If the sum of the interior angles of a regular polygon is 1080°, what is the name of the polygon?
Answer: C
The formula for the sum of the interior angles of an n-sided polygon is (n-2) × 180°.
(n-2) × 180° = 1080°
n - 2 = 1080 / 180
n - 2 = 6
n = 8.
A polygon with 8 sides is an octagon.
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A triangle with sides 5 cm, 12 cm, and 13 cm is a:
Answer: C
We check if the sides satisfy the Pythagorean theorem (a² + b² = c²).
The longest side is 13 cm, so it would be the hypotenuse.
5² + 12² = 25 + 144 = 169.
13² = 169.
Since 5² + 12² = 13², the triangle is a right-angled triangle. This is a common Pythagorean triplet (5, 12, 13).
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How many lines of symmetry does a regular hexagon have?
Answer: C
A regular n-sided polygon has n lines of symmetry.
For a regular hexagon, n=6. It has 6 lines of symmetry.
3 lines pass through opposite vertices.
3 lines pass through the midpoints of opposite sides.
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An angle is equal to one-fifth of its supplement. Find its measure.
Answer: A
Let the angle be x. Its supplement is 180° - x.
x = \(\frac{1}{5}(180° - x)\)
5x = 180° - x
6x = 180°
x = 30°.
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In \(\triangle PQR\) and \(\triangle XYZ\), if \(PQ/XY = QR/YZ = PR/XZ\), which similarity criterion applies?
Answer: C
The condition given is that the ratios of the lengths of the corresponding sides of the two triangles are equal. This is the definition of the SSS (Side-Side-Side) similarity criterion. If this condition is met, the two triangles are similar.
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In a parallelogram, adjacent angles are:
Answer: C
In a parallelogram, consecutive or adjacent angles are supplementary, meaning their sum is 180°. This is because consecutive sides act as a transversal cutting parallel opposite sides.
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The measure of two angles of a triangle are in the ratio 2:3. The third angle is 70°. Find the smallest angle of the triangle.
Answer: C
The sum of angles in a triangle is 180°. The sum of the two unknown angles is 180° - 70° = 110°.
Let the angles be 2x and 3x.
2x + 3x = 110°
5x = 110°
x = 22°.
The angles are 2x = 2(22) = 44° and 3x = 3(22) = 66°. The smallest angle of the triangle is 44°.
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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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