Here are the answers to your questions based on the Triangle Inequality Theorem and related concepts:
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Apply the Triangle Inequality Theorem to determine the only possible whole number measure of the third side of a triangle if the first two sides measure 1 foot and 4 feet. To satisfy the triangle inequality, the third side \( c \) must satisfy:
- \( 1 + 4 > c \) → \( c < 5 \)
- \( 1 + c > 4 \) → \( c > 3 \)
- \( 4 + c > 1 \) → \( c > -3 \) (always true for positive lengths) Therefore, \( 3 < c < 5 \), and the only whole number measure is 4 feet.
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What inequality represents the values that side a could be to create a triangle, if side b = 12 cm and side c = 17 cm? Applying the triangle inequality:
- \( a + 12 > 17 \) → \( a > 5 \)
- \( a + 17 > 12 \) → \( a > -5 \) (always true for positive lengths)
- \( 12 + 17 > a \) → \( a < 29 \) Thus, the inequality is 5 < a < 29.
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How many right angles are possible in a triangle? A triangle can have 1 right angle.
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Given a triangle with side lengths a, b, and c, which conditional statements must be true? The correct statements are: a < b + c and a > b - c.
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Apply the Triangle Inequality Theorem to determine which three side lengths form a triangle.
- For 10, 10, 30: 10 + 10 < 30 (not a triangle)
- For 4, 8, 12: 4 + 8 = 12 (not a triangle)
- For 10, 20, 40: 10 + 20 < 40 (not a triangle)
- For 5, 7, 9: 5 + 7 > 9; 5 + 9 > 7; 7 + 9 > 5 (this is a triangle). Therefore, the only valid set is 5, 7, 9.
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Use the Triangle Angle Sum Theorem to find the largest angle in a triangle with angle measures of (3x), (6x - 5), and (8x + 15). Set up the equation based on the angle sum: \( 3x + (6x - 5) + (8x + 15) = 180 \) Simplifying gives: \( 17x + 10 = 180 \) → \( 17x = 170 \) → \( x = 10 \) Hence the angles are:
- \( 3x = 30 \) degrees
- \( 6x - 5 = 55 \) degrees
- \( 8x + 15 = 95 \) degrees (largest) Therefore, the largest angle is 95 degrees.
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A right triangle has acute angles measuring (3x + 4) degrees and (5x + 6) degrees. What are the two angle measures? Since it is a right triangle, the sum of the two acute angles must be \( 90 \) degrees: \( (3x + 4) + (5x + 6) = 90 \) → \( 8x + 10 = 90 \) → \( 8x = 80 \) → \( x = 10 \) Thus, the angles are:
- \( 3(10) + 4 = 34 \)
- \( 5(10) + 6 = 56 \) So the two angle measures are 34 degrees and 56 degrees.
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Use the Triangle Angle Sum Theorem to find the measure of the angle at point B. I cannot answer this without the measures of the other angles or a visual representation. If you provide more context, I can assist.
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Find the measure of the smallest angle of the triangle. Again, I need more context as the measures are not provided. Please supply the information.
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How many obtuse angles are possible in a triangle? A triangle can have 0 obtuse angles.
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The Triangle Angle Sum Theorem states that the interior angles of a triangle sum to 180 degrees.
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Find the missing angle measure. Without a specific context or image, I cannot proceed. Please provide the angle measures involved.
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The angles of a triangle measure 2x, 3x, and 5x. Using the Triangle Angle Sum Theorem, find the measure of the smallest angle in this triangle. Setting up the equation: \( 2x + 3x + 5x = 180 \) → \( 10x = 180 \) → \( x = 18 \). Therefore, the angles are:
- \( 2x = 36 \)
- \( 3x = 54 \)
- \( 5x = 90 \) The smallest angle is 36 degrees.
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A right triangle has acute angles measuring (3x + 4) degrees and (5x + 6) degrees. What equation could be used to find the value of x? The correct equation is: (3x + 4) + (5x + 6) = 90.
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Two sides of a triangle are 15 inches and 8 inches. Determine the length of the third side from the choices given. Let \( c \) be the length of the third side. Using the Triangle Inequality Theorem,
- \( 15 + 8 > c \) → \( c < 23 \)
- \( 15 + c > 8 \) → \( c > -7 \) (always true)
- \( 8 + c > 15 \) → \( c > 7 \) Thus, \( 7 < c < 23 \). The only choice that fits this is 9 inches.
If you need any further clarification or additional problems revisited, feel free to ask!