Let's solve the problem step by step and fill in the
Statements and
Reasons table to prove that ∠PQS and ∠SQR are complementary, given that ∠PQR is a right angle.
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Given:
- ∠PQR is a right angle
To Prove:
- ∠PQS and ∠SQR are complementary (i.e., their measures add up to 90°)
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We'll now complete the two-column proof:
|
Statements |
Reasons |
|----------------|-------------|
| 1. ∠PQR is a right angle | 1. Given |
| 2. m∠PQR = 90° | 2. Definition of a right angle |
| 3. m∠PQS + m∠SQR = m∠PQR | 3. Angle Addition Postulate (since point S lies on the interior of ∠PQR) |
| 4. m∠PQS + m∠SQR = 90° | 4. Substitution Property of Equality (from statements 2 and 3) |
| 5. ∠PQS and ∠SQR are complementary | 5. Definition of complementary angles (two angles whose measures sum to 90°) |
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✔ Explanation:
-
Step 1: We start with what’s given — that ∠PQR is a right angle.
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Step 2: By definition, a right angle measures exactly 90°.
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Step 3: Since point S lies between rays QP and QR (as shown in the diagram), the angle ∠PQR is split into two smaller angles: ∠PQS and ∠SQR. The
Angle Addition Postulate tells us that the sum of these two smaller angles equals the whole angle.
-
Step 4: Substitute the known value of 90° from Step 2 into the equation from Step 3.
-
Step 5: By the
definition of complementary angles, if two angles add up to 90°, they are complementary.
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✔ Final Answer:
✔ Therefore, ∠PQS and ∠SQR are
complementary.
This completes the proof.
Parent Tip: Review the logic above to help your child master the concept of angle proofs worksheet.
