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1.
Prove that angles opposite to equal sides of an isosceles triangle are equal.2.
In a triangle ABC, E and F respectively are mid-points of equal sides AB and AC of ΔABC. Show that BF = CE.3.
AD is an altitude of an isosceles ΔABC in which AB = AC. Show that AD bisects BC.4.
D is a point on side BC of ΔABC such that AD = AC. Show that AB > AD.5.
In ΔABC, if BC = AB and └B=80� then find the measure of └�A.6.
The angles of a triangle are in the ratio 2:3:4. Find the measure of the angles.7.
In ΔABC, if └A = 80�, └B = 70�, then identify the longest and the shortest side of the triangle.8.
ABCD is a square. P is any point inside it such that, DPQR is another square. Prove that AP = CR.9.
In a ΔABC, if └A = └B, then what is AB : BC10.
Prove that any two sides of a triangle are together greater than twice the median drawn to thethird side.Community Answer
Needed a Test for traingles? Related: Examples NCERT Properties of T...
Properties of Triangles
Triangles are one of the fundamental shapes in geometry. They have three sides and three angles. In this section, we will explore the properties of triangles. Understanding these properties is essential for solving problems related to triangles.
1. Sum of Interior Angles:
The sum of the three interior angles of a triangle is always 180 degrees. This property is known as the Angle Sum Property of a Triangle. It can be represented by the equation: ∠A + ∠B + ∠C = 180°, where ∠A, ∠B, and ∠C are the angles of the triangle.
2. Exterior Angle Property:
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. In other words, the exterior angle is equal to the sum of the opposite interior angles. This property can be expressed as: ∠D = ∠A + ∠B, where ∠D is the exterior angle and ∠A and ∠B are the interior angles.
3. Triangle Inequality Theorem:
According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Mathematically, for a triangle with sides a, b, and c, the theorem can be stated as: a + b > c, b + c > a, and c + a > b.
4. Types of Triangles:
Based on the length of their sides, triangles can be classified into three types:
- Scalene Triangle: A triangle with all three sides of different lengths.
- Isosceles Triangle: A triangle with two sides of equal length.
- Equilateral Triangle: A triangle with all three sides of equal length.
5. Types of Triangles Based on Angles:
Based on the measures of their angles, triangles can be classified into three types:
- Acute Triangle: A triangle with all three angles measuring less than 90 degrees.
- Obtuse Triangle: A triangle with one angle measuring more than 90 degrees.
- Right Triangle: A triangle with one angle measuring exactly 90 degrees.
6. Pythagorean Theorem:
The Pythagorean Theorem states that in a right 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. This theorem is represented by the equation: a² + b² = c², where c is the hypotenuse and a and b are the other two sides.
7. Congruence of Triangles:
Two triangles are said to be congruent if their corresponding sides and angles are equal. Congruent triangles have the same shape and size.
Understanding these properties of triangles is crucial for solving geometric problems involving triangles. By applying these properties, we can determine the measures of angles, lengths of sides, and classify different types of triangles.
Triangles are one of the fundamental shapes in geometry. They have three sides and three angles. In this section, we will explore the properties of triangles. Understanding these properties is essential for solving problems related to triangles.
1. Sum of Interior Angles:
The sum of the three interior angles of a triangle is always 180 degrees. This property is known as the Angle Sum Property of a Triangle. It can be represented by the equation: ∠A + ∠B + ∠C = 180°, where ∠A, ∠B, and ∠C are the angles of the triangle.
2. Exterior Angle Property:
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. In other words, the exterior angle is equal to the sum of the opposite interior angles. This property can be expressed as: ∠D = ∠A + ∠B, where ∠D is the exterior angle and ∠A and ∠B are the interior angles.
3. Triangle Inequality Theorem:
According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Mathematically, for a triangle with sides a, b, and c, the theorem can be stated as: a + b > c, b + c > a, and c + a > b.
4. Types of Triangles:
Based on the length of their sides, triangles can be classified into three types:
- Scalene Triangle: A triangle with all three sides of different lengths.
- Isosceles Triangle: A triangle with two sides of equal length.
- Equilateral Triangle: A triangle with all three sides of equal length.
5. Types of Triangles Based on Angles:
Based on the measures of their angles, triangles can be classified into three types:
- Acute Triangle: A triangle with all three angles measuring less than 90 degrees.
- Obtuse Triangle: A triangle with one angle measuring more than 90 degrees.
- Right Triangle: A triangle with one angle measuring exactly 90 degrees.
6. Pythagorean Theorem:
The Pythagorean Theorem states that in a right 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. This theorem is represented by the equation: a² + b² = c², where c is the hypotenuse and a and b are the other two sides.
7. Congruence of Triangles:
Two triangles are said to be congruent if their corresponding sides and angles are equal. Congruent triangles have the same shape and size.
Understanding these properties of triangles is crucial for solving geometric problems involving triangles. By applying these properties, we can determine the measures of angles, lengths of sides, and classify different types of triangles.
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Needed a Test for traingles? Related: Examples NCERT Properties of Triangles - Triangles, Class 9, Mathematics
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Needed a Test for traingles? Related: Examples NCERT Properties of Triangles - Triangles, Class 9, Mathematics for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about Needed a Test for traingles? Related: Examples NCERT Properties of Triangles - Triangles, Class 9, Mathematics covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Needed a Test for traingles? Related: Examples NCERT Properties of Triangles - Triangles, Class 9, Mathematics.
Needed a Test for traingles? Related: Examples NCERT Properties of Triangles - Triangles, Class 9, Mathematics for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about Needed a Test for traingles? Related: Examples NCERT Properties of Triangles - Triangles, Class 9, Mathematics covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Needed a Test for traingles? Related: Examples NCERT Properties of Triangles - Triangles, Class 9, Mathematics.
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