Which of the following TWO statements are correct regarding unique triangles? A. No triangle can be constructed with side lengths of 15 cm and 21 cm and an included angle measure of \( 160^{\circ} \). B. Exactly 1 triangle can be constructed with side lengths of 15 cm and 21 cm and an included angle measure of \( 160^{\circ} \). C. More than 1 triangle can be constructed with side lengths of 15 cm and 21 cm and an included angle measure of \( 160^{\circ} \). D. No triangle can be constructed with angle measures of \( 35^{\circ} \) and \( 145^{\circ} \) and a side length of 4 inches. E. Exactly 1 triangle can be constructed with angle measures of \( 35^{\circ} \) and \( 145^{\circ} \) and a side length of 4 inches. F. More than 1 triangle can be constructed with angle measures of \( 35^{\circ} \) and \( 145^{\circ} \) and a side length of 4 inches. Which sets of conditions guarantee a unique triangle? Choose THREE that apply. A. Side lengths of 3 inches, 4 inches, and 5 inches. B. Angle measures of \( 25^{\circ}, 75^{\circ} \), and \( 80^{\circ} \). C. Side lengths of 2 inches and 5 inches with an included angle of \( 30^{\circ} \). D. Angle measures of \( 20^{\circ} \) and \( 60^{\circ} \) with an included side that is 9 cm long. E. Side lengths of 4 cm and 6 cm with a non included angle measure of \( 45^{\circ} \).

Let's address both parts of your question separately. --- ### **First Part: Identifying Two Correct Statements Regarding Unique Triangles** **Given Statements:** - **A.** No triangle can be constructed with side lengths of 15 cm and 21 cm and an included angle measure of \( 160^{\circ} \). - **B.** Exactly 1 triangle can be constructed with side lengths of 15 cm and 21 cm and an included angle measure of \( 160^{\circ} \). - **C.** More than 1 triangle can be constructed with side lengths of 15 cm and 21 cm and an included angle measure of \( 160^{\circ} \). - **D.** No triangle can be constructed with angle measures of \( 35^{\circ} \) and \( 145^{\circ} \) and a side length of 4 inches. - **E.** Exactly 1 triangle can be constructed with angle measures of \( 35^{\circ} \) and \( 145^{\circ} \) and a side length of 4 inches. - **F.** More than 1 triangle can be constructed with angle measures of \( 35^{\circ} \) and \( 145^{\circ} \) and a side length of 4 inches. **Analysis:** 1. **Statements B and D are Correct:** - **B.** *Exactly 1 triangle can be constructed with side lengths of 15 cm and 21 cm and an included angle measure of \( 160^{\circ} \).* Using the **SAS (Side-Angle-Side)** condition, when two sides and the included angle are known, a unique triangle is determined, provided the triangle inequality holds. Here, the angle of \( 160^{\circ} \) is valid, ensuring exactly one triangle exists. - **D.** *No triangle can be constructed with angle measures of \( 35^{\circ} \) and \( 145^{\circ} \) and a side length of 4 inches.* The sum of the given angles is \( 35^{\circ} + 145^{\circ} = 180^{\circ} \). Since the sum of angles in a triangle must be exactly \( 180^{\circ} \), having two angles sum to \( 180^{\circ} \) leaves no room for a third angle, making triangle construction impossible. **Incorrect Statements:** - **A & C:** Given side lengths and an included angle via SAS guarantee a unique triangle, so **A** is incorrect (a triangle can be constructed) and **C** is incorrect (only one triangle exists). - **E & F:** Since two angles already sum to \( 180^{\circ} \), no triangle can exist, making **E** and **F** incorrect. --- ### **Second Part: Identifying Three Conditions That Guarantee a Unique Triangle** **Given Conditions:** - **A.** Side lengths of 3 inches, 4 inches, and 5 inches. - **B.** Angle measures of \( 25^{\circ}, 75^{\circ} \), and \( 80^{\circ} \). - **C.** Side lengths of 2 inches and 5 inches with an included angle of \( 30^{\circ} \). - **D.** Angle measures of \( 20^{\circ} \) and \( 60^{\circ} \) with an included side that is 9 cm long. - **E.** Side lengths of 4 cm and 6 cm with a non-included angle measure of \( 45^{\circ} \). **Analysis:** 1. **Conditions A, C, and D Guarantee a Unique Triangle:** - **A.** *Side lengths of 3 inches, 4 inches, and 5 inches.* This is the **SSS (Side-Side-Side)** condition. Knowing all three side lengths (and satisfying the triangle inequality) guarantees a unique triangle. - **C.** *Side lengths of 2 inches and 5 inches with an included angle of \( 30^{\circ} \).* This is the **SAS (Side-Angle-Side)** condition. Two sides and the included angle uniquely determine a triangle. - **D.** *Angle measures of \( 20^{\circ} \) and \( 60^{\circ} \) with an included side that is 9 cm long.* This represents the **ASA (Angle-Side-Angle)** condition. Two angles and the included side uniquely determine a triangle. 2. **Incorrect Conditions:** - **B.** *Angle measures of \( 25^{\circ}, 75^{\circ} \), and \( 80^{\circ} \).* While the angles sum to \( 180^{\circ} \), without any side lengths, infinitely many similar triangles exist. - **E.** *Side lengths of 4 cm and 6 cm with a non-included angle measure of \( 45^{\circ} \).* This is the **SSA (Side-Side-Angle)** condition, which does **not** guarantee a unique triangle. It can result in zero, one, or two possible triangles depending on the specific values. --- ### **Summary of Correct Answers** - **First Part:** **Statements B and D** are correct. - **Second Part:** **Conditions A, C, and D** guarantee a unique triangle.