Math Problem Statement

Complete the proof.

A triangle has three labeled vertices, two labeled points, and two line segments. The bottom side of the triangle is horizontal, starts at vertex P at left and ends at vertex Q at right. Vertex S is above and right of P and above and left of Q. The left side starts at vertex S and ends at vertex P. Between S and P lies point T. The right side starts at vertex S and ends at vertex Q. Between S and Q lies point V. The first line segment starts at vertex Q, goes up and to the left, passes through a second line segment at an unlabeled point, and ends at point T. The second line segment starts at vertex P, goes up and to the right, passes through the first line segment at an unlabeled point, and ends at point V. Given: SP ≅ SQ and ST ≅ SV Prove: △SPV ≅ △SQT and △TPQ ≅ △VQP Statements Reasons 1. SP ≅ SQ; ST ≅ SV 1. Given 2. ---Select--- 2. ---Select--- 3. ---Select--- 3. ---Select--- 4. PV ≅ QT 4. ---Select--- 5. PQ ≅ PQ 5. ---Select--- 6. ---Select--- 6. If segments are ≅, they are = in measure. 7. ---Select--- 7. ---Select--- 8. ---Select--- 8. Substitution 9. ---Select--- 9. Substitution 10. ---Select--- 10. ---Select--- 11. PT ≅ QV 11. ---Select--- 12. ---Select--- 12. ---Select---

Solution

Let's go step by step and complete the proof logically based on the given information. Here's how we can structure the proof:

Given:

  • SPSQSP \cong SQ (The two sides of the triangle are congruent)
  • STSVST \cong SV (Two line segments on each side are congruent)

Proof:

StatementsReasons
1. SPSQSP \cong SQ; STSVST \cong SV1. Given
2. PQ\angle P \cong \angle Q2. Base angles of an isosceles triangle are congruent (△SPQ is isosceles)
3. STVSVT\angle STV \cong \angle SVT3. Vertical angles are congruent
4. PVQTPV \cong QT4. Congruent sides from isosceles triangles △SPV and △SQT
5. PQPQPQ \cong PQ5. Reflexive property of equality
6. SPVSQT△SPV \cong △SQT6. Side-Angle-Side (SAS) Postulate (from steps 1, 2, 4)
7. TPVQTP \cong VQ7. Corresponding parts of congruent triangles (CPCTC from step 6)
8. PQPQPQ \cong PQ8. Reflexive property
9. TPQVQP△TPQ \cong △VQP9. Side-Angle-Side (SAS) Postulate (from steps 4, 5, 7)
10. PTQVPT \cong QV10. Corresponding parts of congruent triangles (CPCTC from step 9)
11. Triangles are congruent11. Conclusion based on above statements

Conclusion:

  • By using the given information and the properties of isosceles triangles, we have proven that SPVSQT△SPV \cong △SQT and TPQVQP△TPQ \cong △VQP using the Side-Angle-Side (SAS) postulate and the property of corresponding parts of congruent triangles.

Would you like more details on a specific part of the proof or clarification on any steps?

Here are some follow-up questions to deepen understanding:

  1. What is the significance of vertical angles in this proof?
  2. How does the SAS Postulate help in proving triangle congruence?
  3. What role does the reflexive property play in triangle congruence proofs?
  4. Why is it important that SPSQSP \cong SQ in this proof?
  5. How do corresponding parts of congruent triangles (CPCTC) help to complete this proof?

Tip: When proving congruence, always check if you can apply the reflexive property to shared sides or angles. This is often a key step!

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Math Problem Analysis

Mathematical Concepts

Triangle Congruence
Isosceles Triangles
Geometric Proofs

Formulas

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Theorems

Side-Angle-Side (SAS) Postulate
Corresponding Parts of Congruent Triangles (CPCTC)
Vertical Angles Theorem

Suitable Grade Level

Grades 9-12