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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The point where the three medians of a triangle meet is called the:
Answer: C
The intersection point of the three medians of a triangle is known as the centroid. The centroid divides each median in the ratio 2:1.
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If the diagonal of a rhombus is equal to its side, then the angles of the rhombus are:
Answer: D
Let the rhombus be ABCD with side length 's'. If the diagonal AC is also 's', then \(\triangle ABC\) has all three sides equal to 's'. This means \(\triangle ABC\) is an equilateral triangle, and all its angles are 60°.
So, \(\angle B = 60°\). The opposite angle, \(\angle D\), is also 60°.
Adjacent angles in a rhombus are supplementary. So, \(\angle A = 180° - 60° = 120°\). The opposite angle, \(\angle C\), is also 120°.
The angles are 60°, 120°, 60°, 120°.
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What is the ratio of the interior angle to the exterior angle of a regular pentagon?
Answer: B
For a regular pentagon, n=5.
The measure of each exterior angle = \(360°/n = 360°/5 = 72°\).
The measure of each interior angle = \(180° - 72° = 108°\).
The ratio of interior angle to exterior angle is 108:72. Dividing both by 36 gives the ratio 3:2.
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The sum of the lengths of any two sides of a triangle is always:
Answer: C
This is the statement of the Triangle Inequality Theorem. For any triangle with side lengths a, b, and c, the following must be true: a + b > c, a + c > b, and b + c > a. In words, the sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
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In \(\triangle PQR\), \(\angle P = 70°\) and \(\angle R = 30°\). Which side of this triangle is the longest?
Answer: C
First, find the third angle, \(\angle Q\).
\(\angle Q = 180° - (70° + 30°) = 180° - 100° = 80°\).
In any triangle, the side opposite the largest angle is the longest side.
The angles are P=70°, Q=80°, R=30°. The largest angle is \(\angle Q = 80°\).
The side opposite to \(\angle Q\) is PR. Therefore, PR is the longest side.
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How many triangles can be formed with sides of length 4 cm, 5 cm, and 10 cm?
Answer: A
To form a triangle, the Triangle Inequality Theorem must be satisfied. The sum of any two sides must be greater than the third side.
Let's check:
4 + 5 > 10? => 9 > 10. This is false.
Since the condition is not met, no triangle can be formed with these side lengths.
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A reflex angle is an angle that is:
Answer: D
A reflex angle is one that turns through more than a straight angle but less than a full turn:
\[ 180^{\circ} < \theta < 360^{\circ} \]
Below 90° an angle is acute, between 90° and 180° obtuse, and exactly 180° straight.
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In an isosceles trapezoid, which of the following is true?
Answer: C
An isosceles trapezoid has one pair of parallel sides, and its two non-parallel sides are equal.
Its diagonals are also equal, and its base angles are equal in pairs - but opposite angles are supplementary rather than equal.
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The point where the three altitudes of a triangle meet is called the:
Answer: D
The three altitudes of a triangle are concurrent at the orthocentre.
The other three classical centres come from different cevians: the medians meet at the centroid, the angle bisectors at the incentre, and the perpendicular bisectors of the sides at the circumcentre.
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