If an angle is its own complement, what is its measure?
Answer: B
Let the angle be x. Its complement is also x.
The sum of complementary angles is 90°.
x + x = 90°
2x = 90°
x = 45°.
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A regular polygon has an interior angle of 150°. How many sides does it have?
Answer: C
If the interior angle is 150°, the exterior angle is 180° - 150° = 30°.
The number of sides 'n' of a regular polygon is given by n = 360° / (exterior angle).
n = 360° / 30° = 12.
The polygon has 12 sides (a dodecagon).
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In triangle PQR, PQ = PR and angle Q = 65°. Find angle P.
Answer: A
Since PQ = PR, the triangle is isosceles. The angles opposite the equal sides are equal. So, angle R = angle Q = 65°.
The sum of angles in a triangle is 180°.
Angle P + Angle Q + Angle R = 180°
Angle P + 65° + 65° = 180°
Angle P + 130° = 180°
Angle P = 50°.
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What is the angle between the hour hand and the minute hand of a clock at 3:30?
Answer: B
The formula for the angle between the hands of a clock is \(|30H - \frac{11}{2}M|\), where H is the hour and M is the minute.
Here, H=3 and M=30.
Angle = \(|30(3) - \frac{11}{2}(30)| = |90 - 11 \times 15| = |90 - 165| = |-75| = 75°\).
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The sides of a triangle are in the ratio 3:4:5. The triangle is:
Answer: B
Let the sides be 3x, 4x, and 5x. We check if they form a Pythagorean triplet.
(3x)² + (4x)² = 9x² + 16x² = 25x².
(5x)² = 25x².
Since (3x)² + (4x)² = (5x)², the sides satisfy the Pythagorean theorem. Therefore, the triangle is a right-angled triangle.
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A triangle can have:
Answer: C
The sum of angles in a triangle is 180°.
Two right angles (90°+90°=180°) would leave 0° for the third angle, which is not possible.
Two obtuse angles (each >90°) would make the sum exceed 180°, which is not possible.
A triangle can always have two acute angles (e.g., in a right-angled triangle or an obtuse-angled triangle). In fact, every triangle must have at least two acute angles.
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The sum of the interior angles of a polygon is 1440°. How many sides does the polygon have?
Answer: C
The formula for the sum of the interior angles of an n-sided polygon is (n-2) × 180°.
(n-2) × 180° = 1440°
n - 2 = 1440 / 180
n - 2 = 8
n = 10.
The polygon has 10 sides (a decagon).
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