The Triangle and its Properties — Important Questions
25 questions
With answersCBSE format
SUMMARY: This chapter explores the fundamental properties and types of triangles, including their angles and sides. KEY TOPICS: types of triangles, angle sum property, exterior angle property, Pythagoras theorem, medians of a triangle, altitudes of a triangle, equilateral triangle, isosceles triangle, scalene triangle, right-angled triangle
What is the sum of the interior angles of a triangle?
A90 degrees
B180 degrees
C270 degrees
D360 degrees
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Correct answer: Option 2 — 180 degrees
Q21 Mark
Which type of triangle has all sides of equal length?
AScalene triangle
BIsosceles triangle
CEquilateral triangle
DRight-angled triangle
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Correct answer: Option 3 — Equilateral triangle
Q31 Mark
In a right-angled triangle, which theorem is used to relate the lengths of the sides?
APythagorean theorem
BTriangle inequality theorem
CAngle sum property
DExterior angle property
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Correct answer: Option 1 — Pythagorean theorem
Q41 Mark
If one angle of a triangle is 70 degrees and another is 50 degrees, what is the measure of the third angle?
A60 degrees
B70 degrees
C80 degrees
D90 degrees
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Correct answer: Option 1 — 60 degrees
Q51 Mark
In triangle ABC, if AD is the median from vertex A to side BC, which of the following is true?
AAD bisects angle A
BAD is equal to side BC
CBD = DC
DAD is perpendicular to BC
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Correct answer: Option 3 — BD = DC
Short Answer Questions5 questions
Q63 Marks
What are the three types of triangles based on their sides?
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The three types of triangles based on their sides are equilateral triangles, isosceles triangles, and scalene triangles. An equilateral triangle has all three sides equal, an isosceles triangle has two sides equal, and a scalene triangle has all sides of different lengths.
Q73 Marks
State the angle sum property of a triangle. How can it be used to find an unknown angle?
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The angle sum property states that the sum of the interior angles of a triangle is always 180 degrees. To find an unknown angle, you can subtract the sum of the known angles from 180 degrees.
Q83 Marks
Explain the exterior angle property of a triangle. How does it relate to the interior angles?
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The exterior angle property states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles. This means that if you know the measures of the interior angles, you can find the exterior angle and vice versa.
Q93 Marks
What is the Pythagorean theorem and in which type of triangle is it applicable?
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The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. It is applicable only in right-angled triangles.
Q103 Marks
Define the terms 'median' and 'altitude' in the context of a triangle. How do they differ?
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A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, while an altitude is a perpendicular segment from a vertex to the line containing the opposite side. The key difference is that a median does not have to be perpendicular, whereas an altitude must be.
Long Answer Questions5 questions
Q116 Marks
Explain the properties of an equilateral triangle. How do the angles and sides compare with other types of triangles? Provide examples to illustrate your points.
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An equilateral triangle is defined as a triangle in which all three sides are of equal length. Consequently, all three interior angles are also equal, each measuring 60 degrees. This property distinguishes equilateral triangles from other types, such as isosceles triangles, which have two equal sides and angles, and scalene triangles, which have no equal sides or angles. For example, if an equilateral triangle has a side length of 6 cm, each angle will measure 60 degrees, showcasing the unique balance of sides and angles in this type of triangle.
Q126 Marks
Describe the angle sum property of triangles. How can this property be used to determine the measure of an unknown angle in a triangle? Provide a specific example.
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The angle sum property states that the sum of the interior angles of a triangle is always 180 degrees. This property can be utilized to find an unknown angle when the measures of the other two angles are known. For instance, in a triangle where one angle measures 50 degrees and another measures 70 degrees, the measure of the third angle can be calculated as follows: 180 - (50 + 70) = 60 degrees. This example illustrates how the angle sum property is fundamental in solving problems related to triangles.
Q136 Marks
What is the exterior angle property of triangles? Explain how this property can be applied to find the measure of an angle in a triangle with an example.
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The exterior angle property states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles. This property can be applied to find an unknown angle in a triangle. For example, consider a triangle where one exterior angle measures 120 degrees, and the two opposite interior angles measure 40 degrees and x degrees. According to the property, we can set up the equation: 120 = 40 + x. Solving for x gives us x = 80 degrees, demonstrating the practical application of the exterior angle property.
Q146 Marks
Define the Pythagorean theorem and explain its significance in right-angled triangles. Provide an example of how to use the theorem to find the length of a side.
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The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem is significant as it provides a method to calculate the length of one side when the lengths of the other two sides are known. For example, if one side measures 3 cm and the other side measures 4 cm, the length of the hypotenuse can be calculated using the formula: c² = a² + b², where c is the hypotenuse. Thus, c² = 3² + 4² = 9 + 16 = 25, leading to c = 5 cm.
Q156 Marks
Discuss the concepts of medians and altitudes in triangles. How do they differ, and what are their respective properties? Provide examples to clarify your explanation.
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Medians and altitudes are both important segments in triangles, but they serve different purposes. A median is a line segment that connects a vertex of the triangle to the midpoint of the opposite side, effectively dividing the triangle into two smaller triangles of equal area. In contrast, an altitude is a perpendicular segment from a vertex to the line containing the opposite side, representing the height of the triangle. For example, in a triangle with vertices A, B, and C, if D is the midpoint of side BC, then AD is the median. If E is the foot of the perpendicular from A to BC, then AE is the altitude. Understanding these concepts is crucial for solving various geometric problems.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): An equilateral triangle has all sides of equal length.
Reason (R): In an equilateral triangle, all angles are also equal and measure 60 degrees.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q171 Mark
Assertion (A): The sum of the interior angles of a triangle is always 180 degrees.
Reason (R): This is known as the angle sum property of triangles.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q181 Mark
Assertion (A): A right-angled triangle can have two obtuse angles.
Reason (R): In a triangle, the sum of the angles must equal 180 degrees.
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Correct answer: Option 4 —
A is false, but R is true.
Q191 Mark
Assertion (A): The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Reason (R): This is a property of exterior angles in triangles.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q201 Mark
Assertion (A): The median of a triangle divides it into two equal areas.
Reason (R): A median connects a vertex to the midpoint of the opposite side.
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Correct answer: Option 2 —
Both A and R are true, but R is not the correct explanation of A.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: An equilateral triangle has all three sides of equal length.
Statement 2: The sum of the interior angles of a triangle is 180 degrees.
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Correct answer: Option 1 —
Both statements are true.
Q221 Mark
Statement 1: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Statement 2: An isosceles triangle has all three angles equal.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q231 Mark
Statement 1: The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Statement 2: A scalene triangle has at least two sides of equal length.
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Correct answer: Option 4 —
Both statements are false.
Q241 Mark
Statement 1: The median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.
Statement 2: All triangles have at least one right angle.
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Correct answer: Option 2 —
Only Statement 1 is true.
Q251 Mark
Statement 1: In any triangle, the longest side is opposite the largest angle.
Statement 2: The altitude of a triangle is always longer than any of its sides.
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Correct answer: Option 4 —
Both statements are false.
