Question

given: △abc with ( a^{2}+b^{2}=c^{2} ) and right △def constructed with legs ( a ) and ( b ) and hypotenuse ( n ) prove: △abc is a right triangle. complete the missing parts of the paragraph proof. proof: we are given ( a^{2}+b^{2}=c^{2} ) for △abc and right △def constructed with legs ( a ) and ( b ) and hypotenuse ( n ). since △def is a right triangle, we know that ( a^{2}+b^{2}=n^{2} ) because of the by substitution, ( c^{2}=n^{2} ) using and the principle root, we can oth sides to get triangles uent. since it is given that is also a right angle by cpctc. therefore, △abc is a right triangle by

Explanation:

1. For the first blank: Since \(\triangle DEF\) is a right - triangle, the equation \(a^{2}+b^{2}=n^{2}\) is based on the Pythagorean theorem (\(a\) and \(b\) are legs and \(n\) is the hypotenuse of a right - triangle).
2. For the second blank: Given \(c^{2}=n^{2}\), taking the square root of both sides gives \(c = n\) (since \(c>0\) and \(n>0\) as side lengths of triangles).
3. For the third blank: In \(\triangle ABC\) and \(\triangle DEF\), \(a=a\), \(b = b\), \(c=n\). By SSS (Side - Side - Side) congruence criterion, \(\triangle ABC\cong\triangle DEF\).
4. For the fourth blank: Since \(\triangle ABC\cong\triangle DEF\) and \(\angle F = 90^{\circ}\) in \(\triangle DEF\), \(\angle C\) (corresponding to \(\angle F\)) is a right angle by CPCTC (Corresponding Parts of Congruent Triangles are Congruent).
5. For the fifth blank: A triangle with one right angle is a right - triangle. The definition of a right - triangle is a triangle that has one \(90^{\circ}\) angle.

Answer:

1. Pythagorean theorem
2. \(c=n\)
3. SSS
4. \(\angle C\)
5. definition of a right - triangle