CBSE Class 7 Mathematics The Triangle And Its Properties Worksheet - Free Printable

To solve the problems presented in the image, we need to analyze each triangle and use geometric properties, trigonometric relationships, or the Pythagorean theorem where applicable. Let's go through each part step by step.

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a)

Given:
- Triangle with angles \(60^\circ\) and \(55^\circ\).
- We need to find the third angle \(x\).

Solution:
The sum of the angles in a triangle is always \(180^\circ\). Therefore:
\[
x = 180^\circ - 60^\circ - 55^\circ = 65^\circ
\]

Answer:
\[
\boxed{65^\circ}
\]

---

b)

Given:
- A right triangle with legs of lengths 12 cm and 15 cm.
- We need to find the hypotenuse \(m\).

Solution:
Using the Pythagorean theorem:
\[
m^2 = 12^2 + 15^2 = 144 + 225 = 369
\]
\[
m = \sqrt{369} = 3\sqrt{41} \approx 19.21 \text{ cm}
\]

Answer:
\[
\boxed{3\sqrt{41} \text{ cm}}
\]

---

c)

Given:
- An isosceles triangle with one angle \(60^\circ\).
- We need to find the other two angles \(y\) and \(x\).

Solution:
In an isosceles triangle, the base angles are equal. Since one angle is \(60^\circ\), the triangle is actually equilateral (all angles are \(60^\circ\)). Therefore:
\[
y = 60^\circ \quad \text{and} \quad x = 60^\circ
\]

Answer:
\[
\boxed{60^\circ, 60^\circ}
\]

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d)

Given:
- A right triangle with legs of lengths 3 cm and 5 cm.
- We need to find the hypotenuse \(b\).

Solution:
Using the Pythagorean theorem:
\[
b^2 = 3^2 + 5^2 = 9 + 25 = 34
\]
\[
b = \sqrt{34} \approx 5.83 \text{ cm}
\]

Answer:
\[
\boxed{\sqrt{34} \text{ cm}}
\]

---

e)

Given:
- An isosceles triangle with a vertex angle of \(110^\circ\).
- We need to find the base angles \(p\) and \(r\).

Solution:
In an isosceles triangle, the base angles are equal. The sum of the angles in a triangle is \(180^\circ\). Therefore:
\[
p + r + 110^\circ = 180^\circ
\]
Since \(p = r\):
\[
2p + 110^\circ = 180^\circ
\]
\[
2p = 70^\circ
\]
\[
p = 35^\circ
\]
Thus, \(r = 35^\circ\).

Answer:
\[
\boxed{35^\circ, 35^\circ}
\]

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f)

Given:
- A right triangle with one leg of length 24 cm and the hypotenuse of length 25 cm.
- We need to find the other leg \(x\).

Solution:
Using the Pythagorean theorem:
\[
25^2 = 24^2 + x^2
\]
\[
625 = 576 + x^2
\]
\[
x^2 = 49
\]
\[
x = 7 \text{ cm}
\]

Answer:
\[
\boxed{7 \text{ cm}}
\]

---

g)

Given:
- A right triangle with one leg of length 5 cm and the hypotenuse of length 13 cm.
- We need to find the other leg \(y\).

Solution:
Using the Pythagorean theorem:
\[
13^2 = 5^2 + y^2
\]
\[
169 = 25 + y^2
\]
\[
y^2 = 144
\]
\[
y = 12 \text{ cm}
\]

Answer:
\[
\boxed{12 \text{ cm}}
\]

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h)

Given:
- A triangle with sides \(5a\), \(3a\), and \(a\).
- We need to determine if this forms a valid triangle.

Solution:
For a triangle to be valid, the sum of the lengths of any two sides must be greater than the length of the third side. Checking the triangle inequality:
1. \(5a + 3a > a \implies 8a > a\) (True)
2. \(5a + a > 3a \implies 6a > 3a\) (True)
3. \(3a + a > 5a \implies 4a > 5a\) (False)

Since the third condition is false, this does not form a valid triangle.

Answer:
\[
\boxed{\text{Not a valid triangle}}
\]

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i)

Given:
- A triangle with sides \(5x\), \(3x\), and \(2x\).
- We need to determine if this forms a valid triangle.

Solution:
Checking the triangle inequality:
1. \(5x + 3x > 2x \implies 8x > 2x\) (True)
2. \(5x + 2x > 3x \implies 7x > 3x\) (True)
3. \(3x + 2x > 5x \implies 5x > 5x\) (False)

Since the third condition is false, this does not form a valid triangle.

Answer:
\[
\boxed{\text{Not a valid triangle}}
\]

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j)

Given:
- An isosceles triangle with sides 5 cm, 5 cm, and an included angle of \(60^\circ\).
- We need to find the third side \(z\).

Solution:
This is an equilateral triangle because it has two equal sides and an included angle of \(60^\circ\). Therefore, all sides are equal:
\[
z = 5 \text{ cm}
\]

Answer:
\[
\boxed{5 \text{ cm}}
\]

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Final Answers:

\[
\boxed{65^\circ, 3\sqrt{41} \text{ cm}, 60^\circ, 60^\circ, \sqrt{34} \text{ cm}, 35^\circ, 35^\circ, 7 \text{ cm}, 12 \text{ cm}, \text{Not a valid triangle}, \text{Not a valid triangle}, 5 \text{ cm}}
\]