QQuestionMathematics
QuestionMathematics
Pedro is going to use SAS to prove that $\triangle P Q R \bullet \triangle S Q R$.
Which of these is a necessary step in Pedro's proof?
A. Prove that $\angle Q P R \geqslant \angle Q S R$ by the Isosceles Triangle Theorem.
B. Prove that $P Q \geqslant S Q$ by CPCTC.
C. Prove that $\angle P Q R \geqslant \angle S Q R$ by vertical angles.
D. Prove that $Q R \geqslant Q R$ by the reflexive property.
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Answer
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Step 1:Let's solve this step by step:
Step 2:: Understand the SAS (Side-Angle-Side) Congruence Theorem
- SAS states that if two triangles have:
Step 3:
Two corresponding sides equal in length
Step 4:
The included angle between those sides is equal - Then the triangles are congruent
Step 5:: Analyze the given diagram
- We see $\triangle PQR$ and $\triangle SQR$
- We want to prove these triangles are congruent using SAS
Step 6:: Examine the given options
Shows $QR = QR$ by the reflexive property
A. Proves an inequality about angles (not helpful for congruence) B. Uses CPCTC (Corresponding Parts of Congruent Triangles are Congruent) after proving congruence C. Another angle inequality (not directly useful for SAS) D.
Step 7:: Recognize the Reflexive Property
- $$QR = QR$$ is always true for any line segment
- The reflexive property states that any quantity is equal to itself
Step 8:: Conclusion
- The reflexive property of $QR = QR$ is a necessary step in proving triangle congruence via SAS
Final Answer
Prove that $QR \geqslant QR$ by the reflexive property.
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