GSEB Class 7 Maths Solutions Chapter 6 The Triangles and Its Properties Exercise 6.3

Question 1. Find the value of the unknown x in the following diagrams.
Answer:
(i) Using the angle sum property of a triangle, we get:
\( 50° + 60° + x = 180° \)
\( 110° + x = 180° \)
\( x = 180° - 110° \)
\( x = 70° \)
Thus, the needed value of x is \( 70° \).
In simple words: For any triangle, all three inside angles always add up to 180 degrees. So, if you know two angles, you can easily find the third one by subtracting their sum from 180.

Exam Tip: Always remember that the sum of angles in any triangle is \( 180° \). Use this fundamental property to find unknown angles.

(ii) Using the angle sum property of a triangle, we get:
\( 30° + 90° + x = 180° \) [The triangle is right-angled at P, so one angle is \( 90° \).]
\( 120° + x = 180° \)
\( x = 180° - 120° \)
\( x = 60° \)
Thus, the needed value of x is \( 60° \).
In simple words: When a triangle has a right angle, that angle is 90 degrees. You can find the last angle by adding the known angles and taking that total away from 180 degrees.

Exam Tip: A square symbol in the corner of a triangle indicates a right angle, which is always \( 90° \).

(iii) Using the angle sum property of a triangle, we get:
\( 30° + 110° + x = 180° \)
\( 140° + x = 180° \)
\( x = 180° - 140° \)
\( x = 40° \)
Thus, the needed value of x is \( 40° \).
In simple words: Sum all the inside angles of a triangle and make sure they equal 180 degrees. This helps you calculate any missing angle.

Exam Tip: Double-check your addition and subtraction to prevent simple calculation errors.

(iv) Using the angle sum property of a triangle, we have:
\( x + x + 50° = 180° \)
\( 2x + 50° = 180° \)
\( 2x = 180° - 50° \)
\( 2x = 130° \)
\( \frac { 2x }{ 2 } = \frac { 130° }{ 2 } \)
\( x = 65° \)
In simple words: When two angles in a triangle are the same (like both 'x' here), you can add them together. Then, use the 180-degree rule to find the value of 'x'.

Exam Tip: If two angles are marked with the same variable (like x), they are equal. This usually means it's an isosceles triangle.

(v) Using the angle sum property of a triangle, we have:
\( x + x + x = 180° \)
\( 3x = 180° \)
\( \frac { 3x }{ 3 } = \frac { 180° }{ 3 } \)
\( x = 60° \)
In simple words: If all three angles in a triangle are the same, it means each angle must be 60 degrees, because 180 divided by 3 is 60.

Exam Tip: An equilateral triangle has all three angles equal to \( 60° \).

(vi) Using the angle sum property of a triangle, we have:
\( x + 2x + 90° = 180° \)
\( 3x + 90° = 180° \)
\( 3x = 180° - 90° \)
\( 3x = 90° \)
\( \frac { 3x }{ 3 } = \frac { 90° }{ 3 } \) [Dividing both sides by 3]
\( x = 30° \)
In simple words: In a right-angled triangle, if one angle is 90 degrees and the other two are in a simple ratio (like 'x' and '2x'), you can combine them, subtract the known angle from 180, and then solve to find 'x'.

Exam Tip: When terms with 'x' are present, combine them first before isolating 'x'.

Question 2. Find the values of the unknown x and y in the following diagrams:
Answer:
(i) Angles y and \( 120° \) form a linear pair.
\( y + 120° = 180° \)
\( y = 180° - 120° = 60° \)
Now, using the angle sum property of a triangle, we have
\( x + y + 50° = 180° \)
\( x + 60° + 50° = 180° \)
\( x + 110° = 180° \)
\( x = 180° - 110° = 70° \)
Thus, \( x = 70° \) and \( y = 60° \).
In simple words: First, use the straight line rule (angles add to 180 degrees) to find 'y'. After that, use the triangle angle rule (angles add to 180 degrees) to find 'x'.

Exam Tip: Always look for linear pairs or vertically opposite angles first, as they often simplify finding initial unknown values.

(ii) Angles y and \( 80° \) are vertically opposite angles, then \( y = 80° \).
Now, using the angle sum property of a triangle, we have
\( x + y + 50° = 180° \) [Using angle sum property]
\( x + 80° + 50° = 180° \)
\( x + 130° = 180° \)
\( x = 180° - 130° = 50° \)
Thus, \( x = 50° \) and \( y = 80° \).
In simple words: Start by using the rule for vertically opposite angles to figure out 'y'. Then, use the rule that all angles inside a triangle add up to 180 degrees to find 'x'.

Exam Tip: Vertically opposite angles are formed when two lines intersect, and they are always equal.

(iii) Using the angle sum property of a triangle, we have
\( 50° + 60° + y = 180° \)
\( y + 110° = 180° \)
\( y = 180° - 110° = 70° \)
Again, x and y form a linear pair.
\( x + y = 180° \)
\( x + 70° = 180° \)
\( x = 180° - 70° = 110° \)
Thus, \( x = 110° \) and \( y = 70° \).
In simple words: First, determine the unknown angle inside the triangle using the 180-degree rule. Next, use the linear pair rule with this interior angle to find the external angle.

Exam Tip: Be careful to distinguish between interior angles (inside the triangle) and exterior angles (forming a straight line with an interior angle).

(iv) x and \( 60° \) angle are vertically opposite angles.
\( x = 60° \)
Now, using the angle sum property of a triangle, we have
\( x + y + 30° = 180° \)
\( 60° + y + 30° = 180° \)
\( y + 90° = 180° \)
\( y = 180° - 90° = 90° \)
Thus, \( x = 60° \) and \( y = 90° \).
In simple words: First, find 'x' using the rule for vertically opposite angles. Then, use the sum of angles in a triangle to calculate 'y'.

Exam Tip: Always clearly identify which angles are vertically opposite or form a linear pair before applying the triangle angle sum property.

(v) y and \( 90° \) are vertically opposite angles, then \( y = 90° \).
Now, using the angle sum property of triangles, we have
\( x + x + y = 180° \)
\( 2x + y = 180° \)
\( 2x + 90° = 180° \)
\( 2x = 180° - 90° \)
\( 2x = 90° \)
\( \frac { 2x }{ 2 } = \frac { 90° }{ 2 } \)
\( x = 45° \)
Thus, \( x = 45° \) and \( y = 90° \).
In simple words: First, find 'y' by recognizing it as a vertically opposite angle. Then, knowing the triangle's angles add to 180 degrees, solve for 'x', remembering that the two 'x' angles are equal.

Exam Tip: For isosceles triangles (where two sides/angles are equal), the angles opposite the equal sides are also equal. This helps in setting up the equation.

(vi) One angle of the triangle = y
Each of the other two angles is equal to their vertically opposite angle x.
Using the angle sum property, we have
\( x + x + y = 180° \)
\( 2x + y = 180° \)
\( 2x + x = 180° \) [since \( x = y \) from vertically opposite angles]
\( 3x = 180° \)
\( \frac { 3x }{ 3 } = \frac { 180° }{ 3 } \)
\( x = 60° \)
But \( y = x \)
Therefore, \( y = 60° \)
Thus, \( x = 60° \) and \( y = 60° \).
In simple words: When a triangle has two equal interior angles ('x'), and the third interior angle ('y') is also equal to 'x' (due to vertically opposite angles), then all three angles are 'x'. This means it's an equilateral triangle, so each angle is 60 degrees.

Exam Tip: If all angles in a triangle are equal, the triangle is equilateral, and each angle measures \( 60° \).