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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The angle which is four times its supplement is:
Answer: B
\[ x = 4(180 - x) \quad\Longrightarrow\quad x = 720 - 4x \]
\[ 5x = 720 \quad\Longrightarrow\quad x = 144^{\circ} \]
Checking: the supplement is 36°, and \( 4 \times 36 = 144 \).
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What is the number of sides of a regular polygon whose each interior angle is 135°?
Answer: B
If the interior angle is 135°, the exterior angle is 180° - 135° = 45°.
The number of sides 'n' of a regular polygon is given by n = 360° / (exterior angle).
n = 360° / 45° = 8.
The polygon is an octagon.
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If one angle of a linear pair is acute, then its other angle will be:
Answer: B
A linear pair of angles are adjacent angles whose non-common sides are opposite rays. Their sum is always 180°.
An acute angle is less than 90°. Let the acute angle be x, where x < 90°.
The other angle will be 180° - x.
Since x < 90°, then 180° - x > 90°. An angle greater than 90° is an obtuse angle.
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An exterior angle of a regular polygon is one-third of its interior angle. How many sides does the polygon have?
Answer: B
Let the exterior angle be x. The interior angle is 3x.
The sum of an interior and its corresponding exterior angle is 180°.
x + 3x = 180°
4x = 180°
x = 45°.
The exterior angle is 45°. The number of sides n = 360° / 45° = 8.
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In triangle ABC, the bisectors of angle B and angle C meet at point O. If angle A = 70°, what is the measure of angle BOC?
Answer: C
In triangle ABC, Angle A + Angle B + Angle C = 180°. So, B + C = 180° - 70° = 110°.
The bisectors of B and C meet at O. In triangle BOC, the angles are Angle BOC, Angle OBC (which is B/2), and Angle OCB (which is C/2).
Angle BOC + B/2 + C/2 = 180°
Angle BOC + (B+C)/2 = 180°
Angle BOC + 110°/2 = 180°
Angle BOC + 55° = 180°
Angle BOC = 125°.
A standard formula for this is Angle BOC = 90° + (Angle A)/2 = 90 + 70/2 = 90 + 35 = 125°.
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The difference between two complementary angles is 40°. Find the angles.
Answer: A
Let the angles be x and y. Since they are complementary, x + y = 90°.
Their difference is x - y = 40°.
Adding the two equations: 2x = 130° => x = 65°.
Substituting x=65° into the first equation: 65° + y = 90° => y = 25°.
The angles are 65° and 25°.
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In \(\triangle ABC\), if the median AD is perpendicular to the side BC, then the triangle is:
Answer: C
A median from a vertex to the opposite side divides the opposite side into two equal parts (BD = DC). If this median is also perpendicular to the side, it acts as an altitude.
In \(\triangle ADB\) and \(\triangle ADC\), we have BD = DC, AD is common, and \(\angle ADB = \angle ADC = 90°\). By the SAS (Side-Angle-Side) congruence rule, \(\triangle ADB \cong \triangle ADC\).
Therefore, the corresponding sides AB and AC must be equal. A triangle with two equal sides is an isosceles triangle.
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