Name the types of following triangles:
(a) Triangle with lengths of sides $ 7 \mathrm{~cm}, 8 \mathrm{~cm} $ and $ 9 \mathrm{~cm} $.
(b) $ \triangle \mathrm{ABC} $ with $ \mathrm{AB}=8.7 \mathrm{~cm}, \mathrm{AC}=7 \mathrm{~cm} $ and $ \mathrm{BC}=6 \mathrm{~cm} $.
(c) $ \triangle \mathrm{PQR} $ such that $ \mathrm{PQ}=\mathrm{QR}=\mathrm{PR}=5 \mathrm{~cm} $.
(d) $ \triangle \mathrm{DEF} $ with $ \mathrm{m} \angle \mathrm{D}=90^{\circ} $
(e) $ \triangle \mathrm{XYZ} $ with $ \mathrm{m} \angle \mathrm{Y}=90^{\circ} $ and $ \mathrm{XY}=\mathrm{YZ} $.
(f) $ \Delta \mathrm{LMN} $ with $ \mathrm{m} \angle \mathrm{L}=30^{\circ}, \mathrm{m} \angle \mathrm{M}=70^{\circ} $ and $ \mathrm{m} \angle \mathrm{N}=80^{\circ} $.

To do:

We have to name the types of triangles given.

Solution:

(a) A scalene triangle is a triangle in which all three sides are of different lengths, and all three angles are of different measures.

Therefore,

Triangle with lengths of sides \( 7 \mathrm{~cm}, 8 \mathrm{~cm} \) and \( 9 \mathrm{~cm} \) is a scalene triangle.

(b) A scalene triangle is a triangle in which all three sides are of different lengths, and all three angles are of different measures.

Therefore,

\( \triangle \mathrm{ABC} \) with \( \mathrm{AB}=8.7 \mathrm{~cm}, \mathrm{AC}=7 \mathrm{~cm} \) and \( \mathrm{BC}=6 \mathrm{~cm} \) is a scalene triangle.

(c) An equilateral triangle is a triangle in which all three sides have the same length.

Therefore,

 \( \triangle \mathrm{PQR} \) such that \( \mathrm{PQ}=\mathrm{QR}=\mathrm{PR}=5 \mathrm{~cm} \) is an equilateral triangle.

(d) A right triangle is a triangle in which one angle is a right angle.

Therefore,

\( \triangle \mathrm{DEF} \) with \( \mathrm{m} \angle \mathrm{D}=90^{\circ} \) is a right-angled triangle.

(e) An isosceles right triangle is defined as a right-angled triangle with an equal base and height.

Therefore,

\( \triangle \mathrm{XYZ} \) with \( \mathrm{m} \angle \mathrm{Y}=90^{\circ} \) and \( \mathrm{XY}=\mathrm{YZ} \) is an Isosceles right triangle.

(f) An acute-angled triangle is a triangle in which all three interior angles are less than $90^o$.
Therefore,

\( \Delta \mathrm{LMN} \) with \( \mathrm{m} \angle \mathrm{L}=30^{\circ}, \mathrm{m} \angle \mathrm{M}=70^{\circ} \) and \( \mathrm{m} \angle \mathrm{N}=80^{\circ} \) is a scalene triangle.