Question

Complete the proof of the Pythagorean theorem.

Answer 1

To Determine:

So, here is the complete proof of the Pythagorean theorem.

Given: Δ ABC is a right triangle, with

a right angle at ∠C

Prove: A²+B² =C²

Answer:

Note: All the answers for the questions marks are filled with in bold text.

Statement

1. ΔABC is a right triangle, with a right angle at ∠C

2. Draw an altitude from point C to line AB

3. ∠CDB and ∠CDA are right angles.

4. ∠BCA ≅ ∠BDC

5. ∠B ≅ ∠B

6. AA Similarity Postulate          

7.   $\frac{a}{x}\:=\:\frac{c}{a}$

8. a² = cx

9. ∠CDA ≅ ∠BCA

10. ∠A ≅ ∠A

11. ΔCBA ~ ΔDBA      

12.  $\frac{b}{y}\:=\:\frac{c}{b}$

13. b² = cy

14. a² + b² = cx + cy

15. $\left(CB\right)^2+\left(CA\right)^2=\left(AB\right)\left(DB+BA\right)$

16. x + y = c

17. a² + b² = c²

Reason

1. Given

2. From a point not on a line, exactly one perpendicular can be drawn through the point to the line.

3. Definition of altitude

4. All right angles are congruent.

5. Reflexive Property

6. AA Similarity Postulate

7. Polygon Similarity Postulate

8. Cross Multiply and Simplify

9. All right angles are Congruent

10. Reflexive Property

11. AA similarity Postulate

12. Polygon Similarity Postulate

13. Cross Multiply and Simplify

14. Addition Property of Equality

15. Distributive Property

16. Segment Addition Postulate

17. Substitution Property

Keywords: statement, reason

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