the given information, the two sides QR and SR are equal. This gives us the second pair of corresponding sides that are equal.
- Hence, QR = SR (given)
Now, we need to prove that the corresponding angles are also equal.
3. In triangle PQR, since PQ = PS (given) and QR = SR (given), we have two sides of triangle PQR equal to two sides of triangle PSR.
- By the side-side-side (SSS) congruence criterion, we can conclude that triangle PQR is congruent to triangle PSR.
- This congruence criterion states that if the three sides of one triangle are equal to the corresponding three sides of another triangle, the two triangles are congruent.
Therefore, we have proved that triangle PQR is congruent to triangle PSR based on the given information and the side-side-side (SSS) congruence criterion.