Question

Complete the proof of the Pythagorean theorem.Given: Δ ABC is a right triangle, witha right angle at ∠CProve: A²+B² =C²Answer:Statement1.ΔABC is a right triangle, with a right angle at ∠C2.Draw an altitude from point C to line AB3.∠CDB and ∠CDA are right angles.4. ∠BCA ≅ ∠BDC5.∠B ≅ ∠B6. ?7.
`(a)/(x) = (c)/(a)`8. a² = cx9.∠CDA ≅ ∠BCA10.∠A ≅ ∠A11. ?12.
`(b)/(y) = (c)/(b)`13. b² = cy14. a² + b² = cx + cy15. ?16. x + y = c17.a² + b² = c²Reason1. Given2. From a point not on a line, exactly one perpendicular can be drawn through the point to the line.3. Definition of altitude4. All right angles are congruent.5. ?6. AA Similarity Postulate7. ?8. ?9. ?10. ?11.AA similarity Postulate12. ?13. ?14. ?15. Distributive Property16. ?17. ?PLEASE FILL IN ALL THE QUESTION MARKS :)

Answer 1

In order to fill each missing statement and reason, let's check for the corresponding part of the missing statement or reason.

For example, reason 5 is missing. The statement 5 is reflexive property of congruency.

In the same way, statement 6 is missing, and the reason is AA Similarity Postulate.

Therefore the statement is Triangles CDB and CDA are similar.

Reason 7 comes from the proportion written in statement 7. So the reason is the proportion between corresponding sides in similar triangles is the same.

(or we can write "corresponding sides of similar triangles are proportional)

Reason 8 comes from the cross product in a fraction equation.

Reason 9 is that all right angles are congruent.

Reason 10 is the reflexive property of congruency.

Statement 11 is that triangles BCA and CDA are similar.

Reason 12 is that the proportion between corresponding sides in similar triangles are the same.

Reason 13 is the cross product in a fraction equation.

Reason 14 is the addition of the equations from statements 8 and 13.

Statement 15 is a² + b² = c(x + y)

Reason 16 is point D is on segment AB (therefore AD + DB = AB)

Reason 17 is the substitution from the equation of statement 16 in the equation of statement 15.