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Isosceles Obtuse Triangle: Properties & Examples

TL;DR

An isosceles obtuse triangle has one obtuse angle (between 90° and 180°) and two equal acute angles, with the two sides forming the obtuse angle equal in length. This article covers the definition, why such a triangle is possible, its properties, the area and perimeter formulas, six worked examples, and the common mistakes.

What Is an Isosceles Obtuse Triangle?

An isosceles obtuse triangle is a triangle that is both isosceles and obtuse at once. Isosceles means two sides are equal (and so the two angles opposite them are equal). Obtuse means one angle is greater than 90° but less than 180°. Put together: the triangle has one obtuse angle and two equal acute angles, with the two equal sides meeting at the obtuse corner.

Because a triangle's three angles add to 180°, only one angle can be obtuse, more than one would already overshoot 180°. That obtuse angle has to be the apex angle, the one between the two equal sides. The remaining two are the base angles, equal to each other and necessarily acute. A common example is angles of 120°, 30°, 30°; another is 100°, 40°, 40°.

You meet this triangle while classifying triangles by both their sides and their angles, which sits in NCERT Class 6, Chapter 5 (Understanding Elementary Shapes) and under CCSS-M 4.G.A.2, where triangles are sorted by angle and by side length.

Why an Isosceles Triangle Can Be Obtuse

This is the question that trips most people up, so it is worth settling directly: can an isosceles triangle be obtuse, and where does the obtuse angle go?

Yes, it can, but only at the apex. Here is the reasoning. The two base angles of an isosceles triangle are equal. Suppose one of them were obtuse, say 100°. Then the other base angle would also be 100° (they are equal), and 100° + 100° = 200° already exceeds 180° before the third angle is even counted. Impossible. So the obtuse angle cannot be a base angle. It must be the single odd angle out, the apex between the two equal sides.

Work the numbers from the other direction. If the apex angle is some obtuse value θ (with 90° < θ < 180°), each base angle is:

base angle=180°−θ2.

When θ is just over 90°, each base angle is just under 45°; when θ approaches 180°, the base angles shrink toward 0°. So in any isosceles obtuse triangle, each base angle is less than 45°. That single fact is a fast way to recognise one.

Properties of the Isosceles Obtuse Triangle

Everything about this triangle flows from "one obtuse apex, two equal sides." The properties worth holding:

• One obtuse angle, two equal acute angles. The obtuse apex angle lies between 90° and 180°; the two base angles are equal and each less than 45°.
• Two equal sides meet at the obtuse angle. These are the legs; the side opposite the obtuse angle is the base, and it is the longest side of the triangle (the longest side always faces the largest angle).
• Exactly one line of symmetry. It runs from the obtuse apex straight down to the midpoint of the base, and it is the perpendicular bisector of the base as well as the bisector of the apex angle.
• Angles still sum to 180°. Like every triangle, no exception for being obtuse or isosceles.

Area and Perimeter of the Isosceles Obtuse Triangle

The formulas are the standard triangle formulas; what matters (per the "derive, don't just list" habit) is knowing what each symbol stands for and why the formula holds.

Perimeter. The perimeter is just the total distance around. With the two equal sides each of length a and the base b:

P=a+a+b=2a+b.

Here a is the length of each equal side and b is the base. Units are simple length units (cm, m).

Area from base and height. Every triangle's area is half its base times its height:

A=12×b×h,

where b is the base and h is the perpendicular height from the obtuse apex down to the base. A caution specific to obtuse triangles: if you instead use one of the equal sides as the base, the corresponding height falls outside the triangle.

Area from three sides (Heron's formula). When you know all three sides but no height, use Heron's formula. With equal sides a, base b, and the semi-perimeter s=2a+b2:

A=s(s−a)(s−a)(s−b).

Heron's formula works for any triangle.

Examples of the Isosceles Obtuse Triangle

With the definition, the why, and the formulas in place, here is the triangle in worked problems, moving from angle identification up to a Heron's-formula area.

Example 1 - The apex angle of an isosceles obtuse triangle is 110°. Find the two base angles.

The base angles are equal and share what is left of 180°:

base angle=180°−110°2=35°.\text{base angle} = \frac{180^{\circ} - 110^{\circ}}{2} = 35^{\circ}.

Final answer: each base angle is 35°.

Example 2 - A student is told a triangle is isosceles with one angle of 100° and is asked to find the other two angles.

The obtuse 100° must be the apex and the two equal base angles share the rest:

base angle=180°−100°2=40°.

Final answer: the angles are 100°, 40°, 40°.

Example 3 - An isosceles obtuse triangle has equal sides of 8 cm each and a base of 15 cm. Find its perimeter.

P=2a+b=31 cm.

Final answer: the perimeter is 31 cm.

Example 4 - An isosceles obtuse triangle has a base of 24 cm and a perpendicular height of 8 cm to that base. Find its area.

A=96 cm².

Final answer: the area is 96 cm².

Example 5 - The apex angle of an isosceles obtuse triangle is 4 times a base angle. Find all three angles, and confirm it is obtuse.

The base angles are 30° each and the apex is 120°.

Final answer: 120°, 30°, 30°. Since 120° > 90°, the triangle is obtuse.

Example 6 - An isosceles obtuse triangle has equal sides of 5 cm and a base of 8 cm. Find its area using Heron's formula.

Final answer: the area is 12 cm².

Why the Isosceles Obtuse Triangle Matters

A triangle that is both obtuse and isosceles is more than a classification puzzle; the shape and its wide, symmetric span turn up wherever a stable, spreading form is needed.

• Roof and bracket design. A low, wide-spanning gable or a shallow support bracket is often an isosceles obtuse triangle.
• Reading shape from angles. This triangle is the cleanest illustration of a rule that runs through all of geometry.
• Where heights go outside. Obtuse triangles are the case that forces students to confront that an altitude can land outside the triangle.
• Tessellation and design. Pairs of isosceles obtuse triangles tile into rhombi.

Key Takeaways

• An isosceles obtuse triangle has one obtuse apex angle and two equal acute base angles, each under 45°.
• The base, opposite the obtuse angle, is the longest side.
• Perimeter is 2a + b; area is ½ × base × height, or Heron's formula when only the three sides are known.

Practice These Problems to Solidify Your Understanding

1. The apex angle of an isosceles obtuse triangle is 130°. Find the two base angles.
2. An isosceles obtuse triangle has equal sides of 9 cm and a base of 16 cm. Find its perimeter.
3. An isosceles obtuse triangle has a base of 20 cm and a height of 6 cm to that base. Find its area.