Question

click the arrow to choose an answer from each menu. triangle pqr is given to be a right triangle, so the pythagorean theorem can be used to determine that choose... . it is given that ( x^2 + y^2 = z^2 ) in triangle abc, so choose... . this information shows that the triangles are congruent, because their three sides are congruent. therefore, to show that triangle pqr is a right triangle, we can use corresponding parts of the congruent triangles to show that choose... .

Explanation:

To solve this, we analyze the right triangle and congruence:

Step 1: Apply Pythagorean Theorem to \( \triangle PQR \)

Since \( \triangle PQR \) is a right triangle, the Pythagorean Theorem gives \( \boldsymbol{a^2 + b^2 = c^2} \) (where \( a, b \) are legs, \( c \) is hypotenuse).

Step 2: Relate to \( \triangle ABC \)

Given \( x^2 + y^2 = z^2 \) in \( \triangle ABC \), and if \( \triangle ABC \cong \triangle PQR \) (by SSS, as all sides match), then corresponding sides satisfy \( x = a \), \( y = b \), \( z = c \).

Step 3: Conclude \( \triangle PQR \) is Right

For \( \triangle PQR \), using the congruent sides and the Pythagorean Theorem (or corresponding angles), we confirm it is a right triangle because the Pythagorean relationship holds (or the right angle from \( \triangle ABC \) corresponds to \( \triangle PQR \)).

Final Answer (Filling the Blanks):

1. First “Choose...” : \( \boldsymbol{a^2 + b^2 = c^2} \) (Pythagorean Theorem for \( \triangle PQR \))
2. Second “Choose...” : \( \boldsymbol{\triangle ABC \cong \triangle PQR} \) (by SSS, since \( x^2 + y^2 = z^2 \) and \( a^2 + b^2 = c^2 \) imply side lengths match)
3. Third “Choose...” : \( \boldsymbol{\angle R \text{ (or the right angle) is } 90^\circ} \) (corresponding parts of congruent triangles, so the right angle from \( \triangle ABC \) maps to \( \triangle PQR \))

Answer:

To solve this, we analyze the right triangle and congruence:

Step 1: Apply Pythagorean Theorem to \( \triangle PQR \)

Since \( \triangle PQR \) is a right triangle, the Pythagorean Theorem gives \( \boldsymbol{a^2 + b^2 = c^2} \) (where \( a, b \) are legs, \( c \) is hypotenuse).

Step 2: Relate to \( \triangle ABC \)

Given \( x^2 + y^2 = z^2 \) in \( \triangle ABC \), and if \( \triangle ABC \cong \triangle PQR \) (by SSS, as all sides match), then corresponding sides satisfy \( x = a \), \( y = b \), \( z = c \).

Step 3: Conclude \( \triangle PQR \) is Right

For \( \triangle PQR \), using the congruent sides and the Pythagorean Theorem (or corresponding angles), we confirm it is a right triangle because the Pythagorean relationship holds (or the right angle from \( \triangle ABC \) corresponds to \( \triangle PQR \)).

Final Answer (Filling the Blanks):

1. First “Choose...” : \( \boldsymbol{a^2 + b^2 = c^2} \) (Pythagorean Theorem for \( \triangle PQR \))
2. Second “Choose...” : \( \boldsymbol{\triangle ABC \cong \triangle PQR} \) (by SSS, since \( x^2 + y^2 = z^2 \) and \( a^2 + b^2 = c^2 \) imply side lengths match)
3. Third “Choose...” : \( \boldsymbol{\angle R \text{ (or the right angle) is } 90^\circ} \) (corresponding parts of congruent triangles, so the right angle from \( \triangle ABC \) maps to \( \triangle PQR \))