How many diagonals does a pentagon have?
Answer: B
A polygon with \( n \) sides has
\[ \frac{n(n-3)}{2} \text{ diagonals} \]
\[ \frac{5(5-3)}{2} = \frac{10}{2} = 5 \]
Each vertex joins to all but itself and its two neighbours, and the division by 2 avoids counting each diagonal from both ends.
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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 \( 90^{\circ} - x \).
\[ x = (90 - x) + 20 \quad\Longrightarrow\quad 2x = 110 \]
\[ x = 55^{\circ} \]
Checking: the complement is 35°, which is indeed 20° less.
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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 three angles add to 180°.
\[ 3x + 5x + 4x = 12x = 180 \]
\[ x = 15 \]
The angles are then 45°, 75° and 60°.
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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 angles of a quadrilateral add to 360°, and the ratio has \( 1+2+3+4 = 10 \) parts.
\[ \text{one part} = \frac{360}{10} = 36^{\circ} \]
\[ \text{largest} = 4 \times 36 = 144^{\circ} \]
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The perimeter of an equilateral triangle is 60 cm. What is the length of one side?
Answer: C
All three sides of an equilateral triangle are equal.
\[ \frac{60}{3} = 20 \text{ cm} \]
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What is the measure of an exterior angle of a regular decagon (10-sided polygon)?
Answer: B
The sum of the exterior angles of any convex polygon is always 360°.
For a regular n-sided polygon, each exterior angle has the same measure.
Measure of one exterior angle = 360° / n.
For a decagon, n=10.
Exterior angle = 360° / 10 = 36°.
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In an isosceles triangle, the vertex angle is 40°. What are the measures of the base angles?
Answer: D
The base angles of an isosceles triangle are equal, and all three add to 180°.
\[ 2b = 180 - 40 = 140 \quad\Longrightarrow\quad b = 70^{\circ} \]
So the base angles are 70° and 70°.
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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.
\[ 3x - 10 = 2x + 15 \quad\Longrightarrow\quad x = 25 \]
Both angles then measure 65°, as they must.
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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
An exterior angle equals the sum of the two interior opposite angles.
\[ 130^{\circ} = 60^{\circ} + \theta \quad\Longrightarrow\quad \theta = 70^{\circ} \]
Equivalently, the adjacent interior angle is 50°, and \( 180 - 50 - 60 = 70 \).
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