CBSE Class 7 Mathematics Triangles and Its Properties Assignment Set 09

Question 1. Can a \( \Delta \) have :
(a) All angles equal to 60°
(b) All angles more than 60°
(c) All angles less than 60°
(d) Two right angles
(e) Two acute angles
(f) Two obtuse angles.
Answer:
(a) Yes. An equilateral triangle has three angles that are all exactly \( 60^{\circ} \) because \( 60^{\circ} + 60^{\circ} + 60^{\circ} = 180^{\circ} \).
(b) No. If each angle is bigger than \( 60^{\circ} \), the total sum will be more than \( 180^{\circ} \), which is impossible.
(c) No. If each angle is smaller than \( 60^{\circ} \), the total sum will be less than \( 180^{\circ} \), which is not allowed.
(d) No. Two right angles would make \( 90^{\circ} + 90^{\circ} = 180^{\circ} \), leaving no degrees for the third angle.
(e) Yes. A triangle can easily have two angles that are less than \( 90^{\circ} \). Every triangle has at least two of these.
(f) No. Two obtuse angles would add up to more than \( 180^{\circ} \) on their own.
In simple words: The three angles inside any triangle must always add up to exactly \( 180^{\circ} \). You can never have a sum that is smaller or larger than this magic number.

Exam Tip: Remember that every triangle must contain at least two acute angles. Always use the angle sum rule to check your answers.

Question 2. One of the angle of a \( \Delta \) is 110° And other two angles are equal. Find the measurement of equal angles.
Answer: Let the measure of each equal angle be \( a \). By using the angle sum property of a triangle:
\( 110^{\circ} + a + a = 180^{\circ} \)
\( \implies 2a = 180^{\circ} - 110^{\circ} \)
\( \implies 2a = 70^{\circ} \)
\( \implies a = 35^{\circ} \)
So, the size of each equal angle is \( 35^{\circ} \).
In simple words: All three angles in a triangle must add up to \( 180^{\circ} \). Subtract the known angle from \( 180^{\circ} \), then divide the rest equally between the other two angles.

Exam Tip: Always state the angle sum property of a triangle clearly in your steps to secure full marks.

Question 3. One of the Exterior angle of a \( \Delta \) is 130°. If the interior opposite angles are equal, find each angle of the \( \Delta \).
Answer: Let each of the equal interior opposite angles be \( b \). Using the exterior angle property, we know that the exterior angle equals the sum of its interior opposite angles:
\( b + b = 130^{\circ} \)
\( \implies 2b = 130^{\circ} \)
\( \implies b = 65^{\circ} \)
So, the two equal opposite angles are \( 65^{\circ} \) each. The third interior angle is:
\( 180^{\circ} - 130^{\circ} = 50^{\circ} \)
Therefore, the angles of the triangle are \( 65^{\circ} \), \( 65^{\circ} \), and \( 50^{\circ} \).
In simple words: The outside angle is equal to the two inside opposite angles added together. Since those two are equal, we just split the outside angle in half.

Exam Tip: Do not forget to find all three angles of the triangle, as the question asks for each angle of the triangle.

Question 4. Find the measurement of each angle of an Equilateral \( \Delta \)
Answer: Let each angle of the equilateral triangle be \( c \). Since all sides of an equilateral triangle are equal, all its angles are also equal:
\( c + c + c = 180^{\circ} \)
\( \implies 3c = 180^{\circ} \)
\( \implies c = 60^{\circ} \)
Thus, every angle of an equilateral triangle is always \( 60^{\circ} \).
In simple words: Since all three angles of this triangle are the same, we divide the total of \( 180^{\circ} \) into three equal parts. Each part is \( 60^{\circ} \).

Exam Tip: Remember that an equilateral triangle always has angles of exactly \( 60^{\circ} \), no matter how big or small its sides are.

Question 5. Find x, y.
Answer: Since the interior angle \( x \) and the exterior angle of \( 120^{\circ} \) lie on a straight line, they form a linear pair:
\( x + 120^{\circ} = 180^{\circ} \)
\( \implies x = 180^{\circ} - 120^{\circ} \)
\( \implies x = 60^{\circ} \)
Now, in \( \Delta ABC \), the sum of all interior angles is \( 180^{\circ} \):
\( 45^{\circ} + y + x = 180^{\circ} \)
\( \implies 45^{\circ} + y + 60^{\circ} = 180^{\circ} \)
\( \implies y + 105^{\circ} = 180^{\circ} \)
\( \implies y = 75^{\circ} \)
Thus, \( x = 60^{\circ} \) and \( y = 75^{\circ} \).
In simple words: The angle inside and the angle outside at corner C make a straight line, so they add up to \( 180^{\circ} \). After finding \( x \), use the \( 180^{\circ} \) total rule to find \( y \).

Exam Tip: A quick way to check your work is to make sure the exterior angle (\( 120^{\circ} \)) equals the sum of the two opposite interior angles (\( 45^{\circ} + 75^{\circ} \)).

