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