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To work with the two larger triangles △ PUX and △ QSY, we might have to first consider the two pairs of smaller triangles that can be identified in the diagram.
See solution.
To prove that △ PUX ≅ △ QSY, we start by separating the two triangles.
Having separated the triangles, we notice that △ PRU and △ QVS have two pairs of congruent corresponding angles, ∠ PRU≅ ∠ QVS and ∠ PUR≅ ∠ QSV If we can show that the triangles also have a congruent corresponding side, we have enough information to prove congruence. Let's look at the triangles when they are not separated.
&RU=RS+SU &VS=VU+SU Since SU is shared between the triangles, the Reflexive Property of Congruence tells us that this side is congruent. Also, we know that RS≅ VU which means we can use the Substitution Property of Equality to rewrite the first equation: &RU= VU+SU &VS=VU+SU Finally, by the Transitive Property of Equality, we are able to show that RU≅ VU: RU=VS ⇔ RU≅ VS Now we have enough information to prove that △ PRU ≅ △ QVS by the ASA Congruence Theorem. Let's separate the triangles once more.
From the diagram, we see that ∠ URP and ∠ URX form a linear pair and so do ∠ SVQ and ∠ SVY. By the Linear Pair Postulate we know these two pairs of angles are supplementary. Since ∠ SVQ≅ ∠ URP, we can by the Congruence Supplements Theorem prove the following. ∠ URX≅ ∠ SVY Let's add this information to the diagram.
Now we have enough information to prove that △ URX ≅ △ SVY by the ASA Congruence Theorem. Finally, we will mark all congruent parts in the two triangles.
From the diagram above, we have enough information to prove congruence between △ PUX and △ QSY. Let's remove all unnecessary parts and only focus on the information we need.
Now we can prove congruence by the AAS Congruence Theorem.
Statement
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Reason
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1. &∠ URP ≅ ∠ SVQ & ∠ PUR ≅ ∠ XUR ≅ ∠ QSV ≅ ∠ YSV & RS≅ VU
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1. Given
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2. SU≅ SU
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2. Reflexive Property of Congruence
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3. RS = VU, SU=SU
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3. Definition of congruent segments
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4. &RU=RS+SU & VS=VU+SU
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4. Segment Addition Postulate
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5. &RU=RS+SU & VS=RS+SU
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5. Substitution Property of Equality
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6. RU=SV
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6. Transitive Property of Equality
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7. RU≅ SV
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7. Definition of congruent segments
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8. △ PRU ≅ QVS
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8. ASA Congruence Theorem
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9. &∠ URP and ∠ URX form a linear pair &∠ SVQ and ∠ SVY form a linear pair
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9. Definition of a linear pair as seen in the diagram
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10. &∠ URP and ∠ URX are supplementary &∠ SVQ and ∠ SVY are supplementary
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10. Linear Pair Postulate
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11. ∠ URX ≅ ∠ SVY
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11. Congruence Supplements Theorem
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12. △ URX ≅ △ SVY
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12. ASA Congruence Theorem
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13. &PU≅ SQ &∠ X ≅ ∠ Y, ∠ P ≅ ∠ Q
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13. Corresponding parts of congruent triangles are congruent
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14. △ PUX ≅ △ QSY
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14. AAS Congruence Theorem
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