The Triangles And Its Properties: CBSE Class 7 Maths

What is a Triangle? And Its Basic Properties

A triangle is a simple closed figure formed by three line segments. It's one of the most fundamental shapes in geometry. Every triangle has three important parts:

1. Vertices: These are the three points where the sides meet. We usually label them with capital letters, like A, B, C.
2. Sides: These are the three line segments that form the triangle. For a triangle ABC, the sides are AB, BC, and CA.
3. Angles: These are formed at each vertex by the meeting of two sides. A triangle has three interior angles, denoted as ∠A, ∠B, ∠C.

Classification of Triangles

Triangles can be classified in two main ways:

1. Based on Side Lengths:

• Scalene Triangle: All three sides have different lengths, and all three angles are different.
• Isosceles Triangle: Exactly two sides are equal in length, and the angles opposite to these equal sides are also equal.
• Equilateral Triangle: All three sides are equal in length, and all three angles are also equal (each measuring 60°).

2. Based on Angle Measures:

• Acute-angled Triangle: All three interior angles are acute (less than 90°).
• Right-angled Triangle: Exactly one interior angle is a right angle (equal to 90°). The side opposite the right angle is called the hypotenuse.
• Obtuse-angled Triangle: Exactly one interior angle is obtuse (greater than 90°).

The Angle Sum Property of a Triangle

One of the most important properties of any triangle is that the sum of the measures of its three interior angles is always 180 degrees. If a triangle has angles ∠A, ∠B, and ∠C, then ∠A + ∠B + ∠C = 180°.

Important Terms Related to Triangles

Median

A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. Every triangle has three medians, and they all intersect at a single point called the centroid.

Altitude

An altitude of a triangle is a perpendicular line segment from a vertex to the opposite side (or to the line containing the opposite side). It represents the height of the triangle from that vertex. Every triangle has three altitudes, and they intersect at a single point called the orthocentre.

Exterior Angle

When one side of a triangle is extended, the angle formed outside the triangle, adjacent to an interior angle, is called an exterior angle.

Interior Opposite Angles

For any exterior angle of a triangle, the two interior angles that are not adjacent to it are called its interior opposite angles.

Understanding the Exterior Angle Property of a Triangle

1. What is the Property? — The Exterior Angle Property states that an exterior angle of a triangle is equal to the sum of its two interior opposite angles. This is a very useful property for finding unknown angles.
2. Identify the Angles — Consider a triangle ABC. If you extend side BC to a point D, then ∠ACD is an exterior angle. The interior angles opposite to ∠ACD are ∠BAC (or ∠A) and ∠ABC (or ∠B). The interior angle adjacent to ∠ACD is ∠ACB (or ∠C).
3. Apply the Property — According to the property, the exterior angle ∠ACD will be equal to the sum of the two interior opposite angles: ∠ACD = ∠BAC + ∠ABC. Example: In a triangle, if ∠A = 50° and ∠B = 60°, then the exterior angle at C (formed by extending BC) would be 50° + 60° = 110°.
4. Solve for Unknowns — You can use this property to find an unknown interior angle if you know the exterior angle and one interior opposite angle, or to find the exterior angle if you know both interior opposite angles. Remember that the exterior angle and its adjacent interior angle form a linear pair, meaning their sum is 180°.

The Triangle Inequality Property

• Concept: For any three given line segments to form a triangle, the sum of the lengths of any two sides must always be greater than the length of the third side. This ensures that the sides can 'meet' to form a closed shape. Let the sides of a triangle be 'a', 'b', and 'c'. Then, the following three conditions must be true: 1. a + b > c 2. b + c > a 3. c + a > b
• Example 1: Can a triangle be formed with side lengths 3 cm, 4 cm, and 5 cm? Check 1: 3 + 4 = 7. Is 7 > 5? Yes. Check 2: 4 + 5 = 9. Is 9 > 3? Yes. * Check 3: 5 + 3 = 8. Is 8 > 4? Yes. Since all three conditions are met, yes, a triangle can be formed with these side lengths. (This is a right-angled triangle, often called a 3-4-5 triangle!)
• Example 2: Can a triangle be formed with side lengths 2 cm, 3 cm, and 6 cm? * Check 1: 2 + 3 = 5. Is 5 > 6? No, 5 is not greater than 6. Since one condition is not met, no, a triangle cannot be formed with these side lengths. The two shorter sides aren't long enough to meet if the third side is 6 cm.

Exam Tip: Avoiding Common Mistakes with Triangles

When solving problems related to triangles, students often make a few common errors. Be careful with these:

1. Confusing Medians and Altitudes: Remember, a median goes to the midpoint of the opposite side, while an altitude is perpendicular to the opposite side. They are different unless the triangle is equilateral or isosceles (for specific vertices).
2. Incorrectly Applying Exterior Angle Property: Ensure you sum the interior opposite angles, not the adjacent interior angle. The exterior angle and its adjacent interior angle always add up to 180° (linear pair).
3. Forgetting Angle Sum Property: Always verify that the three interior angles of any triangle add up to exactly 180°. This is a fundamental check for many problems.
4. Triangle Inequality: Don't just check one pair of sides. All three sums (a+b>c, b+c>a, c+a>b) must hold true for a triangle to be possible.

Practice Questions with Solutions

• Q: In triangle PQR, ∠P = 70° and ∠Q = 50°. Find the measure of ∠R. A: Step 1: Recall the Angle Sum Property of a triangle, which states that the sum of all three interior angles is 180°. Step 2: So, ∠P + ∠Q + ∠R = 180°. Step 3: Substitute the given values: 70° + 50° + ∠R = 180°. Step 4: Add the known angles: 120° + ∠R = 180°. Step 5: Isolate ∠R: ∠R = 180° - 120°. Final answer: ∠R = 60°.
• Q: An exterior angle of a triangle is 110°. Its interior opposite angles are in the ratio 2:3. Find the measure of these two interior opposite angles. A: Step 1: Let the two interior opposite angles be 2x and 3x. Step 2: According to the Exterior Angle Property, the exterior angle is equal to the sum of the two interior opposite angles. So, 2x + 3x = 110°. Step 3: Combine like terms: 5x = 110°. Step 4: Solve for x: x = 110° / 5 = 22°. Step 5: Calculate the measures of the angles: First angle = 2x = 2 22° = 44°. Second angle = 3x = 3 22° = 66°. Final answer: The two interior opposite angles are 44° and 66°.
• Q: Can a triangle have sides with lengths 7 cm, 8 cm, and 16 cm? Justify your answer. A: Step 1: Apply the Triangle Inequality Property. The sum of any two sides must be greater than the third side. Step 2: Check the first condition: 7 cm + 8 cm = 15 cm. Is 15 cm > 16 cm? No, it is not. Step 3: Since one condition fails, we do not need to check the others. Final answer: No, a triangle cannot be formed with side lengths 7 cm, 8 cm, and 16 cm because the sum of the two shorter sides (7+8=15) is not greater than the longest side (16).
• Q: In triangle XYZ, M is the midpoint of side YZ. What is the line segment XM called? A: Step 1: Recall the definition of a median. Step 2: A median connects a vertex to the midpoint of the opposite side. Step 3: Here, XM connects vertex X to the midpoint M of the opposite side YZ. Final answer: The line segment XM is called a median of triangle XYZ.

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