Question

prove that $\triangle abc$ is a right triangle. select the correct answer from each drop - down menu.
$\overline{ab}$ is congruent to $\overline{de}$ because segment $de$ was constructed so that $de = ab$. $\overline{bc}$ is congruent to $\overline{ef}$ because segment $ef$ was constructed so that $ef = bc$. since $\triangle def$ is a right triangle, $de^{2}+ef^{2}=df^{2}$ by the pythagorean theorem. we are given that $ab^{2}+bc^{2}=ac^{2}$. since $de\cong ab$ and $ef\cong bc$, $de^{2}+ef^{2}=ac^{2}$ by the substitution property of equality. also, $df^{2}=ac^{2}$ by the

division property of equality
substitution property of equality
pythagorean theorem

taking the square root of both sides of the equation gives $df$

the definition of congruence. applying the
$\cong e$. therefore $\angle b$ is a right angle and $\triangle abc$ is a right triangle.

question 2
suppose triangles $p$, $q$, and $r$ have sides with the given measurements.

• triangle $p$: 12, 24, and 30
• triangle $q$: 9, 40, and 41
• triangle $r$: 5, 18, and 21

which triangle is a right triangle? explain your reasoning.

Explanation:

Answer:

1. Pythagorean theorem; substitution property of equality; transitive property of equality; $DF = AC$; SSS triangle congruence theorem
2. triangle Q; $9^2 + 40^2 = 81 + 1600 = 1681 = 41^2$