Question 6. Find x, y.
Answer: Since the interior angle \( y \) and the exterior angle of \( 150^{\circ} \) lie on a straight line, they form a linear pair:
\( y + 150^{\circ} = 180^{\circ} \)
\( \implies y = 180^{\circ} - 150^{\circ} \)
\( \implies y = 30^{\circ} \)
In the right-angled triangle \( \Delta ABC \), the sum of angles is \( 180^{\circ} \) and the angle at B is \( 90^{\circ} \):
\( x + 90^{\circ} + y = 180^{\circ} \)
\( \implies x + 90^{\circ} + 30^{\circ} = 180^{\circ} \)
\( \implies x + 120^{\circ} = 180^{\circ} \)
\( \implies x = 60^{\circ} \)
Thus, \( x = 60^{\circ} \) and \( y = 30^{\circ} \).
In simple words: The angle inside and the angle outside at corner C lie on a straight line and add up to \( 180^{\circ} \). Once you find \( y = 30^{\circ} \), use the \( 180^{\circ} \) total rule to find \( x \).

Exam Tip: When a square symbol is shown at a corner, it always means a right angle (\( 90^{\circ} \)). Do not wait for the question to say it!

Question 7. Find x, y.
Answer: In \( \Delta ABC \), the sum of all interior angles is \( 180^{\circ} \):
\( 55^{\circ} + 60^{\circ} + y = 180^{\circ} \)
\( \implies 115^{\circ} + y = 180^{\circ} \)
\( \implies y = 65^{\circ} \)
Since the line DAB is a straight line, the angles \( x \) and \( 55^{\circ} \) form a linear pair:
\( x + 55^{\circ} = 180^{\circ} \)
\( \implies x = 180^{\circ} - 55^{\circ} \)
\( \implies x = 125^{\circ} \)
Thus, \( x = 125^{\circ} \) and \( y = 65^{\circ} \).
In simple words: The three angles inside the triangle must add up to \( 180^{\circ} \), which gives \( y = 65^{\circ} \). Since DAB is a straight line, \( x \) and \( 55^{\circ} \) add up to \( 180^{\circ} \) as well.

Exam Tip: When a side is extended, the interior and exterior angles at that vertex form a straight line and always add up to \( 180^{\circ} \).

Question 8. Find x, y.
Answer: At vertex C, the angle inside the triangle and the opposite angle \( x \) are vertically opposite angles. Since vertically opposite angles are equal:
\( x = 40^{\circ} \)
Now, in \( \Delta ABC \), the sum of all interior angles is \( 180^{\circ} \):
\( 70^{\circ} + y + 40^{\circ} = 180^{\circ} \)
\( \implies y + 110^{\circ} = 180^{\circ} \)
\( \implies y = 180^{\circ} - 110^{\circ} \)
\( \implies y = 70^{\circ} \)
Thus, \( x = 40^{\circ} \) and \( y = 70^{\circ} \).
In simple words: Angles that are directly opposite each other across an intersection are always equal. This makes \( x = 40^{\circ} \). Then, use the \( 180^{\circ} \) sum rule inside the triangle to find \( y \).

Exam Tip: Vertically opposite angles are equal. Identifying them first can instantly simplify complex multi-step angle problems.

Question 9. Find x, y, z.
Answer: Since C lies on the straight line segment BE, the angles on either side form a linear pair:
\( 120^{\circ} + x = 180^{\circ} \)
\( \implies x = 180^{\circ} - 120^{\circ} \)
\( \implies x = 60^{\circ} \)
Now, in \( \Delta ACE \), the sum of all interior angles is \( 180^{\circ} \):
\( x + y + 40^{\circ} = 180^{\circ} \)
\( \implies 60^{\circ} + y + 40^{\circ} = 180^{\circ} \)
\( \implies y + 100^{\circ} = 180^{\circ} \)
\( \implies y = 80^{\circ} \)
Since CAD is a straight line, the angles \( z \) and \( 40^{\circ} \) form a linear pair:
\( z + 40^{\circ} = 180^{\circ} \)
\( \implies z = 180^{\circ} - 40^{\circ} \)
\( \implies z = 140^{\circ} \)
Thus, \( x = 60^{\circ} \), \( y = 80^{\circ} \), and \( z = 140^{\circ} \).
In simple words: The line at the bottom and the line running up through A are both straight lines. Use the \( 180^{\circ} \) straight-line rule to find \( x \) and \( z \), and then the triangle rule for \( y \).

Exam Tip: Be careful to identify which lines are straight and continuous so you can apply the linear pair rule correctly.

Question 10. Find x, y.
Answer: In this figure, we can solve for \( x \) and \( y \) by using the exterior angle theorem:
1. For the left triangle, the exterior angle \( x \) at vertex D is equal to the sum of its two interior opposite angles, which are \( 40^{\circ} \) and \( 35^{\circ} \):
\( x = 40^{\circ} + 35^{\circ} \)
\( \implies x = 75^{\circ} \)
2. For the main triangle, the exterior angle \( y \) at vertex A is equal to the sum of the two interior opposite angles at C and E, which are \( 40^{\circ} \) and \( 45^{\circ} \):
\( y = 40^{\circ} + 45^{\circ} \)
\( \implies y = 85^{\circ} \)
Thus, \( x = 75^{\circ} \) and \( y = 85^{\circ} \).
In simple words: An outside angle of a triangle is always equal to the two inside opposite angles added together. We use this handy rule to find both \( x \) and \( y \) directly.

Exam Tip: The exterior angle theorem is extremely powerful and saves you from having to calculate intermediate angles first.