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
\[ x = \frac{1}{5}(180 - x) \quad\Longrightarrow\quad 5x = 180 - x \]
\[ 6x = 180 \quad\Longrightarrow\quad x = 30^{\circ} \]
Checking: the supplement is 150°, five times as large.
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What type of triangle is formed by sides of length 7, 8, and 13?
Answer: C
We use the converse of the Pythagorean theorem. Let c be the longest side.
If a² + b² > c², the triangle is acute.
If a² + b² = c², the triangle is right-angled.
If a² + b² < c², the triangle is obtuse.
Here, a=7, b=8, c=13.
a² + b² = 7² + 8² = 49 + 64 = 113.
c² = 13² = 169.
Since 113 < 169 (a² + b² < c²), the triangle is obtuse-angled.
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In a parallelogram, adjacent angles are:
Answer: C
Adjacent angles of a parallelogram lie between the same pair of parallel sides, so they are supplementary:
\[ \angle A + \angle B = 180^{\circ} \]
Opposite angles are the equal ones - the pair this question is testing against.
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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
\[ x = 90 - x \quad\Longrightarrow\quad 2x = 90 \]
\[ x = 45^{\circ} \]
An angle that is its own supplement would be 90° instead.
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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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