Exercise 3 Page 635

How should we prove congruence?

We have not been given any information about the angles of PQUT. Additionally, we do not know if any sides are parallel which would allow us to relate, for example, alternate interior angles. Therefore, we can assume that congruence has to be proved by the SSS Congruence Theorem.

Plan for proving congruence by SSS

Our plan includes 6 general steps.

Show that PU and PU as well as RS and RS are congruent by the Reflexive Property of Congruence.
Show that ∠ URT and ∠ PSQ are right angles by the Right Angles Congruence Theorem.
Show that PS and RU are congruent by the Segment Addition Postulate.
Show that △ TRU and △ QSP are congruent by the HL Congruence Theorem.
Show that △ PRT and △ USP are congruent by the SAS Congruence Theorem.
Show that △ PTU and △ UQP are congruent by the SSS Congruence Theorem.

Two-column proof

Finally, we will prove △ PTU ≅ △ UQP using a two-column proof.

Statement	Reason
1. &PQ≅ TU, PR≅ SU & PU⊥ RT, PU⊥ QS & ∠ PRT and ∠ USQ are right angles	1. Given
2. PU≅ PU, RS≅ RS	2. Reflexive Property of Congruence
3. ∠ URT and ∠ PSQ are right angles	3. Definition of perpendicular lines
4. ∠ URT≅ ∠ PSQ ≅ ∠ USQ ≅ ∠ PRT	4. Right Angles Congruence Theorem
5. & PS=PR+RS & UR=US+RS	5. Segment Addition Postulate
6. & PS=US+RS & UR=US+RS	6. Substitution Property of Equality
7. PS=UR	7. Transitive Property of Equality
8. PS≅ UR	8. Definition of congruent segments
9. △ TRU ≅ △ QSP	9. HL Congruence Theorem
10. TR≅ QS	10. Corresponding parts of congruent triangles are congruent
11. △ PRT≅ △ USP	11. SAS Congruence Theorem
12. PT≅ UQ	12. Corresponding parts of congruent triangles are congruent
13. △ PTU≅ △ UQP	13. SSS Congruence Theorem

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How should we prove congruence?

Plan for proving congruence by SSS

Two-column proof