Why Every Triangle's Angles Add Up to the Same Number
Learn to classify triangles by sides and angles, use the angle-sum and exterior-angle properties, construct triangles from SSS, SAS and ASA, and name every quadrilateral.
How are triangles classified?
A triangle can be described in two independent ways: by the lengths of its sides and by the sizes of its angles. Every triangle therefore carries two labels at once — a triangle can be both isosceles and right-angled, for instance.
This page covers everything in the ICSE Class 6 Mathematics chapter on triangles and quadrilaterals: the two classifications, the angle-sum and exterior-angle properties, the three standard constructions, and the family of quadrilaterals.
This page covers everything in the ICSE Class 6 Mathematics chapter on triangles and quadrilaterals: the two classifications, the angle-sum and exterior-angle properties, the three standard constructions, and the family of quadrilaterals.
What are scalene, isosceles and equilateral triangles?
By sides: a scalene triangle has all three sides of different lengths, an isosceles triangle has exactly two sides equal, and an equilateral triangle has all three sides equal.
By angles: an acute-angled triangle has all three angles less than , a right-angled triangle has one angle of exactly , and an obtuse-angled triangle has one angle greater than .
The two classifications interact. In an isosceles triangle the angles opposite the equal sides are also equal, and in an equilateral triangle all three angles are equal — each one , since they must share .
For example, the triangular frame under a bicycle seat is usually scalene, while a road warning sign is an equilateral triangle.
A triangle can have at most one angle of or more. Two right angles would already use up the whole , leaving nothing for the third angle — so "right-angled isosceles" is possible, but "two right angles" is not.
By angles: an acute-angled triangle has all three angles less than , a right-angled triangle has one angle of exactly , and an obtuse-angled triangle has one angle greater than .
The two classifications interact. In an isosceles triangle the angles opposite the equal sides are also equal, and in an equilateral triangle all three angles are equal — each one , since they must share .
For example, the triangular frame under a bicycle seat is usually scalene, while a road warning sign is an equilateral triangle.
A triangle can have at most one angle of or more. Two right angles would already use up the whole , leaving nothing for the third angle — so "right-angled isosceles" is possible, but "two right angles" is not.
Formula
What are the angle-sum and exterior-angle properties?
Two properties let you find unknown angles in any triangle:
Worked example with the angle sum. Two angles of a triangle are and , so
Worked example with the exterior angle. If the two interior opposite angles are and , the exterior angle is
You can check this against the angle sum: the third interior angle is , and that angle plus its exterior angle gives , a linear pair as expected.
The exterior-angle property is simply a shortcut built from the angle sum and the straight line — which is why the two can always be used to verify each other.
Worked example with the angle sum. Two angles of a triangle are and , so
Worked example with the exterior angle. If the two interior opposite angles are and , the exterior angle is
You can check this against the angle sum: the third interior angle is , and that angle plus its exterior angle gives , a linear pair as expected.
The exterior-angle property is simply a shortcut built from the angle sum and the straight line — which is why the two can always be used to verify each other.
How do you construct a triangle with ruler and compasses?
Three sets of measurements each fix a unique triangle, and each has its own method.
SSS (three sides given): draw the longest side as the base. Open the compasses to the second side and draw an arc from one end; open to the third side and draw an arc from the other end. Where the arcs cross is the third vertex.
SAS (two sides and the angle between them): draw one side, construct the given angle at one end with a protractor, mark the second side's length along that arm, and join up.
ASA (two angles and the side between them): draw the given side, construct one angle at each end, and extend both arms until they meet.
For example, to construct a triangle with sides 5 cm, 4 cm and 3 cm, draw the 5 cm base, then strike arcs of 4 cm and 3 cm from its two ends.
One caution with SSS: the two shorter sides must together exceed the longest, or the arcs will never meet. Sides of 2 cm, 3 cm and 9 cm cannot form a triangle at all.
SSS (three sides given): draw the longest side as the base. Open the compasses to the second side and draw an arc from one end; open to the third side and draw an arc from the other end. Where the arcs cross is the third vertex.
SAS (two sides and the angle between them): draw one side, construct the given angle at one end with a protractor, mark the second side's length along that arm, and join up.
ASA (two angles and the side between them): draw the given side, construct one angle at each end, and extend both arms until they meet.
