Question

Which of the following triangles are isosceles as well as obtuse-angled triangles?

Answer 1

An obtuse angled triangle is the triangle in which one of the angles is greater than ${90}^{\circ }.$
An isosceles triangle is the triangle in which two sides are equal.
1. Fig 1:
$\Delta PQR$ is isosceles $[\because PQ=PR]$
$\Rightarrow \angle Q=\angle R$
[$\because$ angles opposite to equal sides of a triangle are equal]
$\angle P+\angle Q+\angle R={180}^{\circ }$
[angle sum property of a triangle]
$\angle P+{25}^{\circ }+{25}^{\circ }={180}^{\circ }$
$\Rightarrow \angle P={180}^{\circ }-{50}^{\circ }={130}^{\circ }$
$\Delta PQR$ is an obtuse angled triangle as one of the angles measures 130°.
2. Fig 2:
$\Delta ABC$ is isosceles $[\because AB=AC]$
Similarly as above, we can find the angles of this triangle.
$\angle A={35}^{\circ },\angle B=\angle C={72.5}^{\circ }$
Since all angles are less than ${90}^{\circ }$, $\Delta ABC$ is an acute angled triangle.
3. Fig 3:
$\Delta XYZ$ is an isosceles as well as an obtuse angled triangle as angle Y measures 110°.
4. Fig 4:
$\Delta MNO$ is an isosceles as well as a right angled triangle.
Hence, only Fig 1 and Fig 3 are isosceles as well as obtuse angled triangles.