SUMMARY: The chapter on Triangles in Class 10 Mathematics focuses on the properties and criteria for the congruence and similarity of triangles, along with related theorems and applications. KEY TOPICS: congruence of triangles, similarity of triangles, Pythagoras theorem, criteria for similarity, basic proportionality theorem, area of similar triangles, right-angled triangles, applications of similarity, criteria for congruence, properties of triangles
In triangle ABC, if angle A = 60° and angle B = 70°, what is the measure of angle C?
A50°
B60°
C70°
D80°
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Correct answer: Option 1 — 50°
Q21 Mark
Which of the following is a property of similar triangles?
ATheir corresponding angles are equal.
BTheir corresponding sides are equal.
CTheir areas are equal.
DTheir perimeters are equal.
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Correct answer: Option 1 — Their corresponding angles are equal.
Q31 Mark
In triangle PQR, if PQ = 8 cm, QR = 6 cm, and PR = 10 cm, which of the following statements is true?
ATriangle PQR is a right triangle.
BTriangle PQR is an equilateral triangle.
CTriangle PQR is an isosceles triangle.
DTriangle PQR is a scalene triangle.
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Correct answer: Option 4 — Triangle PQR is a scalene triangle.
Q41 Mark
If two triangles are congruent, which of the following is NOT necessarily true?
ATheir corresponding angles are equal.
BTheir corresponding sides are equal.
CThey have the same area.
DThey are similar triangles.
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Correct answer: Option 4 — They are similar triangles.
Q51 Mark
In triangle XYZ, if the lengths of sides XY, YZ, and XZ are in the ratio 3:4:5, what type of triangle is it?
AAcute triangle
BObtuse triangle
CRight triangle
DEquilateral triangle
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Correct answer: Option 3 — Right triangle
Short Answer Questions5 questions
Q63 Marks
What is the sum of the interior angles of a triangle?
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The sum of the interior angles of a triangle is always 180 degrees.
Q73 Marks
Define an isosceles triangle and give an example.
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An isosceles triangle is a triangle with at least two sides of equal length. An example is a triangle with sides measuring 5 cm, 5 cm, and 8 cm.
Q83 Marks
State the Pythagorean theorem and its application in right-angled triangles.
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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 used to find the length of a side when the lengths of the other two sides are known.
Q93 Marks
Explain the criteria for similarity of triangles.
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Two triangles are similar if their corresponding angles are equal and the lengths of their corresponding sides are in proportion. This can be established using criteria such as AA (Angle-Angle), SSS (Side-Side-Side), or SAS (Side-Angle-Side).
Q103 Marks
How can you prove that the angles opposite to equal sides of an isosceles triangle are equal?
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To prove that the angles opposite to equal sides of an isosceles triangle are equal, we can use the properties of congruent triangles. By drawing a perpendicular from the vertex angle to the base, we create two congruent triangles, which show that the angles opposite the equal sides are equal by the converse of the Isosceles Triangle Theorem.
Long Answer Questions5 questions
Q116 Marks
Prove that the sum of the angles in a triangle is 180 degrees using the properties of parallel lines and transversals.
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To prove that the sum of the angles in a triangle is 180 degrees, consider triangle ABC. Extend the line segment BC to a point D. Draw a line parallel to line AC through point B. According to the properties of parallel lines, the angle formed at point B (angle ABC) is equal to the angle formed at point D (angle ACD). Similarly, the angle at point C (angle ACB) is equal to the angle formed at point E (angle ABD) where line AB intersects the parallel line at point E. Therefore, angle ABC + angle ACB + angle A = angle ACD + angle ABD + angle A. Since angle ACD + angle ABD + angle A equals 180 degrees, we conclude that the sum of the angles in triangle ABC is 180 degrees.
Q126 Marks
In triangle ABC, if angle A = 40 degrees and angle B = 60 degrees, find angle C and justify your answer using the properties of triangles.
