Which of the following TWO statements about triangles are true? A. There is no triangle with side lengths of 14 inches, 17 inches, and 32 inches. B. There is exactly 1 triangle with side lengths of 14 inches, 17 inches, and 32 inches. C. There is more than 1 triangle with side lengths of 14 inches, 17 inches, and 32 inches. D. There is no triangle with the angle measures of \( 67^{\circ} \) and \( 100^{\circ} \). E. There is exactly 1 triangle with the angle measures of \( 67^{\circ} \) and \( 100^{\circ} \). F. There is more than 1 triangle with the angle measures of \( 67^{\circ} \) and \( 100^{\circ} \). A triangle has a side length of 13 units and 22 units. Which measurements could be a possible length of the third side? Choose THREE that apply. A. 5 units B. 9.1 units C. 15 units D. 26.5 units E. 35 units F. 44.2 units

Let's analyze each part of your question step by step. --- ### **Part 1: True Statements About Triangles** **Statements:** - **A.** There is no triangle with side lengths of 14 inches, 17 inches, and 32 inches. - **B.** There is exactly 1 triangle with side lengths of 14 inches, 17 inches, and 32 inches. - **C.** There is more than 1 triangle with side lengths of 14 inches, 17 inches, and 32 inches. - **D.** There is no triangle with the angle measures of \( 67^{\circ} \) and \( 100^{\circ} \). - **E.** There is exactly 1 triangle with the angle measures of \( 67^{\circ} \) and \( 100^{\circ} \). - **F.** There is more than 1 triangle with the angle measures of \( 67^{\circ} \) and \( 100^{\circ} \). **Analysis:** 1. **For Statements A, B, and C (Side Lengths Analysis):** - **Triangle Inequality Theorem:** The sum of any two side lengths of a triangle must be greater than the third side. - **Check for 14, 17, 32 inches:** - \( 14 + 17 = 31 \) which is **less than** 32. - This violates the triangle inequality. - **Conclusion:** - **Statement A** is **True**. - **Statements B and C** are **False** because no triangle can be formed with these side lengths. 2. **For Statements D, E, and F (Angle Measures Analysis):** - **Angle Sum Property:** The sum of the angles in a triangle is \( 180^{\circ} \). - Given angles: \( 67^{\circ} \) and \( 100^{\circ} \). - Sum of given angles: \( 67 + 100 = 167^{\circ} \). - Third angle: \( 180 - 167 = 13^{\circ} \). - **Uniqueness:** Once two angles are specified, the third is uniquely determined. - **Conclusion:** - **Statement E** is **True**. - **Statements D and F** are **False** because exactly one unique triangle can be formed with these angle measures. ****True Statements:** **A** and **E**. --- ### **Part 2: Possible Lengths for the Third Side** **Given:** - Two sides of a triangle: 13 units and 22 units. **Possible Third Side (x):** - **Triangle Inequality Conditions:** 1. \( x > |22 - 13| \) → \( x > 9 \) 2. \( x < 22 + 13 \) → \( x < 35 \) - **Thus, the third side must satisfy:** \( 9 < x < 35 \) **Options:** - **A.** 5 units → **Invalid** (5 < 9) - **B.** 9.1 units → **Valid** (9.1 > 9 and < 35) - **C.** 15 units → **Valid** (15 > 9 and < 35) - **D.** 26.5 units → **Valid** (26.5 > 9 and < 35) - **E.** 35 units → **Invalid** (35 is not less than 35) - **F.** 44.2 units → **Invalid** (44.2 > 35) ****Valid Options for the Third Side:** **B. 9.1 units**, **C. 15 units**, and **D. 26.5 units** --- **Summary of Correct Answers:** 1. **First Question (True Statements):** **A** and **E** 2. **Second Question (Possible Third Side Lengths):** **B**, **C**, and **D**