To solve this we use the Hinge Theorem SAS Inequality Theore...

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### Correct Answer: \( 0 < x < 80 \) ### Detailed Steps for Solving the Problem: To solve this, we use the **Hinge Theorem** (SAS Inequality Theorem), which relates the lengths of sides and included angles in two triangles with two congruent sides. #### Step 1: Identify the Triangles and Congruent Sides The diagram shows two triangles with: - Two sides of length \( 7 \) (congruent in both triangles). - Third sides: \( 13 \) (for the triangle with included angle \( \boldsymbol{\left(\frac{1}{8}x\right)^\circ} \)) and \( 21 \) (for the triangle with included angle \( \boldsymbol{10^\circ} \)). #### Step 2: Apply the Hinge Theorem The Hinge Theorem states: *If two sides of one triangle are congruent to two sides of another triangle, but the third side of the first triangle is shorter, then its included angle is smaller than the included angle of the second triangle.* Here: - The third side \( 13 \) (of the first triangle) is shorter than \( 21 \) (of the second triangle). - Thus, the included angle \( \left(\frac{1}{8}x\right)^\circ \) (for the triangle with third side \( 13 \)) must be smaller than the included angle \( 10^\circ \) (for the triangle with third side \( 21 \)). #### Step 3: Set Up and Solve the Inequality From the Hinge Theorem: \[ 10^\circ > \left(\frac{1}{8}x\right)^\circ \] Multiply both sides by \( 8 \) to solve for \( x \): \[ 80 > x \] Additionally, the angle \( \left(\frac{1}{8}x\right)^\circ \) must be positive (since angles in a triangle are positive): \[ \frac{1}{8}x > 0 \implies x > 0 \] #### Step 4: Combine the Inequalities From \( x > 0 \) and \( x < 80 \), we get: \[ 0 < x < 80 \] ### Relevant Knowledge Points: Hinge Theorem (SAS Inequality Theorem) The Hinge Theorem states: - If two sides of one triangle are congruent to two sides of another triangle, and the **included angle** of the first triangle is **larger** than the included angle of the second triangle, then the **third side** of the first triangle is **longer** than the third side of the second triangle. - Conversely, if the third side of the first triangle is longer, then its included angle is larger. ### Explanation of the Hinge Theorem Imagine two triangles with two pairs of congruent sides (e.g., \( AB = DE \) and \( AC = DF \)). The "hinge" is the included angle (e.g., \( \angle BAC \) and \( \angle EDF \)). If \( \angle BAC > \angle EDF \), then the side opposite the larger angle (\( BC \)) will be longer than the side opposite the smaller angle (\( EF \)). In our problem: - Both triangles have two sides of length \( 7 \) (congruent). - The third side of the first triangle is \( 13 \) (shorter), and its included angle is \( \left(\frac{1}{8}x\right)^\circ \). - The third side of the second triangle is \( 21 \) (longer), and its included angle is \( 10^\circ \). By the Hinge Theorem, since \( 21 > 13 \) (longer third side), the included angle for \( 21 \) (\( 10^\circ \)) must be larger than the included angle for \( 13 \) (\( \left(\frac{1}{8}x\right)^\circ \)). This gives \( 10 > \frac{1}{8}x \), leading to \( x < 80 \). The angle must also be positive (\( \frac{1}{8}x > 0 \)), so \( x > 0 \). Thus, \( 0 < x < 80 \).

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