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Note that A, B and Q are all radii of identical arcs.
See solution
Let's use the plans step-by-step to prove the construction is valid.
The two triangles share PQ as a side so we know this side is congruent by the Reflexive Property of Congruence. Additionally, A and B on the horizontal line have been marked by drawing an arc with P as the center. Therefore, PA and PB are congruent since they are the radii of the same arc.
Similarly, Q has been marked by drawing two additional arcs with centers in A an B using the same compass settings. Therefore, AQ and BQ are congruent since they are the radii of identical arcs.
Now we can prove that △ APQ ≅ △ BPQ by the SSS Congruence Theorem.
Knowing that △ APQ ≅ △ BPQ, we also know that ∠ APQ ≅ ∠ BPQ because these are congruent corresponding angles. Also, △ APQ and △ BPQ share PM as a side, which means this side is congruent in our triangles.
Now we can prove that △ APM ≅ △ BPM by the SAS Congruence Theorem
As △ AMP ≅ △ BMP, we know that ∠ AMP ≅ ∠ BMP as these are congruent corresponding angles. Examining the diagram, we also see that ∠ AMP and ∠ BMP forms a linear pair which means they are supplementary angles: m∠ AMP+m∠ BMP=180^(∘) Since these angles are congruent, they are right angles. Thus, the constructions is correct.
Let's show this as a two-column proof as well.
Statement
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Reason
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1. AP≅BP, AQ≅BQ
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1. Given
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2. PQ≅ PQ
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2. Reflexive Property of Congruence
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3. △ APQ ≅ △ BPQ
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3. SSS Congruence Theorem
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4. ∠ APQ ≅ ∠ BPQ
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4. Corresponding parts of congruent triangles are congruent
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5. PM≅ PM
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5. Reflexive Property of Congruence
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6. △ APM ≅ △ BPM
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6. SAS Congruence Theorem
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7. ∠ AMP ≅ ∠ BMP
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7. Corresponding parts of congruent triangles are congruent
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8. ∠ AMP and ∠ BMP form a linear pair
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8. Definition of a linear pair
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9. MP ⊥ AB
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9. Linear Pair Perpendicular Theorem
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10. ∠ AMP and ∠ BMP are right angles
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10. Definition of perpendicular lines
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