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In triangle ABC, we know that the sum of the angles must equal 180 degrees. Given that angle A is 40 degrees and angle B is 60 degrees, we can find angle C by using the equation: angle A + angle B + angle C = 180 degrees. Substituting the known values, we have 40 + 60 + angle C = 180. Simplifying this, we find that angle C = 180 - 100 = 80 degrees. Thus, angle C is 80 degrees, which is justified by the fundamental property that the sum of the interior angles of a triangle is always 180 degrees.
Q136 Marks
Using the Pythagorean theorem, demonstrate that a triangle with sides of lengths 3 cm, 4 cm, and 5 cm is a right triangle.
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To demonstrate that a triangle with sides of lengths 3 cm, 4 cm, and 5 cm is a right triangle, we apply the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides. Here, we identify 5 cm as the hypotenuse. Thus, we calculate: 5^2 = 25, and 3^2 + 4^2 = 9 + 16 = 25. Since both sides of the equation are equal (25 = 25), we conclude that the triangle with sides 3 cm, 4 cm, and 5 cm is indeed a right triangle.
Q146 Marks
Explain the criteria for similarity of triangles and provide an example to illustrate your explanation.
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The criteria for similarity of triangles include the Angle-Angle (AA) criterion, the Side-Side-Side (SSS) criterion, and the Side-Angle-Side (SAS) criterion. According to the AA criterion, if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. For example, consider triangles ABC and DEF where angle A = angle D and angle B = angle E. By the AA criterion, triangles ABC and DEF are similar, meaning their corresponding sides are in proportion. If side AB is 4 cm and side DE is 8 cm, then the ratio of the sides is 1:2, confirming the similarity of the triangles.
Q156 Marks
A triangle has sides of lengths 7 cm, 24 cm, and 25 cm. Determine whether it is a right triangle and justify your answer using the converse of the Pythagorean theorem.
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To determine if the triangle with sides of lengths 7 cm, 24 cm, and 25 cm is a right triangle, we can use the converse of the Pythagorean theorem. According to this theorem, if the square of the longest side is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. Here, we identify 25 cm as the longest side. We calculate: 25^2 = 625 and 7^2 + 24^2 = 49 + 576 = 625. Since both sides of the equation are equal (625 = 625), we conclude that the triangle with sides 7 cm, 24 cm, and 25 cm is indeed a right triangle, confirming our result using the converse of the Pythagorean theorem.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): In a triangle, the sum of the lengths of any two sides is greater than the length of the third side.
Reason (R): This property is known as the Triangle Inequality Theorem.
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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): If two angles of a triangle are equal, then the sides opposite to those angles are also equal.
Reason (R): This is a property of isosceles 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): In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Reason (R): This is known as the Pythagorean theorem, which applies to all triangles.
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Correct answer: Option 3 —
A is true, but R is false.
Q191 Mark
Assertion (A): The area of a triangle can be calculated using the formula A = 1/2 * base * height.
Reason (R): This formula is applicable only to right-angled triangles.
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Correct answer: Option 3 —
A is true, but R is false.
Q201 Mark
Assertion (A): The angles of a triangle always add up to 180 degrees.
Reason (R): This is true for all triangles regardless of their type.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: The sum of the angles in a triangle is 180 degrees.
Statement 2: A triangle can have two right angles.
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Correct answer: Option 2 —
Only Statement 1 is true.
Q221 Mark
Statement 1: In an isosceles triangle, the angles opposite to the equal sides are equal.
Statement 2: The area of a triangle can be calculated using the formula A = 1/2 * base * height.
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Correct answer: Option 1 —
Both statements are true.
Q231 Mark
Statement 1: All triangles are similar if they have the same perimeter.
Statement 2: The Pythagorean theorem applies only to right-angled triangles.
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Correct answer: Option 1 —
Both statements are true.
Q241 Mark
Statement 1: The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
Statement 2: A triangle can have more than one obtuse angle.
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Correct answer: Option 4 —
Both statements are false.
Q251 Mark
Statement 1: In a right triangle, the hypotenuse is always the longest side.
Statement 2: The centroid of a triangle is the point where the three medians intersect.
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Correct answer: Option 1 —
Both statements are true.
