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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An angle is 20° more than its complement. Find the angle.
Answer: C
Let the angle be x. Its complement is 90° - x.
According to the problem, x = (90° - x) + 20°.
x = 110° - x
2x = 110°
x = 55°
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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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The supplement of an angle is 110°. Find the angle.
Answer: B
Two angles are supplementary if their sum is 180°.
Let the angle be x.
x + 110° = 180°
x = 180° - 110° = 70°
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What is the number of sides of a regular polygon whose each interior angle is 135°?
Answer: B
If the interior angle is 135°, the exterior angle is 180° - 135° = 45°.
The number of sides 'n' of a regular polygon is given by n = 360° / (exterior angle).
n = 360° / 45° = 8.
The polygon is an octagon.
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Two parallel lines are intersected by a transversal. If one of the interior angles is 57°, what is the measure of its co-interior angle?
Answer: C
Co-interior angles (or consecutive interior angles) are on the same side of the transversal and between the parallel lines. They are supplementary, meaning their sum is 180°.
Let the other co-interior angle be x.
x + 57° = 180°
x = 180° - 57° = 123°.
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In an isosceles trapezoid, which of the following is true?
Answer: C
An isosceles trapezoid is a quadrilateral with a pair of parallel sides and a pair of equal non-parallel sides. Its diagonals are also equal in length, and base angles are equal.
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In \(\triangle ABC\), the side BC is extended to D. If \(\angle ACD = 120°\) and \(\angle ABC = 45°\), find \(\angle BAC\).
Answer: C
The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Here, \(\angle ACD\) is the exterior angle, and \(\angle ABC\) and \(\angle BAC\) are the interior opposite angles.
\(\angle ACD = \angle ABC + \angle BAC\)
120° = 45° + \(\angle BAC\)
\(\angle BAC = 120° - 45° = 75°\).
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A polygon with 8 sides is called a/an:
Answer: C
A polygon is named based on its number of sides.
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What is the sum of the interior angles of a triangle?
Answer: B
The sum of the measures of the interior angles of any triangle is always 180 degrees. This is a fundamental theorem of Euclidean geometry.
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