Given: S is the midpoint of QR, QR ⊥PS and RSP and QSP are right angles
Prove: PR is congruent to PQ
I just need help starting it. Would I chop it up? For example
S is the midpoint of QR / Given
QS = SR / definition of mid point
PS = PS/ reflexive property
Then put in the perpendicular or right angles?
the two right angles are congruent
PS ≅ PS
QS ≅ PS
Now use SAS
Yes, you can start the proof by breaking it down and stating the given information. Here's a step-by-step breakdown of the proof:
1. Given: S is the midpoint of QR
This means QS = SR. (Definition of a midpoint)
2. Given: QR ⊥ PS
This means that QR is perpendicular to PS.
3. Given: RSP and QSP are right angles
This means that angle RSP and angle QSP are right angles.
Now, you can proceed with the proof by using these given statements as well as some additional logical steps.
4. QS = SR (Given)
PS = PS (Reflexive Property)
5. QS = SR, QR ⊥ PS, RSP and QSP are right angles
Triangle QSR is a right triangle. (Definition of a right triangle)
6. QS = SR, QR ⊥ PS, RSP and QSP are right angles
Triangle PQS is a right triangle. (Definition of a right triangle)
7. Triangle QSR is congruent to triangle PQS.
Angle-Side-Angle (ASA) congruence theorem
8. PR = PQ (Corresponding parts of congruent triangles are congruent)
Therefore, PR is congruent to PQ. (QED)
Yes, that's a good approach. Here's how you can start the proof:
Statements:
1. S is the midpoint of QR. (Given)
2. QR ⊥ PS. (Given)
3. ∠RSP = 90°. (Given)
4. ∠QSP = 90°. (Given)
Reasons:
1. Definition of midpoint.
2. Given.
3. Given.
4. Given.
Now, to prove that PR is congruent to PQ, you can use the right angles and the fact that S is the midpoint of QR.
To continue the proof, you can use the fact that when two lines are perpendicular, the opposite sides of the intersection are congruent. So, we can say that PR is congruent to SR and PQ is congruent to QS.
Here's how to write it in your proof:
5. ∠RSP and ∠QSP are right angles. (Given)
6. QR ⊥ PS implies PR ≅ SR and PQ ≅ QS. (Perpendicular lines form congruent opposite segments)
Now, you have established that PR is congruent to SR and PQ is congruent to QS. Since S is the midpoint of QR (given), it follows that QR is divided into two congruent segments, PR and SQ (Definition of Midpoint).
7. S is the midpoint of QR implies PR ≅ SQ. (Definition of Midpoint)
Finally, using the transitive property of congruence, we can conclude that PR is congruent to PQ.
8. PR ≅ SR, PQ ≅ QS, and PR ≅ SQ implies PR ≅ PQ. (Transitive Property of Congruence)
Now you have completed the proof, showing that PR is congruent to PQ based on the given statements and using the properties and definitions of congruent segments.