How many diagonals does a pentagon have?
Answer: B
The formula for the number of diagonals in an n-sided polygon is \(\frac{n(n-3)}{2}\).
For a pentagon, n=5.
Number of diagonals = \(\frac{5(5-3)}{2} = \frac{5 \times 2}{2} = 5\).
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What is the measure of a straight angle?
Answer: B
A straight angle is an angle formed by a straight line. Its measure is always 180 degrees.
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In triangle ABC, angle A = 3x, angle B = 5x, and angle C = 4x. What is the value of x?
Answer: B
The sum of angles in a triangle is 180°.
3x + 5x + 4x = 180°
12x = 180°
x = 180° / 12 = 15.
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The angles of a quadrilateral are in the ratio 1:2:3:4. What is the measure of the largest angle?
Answer: D
The sum of the interior angles of a quadrilateral is 360°.
Let the angles be x, 2x, 3x, and 4x.
x + 2x + 3x + 4x = 360°
10x = 360°
x = 36°.
The largest angle is 4x = 4 × 36° = 144°.
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A line segment has no definite:
Answer: B
In Euclidean geometry, a line or line segment is a one-dimensional object. It has a definite length and two distinct endpoints. However, it is considered to have no breadth or thickness.
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What is the complement of 63°?
Answer: A
Complementary angles add up to 90°.
The complement of 63° is 90° - 63° = 27°.
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The sum of the lengths of any two sides of a triangle is always:
Answer: C
This is the statement of the Triangle Inequality Theorem. For any triangle with side lengths a, b, and c, the following must be true: a + b > c, a + c > b, and b + c > a. In words, the sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
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Two lines intersect at a point. If one pair of vertically opposite angles are (3x - 10)° and (2x + 15)°, find the value of x.
Answer: C
Vertically opposite angles are equal when two lines intersect.
3x - 10 = 2x + 15
3x - 2x = 15 + 10
x = 25.
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The lengths of two sides of a triangle are 8 cm and 11 cm. The length of the third side must be:
Answer: C
The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let the third side be x.
x + 8 > 11 => x > 3
8 + 11 > x => 19 > x
So, the length of the third side must be between 3 cm and 19 cm.
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An exterior angle of a triangle is 130° and one of the interior opposite angles is 60°. Find the other interior opposite angle.
Answer: C
The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Let the other interior opposite angle be x.
130° = 60° + x
x = 130° - 60° = 70°.
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