For example, to construct a triangle with sides 5 cm, 4 cm and 3 cm, draw the 5 cm base, then strike arcs of 4 cm and 3 cm from its two ends.
One caution with SSS: the two shorter sides must together exceed the longest, or the arcs will never meet. Sides of 2 cm, 3 cm and 9 cm cannot form a triangle at all.
What are the parts and types of a quadrilateral?
A quadrilateral is a closed figure with four sides. It has four vertices, four angles, and two diagonals joining opposite vertices. Sides sharing a vertex are adjacent; sides that do not meet are opposite.
Its angles always total , which you can verify by drawing one diagonal: the quadrilateral splits into two triangles, and .
The family is distinguished by properties:
- square: four equal sides, four right angles
- rectangle: opposite sides equal, four right angles
- parallelogram: opposite sides parallel and equal
- rhombus: four equal sides, opposite angles equal
- trapezium: exactly one pair of parallel sides
- kite: two pairs of adjacent sides equal
The relationships nest, and that is what questions probe. Every square is also a rectangle and a rhombus, because it satisfies both definitions — but a rectangle is not necessarily a square.
Its angles always total , which you can verify by drawing one diagonal: the quadrilateral splits into two triangles, and .
The family is distinguished by properties:
- square: four equal sides, four right angles
- rectangle: opposite sides equal, four right angles
- parallelogram: opposite sides parallel and equal
- rhombus: four equal sides, opposite angles equal
- trapezium: exactly one pair of parallel sides
- kite: two pairs of adjacent sides equal
The relationships nest, and that is what questions probe. Every square is also a rectangle and a rhombus, because it satisfies both definitions — but a rectangle is not necessarily a square.
Exam tip
Exam tip: checking whether three lengths can form a triangle
Students attempt constructions with impossible measurements and then assume their compass work went wrong.
Before drawing, add the two shorter sides and compare with the longest. The sum must be greater than the longest side. For 2 cm, 3 cm and 9 cm, , which is less than 9, so no triangle exists and no amount of careful drawing will produce one.
The same check catches trick questions in angle work: the three angles given must total exactly . If a question offers , and , that sums to and describes no triangle — say so rather than forcing an answer.
Before drawing, add the two shorter sides and compare with the longest. The sum must be greater than the longest side. For 2 cm, 3 cm and 9 cm, , which is less than 9, so no triangle exists and no amount of careful drawing will produce one.
The same check catches trick questions in angle work: the three angles given must total exactly . If a question offers , and , that sums to and describes no triangle — say so rather than forcing an answer.
Did you know
Why does a quadrilateral's angle sum come to 360 degrees?
Draw a single diagonal and the four-sided figure becomes two triangles. Each triangle contributes , and every one of those angles lies inside the original quadrilateral, so the total is .
The same trick extends to any polygon: split it into triangles from one vertex, count them, and multiply by . It is the reason all those angle-sum rules are really one rule.
The same trick extends to any polygon: split it into triangles from one vertex, count them, and multiply by . It is the reason all those angle-sum rules are really one rule.
Key takeaways
Triangles and quadrilaterals in 30 seconds
- Triangles are classified by sides (scalene, isosceles, equilateral) and independently by angles (acute, right, obtuse); at most one angle can be or more.
- The three angles of a triangle total , and an exterior angle equals the sum of the two interior opposite angles.
- SSS, SAS and ASA each fix a unique triangle; for SSS the two shorter sides must together exceed the longest.
- A quadrilateral has four sides, two diagonals, and angles totalling , provable by splitting it into two triangles.
- Square, rectangle, parallelogram, rhombus, trapezium and kite are told apart by their sides, angles and parallels — and a square qualifies as both a rectangle and a rhombus.
You will remember all of this far better after answering five questions on it than after reading it twice.
- The three angles of a triangle total , and an exterior angle equals the sum of the two interior opposite angles.
- SSS, SAS and ASA each fix a unique triangle; for SSS the two shorter sides must together exceed the longest.
- A quadrilateral has four sides, two diagonals, and angles totalling , provable by splitting it into two triangles.
- Square, rectangle, parallelogram, rhombus, trapezium and kite are told apart by their sides, angles and parallels — and a square qualifies as both a rectangle and a rhombus.
You will remember all of this far better after answering five questions on it than after reading it twice.