The Angle Outside the Triangle Tells You What's Inside
Learn to use the angle sum and the exterior angle rule, construct triangles from SAS and ASA, and draw an altitude even when it falls outside the triangle.
How can an angle outside a triangle tell you about the inside?
Extend one side of a triangle and the angle formed outside — the exterior angle — is exactly equal to the two interior angles it does not touch, added together. So measuring outside tells you about inside, with no work in between.
This page covers everything in the CBSE Class 7 Mathematics chapter's second half: the angle sum property, the exterior angle property, constructing triangles from SAS and ASA, and drawing altitudes.
This page covers everything in the CBSE Class 7 Mathematics chapter's second half: the angle sum property, the exterior angle property, constructing triangles from SAS and ASA, and drawing altitudes.
Formula
What is the angle sum property?
The three angles of any triangle always add to the same total:
This holds for every triangle — large, small, scalene or equilateral.
Worked example. Two angles are and , so
In an equilateral triangle all three are equal, so each is .
In an isosceles triangle the angles opposite the equal sides are equal. If the unequal angle is , the other two share the remaining , giving each.
Two consequences follow directly and are often examined. A triangle can have at most one angle of or more, since two would already use the whole . And the three angles can never include a angle, which is the collinear case from the previous part — no triangle at all.
This holds for every triangle — large, small, scalene or equilateral.
Worked example. Two angles are and , so
In an equilateral triangle all three are equal, so each is .
In an isosceles triangle the angles opposite the equal sides are equal. If the unequal angle is , the other two share the remaining , giving each.
Two consequences follow directly and are often examined. A triangle can have at most one angle of or more, since two would already use the whole . And the three angles can never include a angle, which is the collinear case from the previous part — no triangle at all.
Formula
What is the exterior angle property?
Extend one side of a triangle beyond a vertex, and the angle between that extension and the adjacent side is an exterior angle. The rule is:
"Interior opposite" means the two angles not next to the exterior angle.
Worked example. The two interior opposite angles are and , so the exterior angle is
You can verify it with the angle sum. The third interior angle is , and that angle plus its exterior angle gives — a linear pair, exactly as expected.
That check shows the exterior angle rule is not a separate fact. It follows from the angle sum plus the straight line, which is why the two can always be used to confirm each other.
It also means an exterior angle is always greater than either interior opposite angle taken alone, since it equals their sum.
"Interior opposite" means the two angles not next to the exterior angle.
Worked example. The two interior opposite angles are and , so the exterior angle is
You can verify it with the angle sum. The third interior angle is , and that angle plus its exterior angle gives — a linear pair, exactly as expected.
That check shows the exterior angle rule is not a separate fact. It follows from the angle sum plus the straight line, which is why the two can always be used to confirm each other.
It also means an exterior angle is always greater than either interior opposite angle taken alone, since it equals their sum.
How do you construct a triangle from SAS and ASA?
Each set of measurements has its own method, and each fixes exactly one triangle.
SAS — two sides and the angle between them. To build a triangle with AB = 5 cm, and BC = 6 cm: draw BC = 6 cm, construct a angle at B with a protractor, mark 5 cm along that arm to get A, then join AC.
ASA — two angles and the side between them. To build a triangle with BC = 6 cm, and : draw BC = 6 cm, construct at B and at C, and extend both arms until they meet at A.
The word included is what these depend on. In SAS the angle must lie between the two given sides, and in ASA the side must lie between the two given angles — otherwise the measurements may not fix a single triangle.
For ASA there is a quick check before you start: the two given angles must add to **less than **, or the arms will never meet. Angles of and describe no triangle.
SAS — two sides and the angle between them. To build a triangle with AB = 5 cm, and BC = 6 cm: draw BC = 6 cm, construct a angle at B with a protractor, mark 5 cm along that arm to get A, then join AC.
ASA — two angles and the side between them. To build a triangle with BC = 6 cm, and : draw BC = 6 cm, construct at B and at C, and extend both arms until they meet at A.
The word included is what these depend on. In SAS the angle must lie between the two given sides, and in ASA the side must lie between the two given angles — otherwise the measurements may not fix a single triangle.
For ASA there is a quick check before you start: the two given angles must add to **less than **, or the arms will never meet. Angles of and describe no triangle.
How do you draw an altitude, including in an obtuse triangle?
An altitude is the perpendicular from a vertex to the opposite side, and it measures the triangle's height from that vertex.
To draw it, place a set-square so one edge lies along the opposite side, slide it until the perpendicular edge passes through the vertex, and draw the line. Mark the right angle where it meets the side.
Every triangle has three altitudes, one from each vertex.
In an acute-angled triangle all three fall inside. In a right-angled triangle, two of them are the legs themselves, since those already meet at .
In an obtuse-angled triangle, something surprising happens: two of the altitudes fall outside the triangle. To draw them you must extend the opposite side beyond the triangle and drop the perpendicular onto that extension.
That is the case students get wrong, because they assume a height must sit inside the shape. It need not — the altitude is defined by being perpendicular to the opposite side, wherever that line has to be extended to.
To draw it, place a set-square so one edge lies along the opposite side, slide it until the perpendicular edge passes through the vertex, and draw the line. Mark the right angle where it meets the side.
Every triangle has three altitudes, one from each vertex.
In an acute-angled triangle all three fall inside. In a right-angled triangle, two of them are the legs themselves, since those already meet at .
In an obtuse-angled triangle, something surprising happens: two of the altitudes fall outside the triangle. To draw them you must extend the opposite side beyond the triangle and drop the perpendicular onto that extension.
That is the case students get wrong, because they assume a height must sit inside the shape. It need not — the altitude is defined by being perpendicular to the opposite side, wherever that line has to be extended to.
Exam tip
Exam tip: checking your angles add to 180
Every triangle-angle answer carries its own check, and it takes one line.
Once you have found the unknown angle, add all three and confirm the total is exactly . If it is not, the arithmetic went wrong and you can fix it before moving on.
For exterior angle questions, use the second check described above: the exterior angle and its adjacent interior angle must form a linear pair adding to .
And read carefully which angles are "interior opposite". Students frequently add the exterior angle to the adjacent interior angle instead of the two opposite ones — a mistake that a quick check would catch immediately.
Once you have found the unknown angle, add all three and confirm the total is exactly . If it is not, the arithmetic went wrong and you can fix it before moving on.
For exterior angle questions, use the second check described above: the exterior angle and its adjacent interior angle must form a linear pair adding to .
And read carefully which angles are "interior opposite". Students frequently add the exterior angle to the adjacent interior angle instead of the two opposite ones — a mistake that a quick check would catch immediately.
Did you know
Why is an exterior angle always bigger than either opposite angle?
Because it equals the two of them added together, and both are positive.
An exterior angle of made from interior opposites of and must exceed each of them individually, since adding a positive amount to either one is what produced it.
That gives a fast sanity check: if your exterior angle comes out smaller than one of the interior opposite angles, you have added the wrong pair.
An exterior angle of made from interior opposites of and must exceed each of them individually, since adding a positive amount to either one is what produced it.
That gives a fast sanity check: if your exterior angle comes out smaller than one of the interior opposite angles, you have added the wrong pair.
Key takeaways
Triangle angles and constructions: quick revision
- The three angles of any triangle add to , so at most one can be or more.
- An exterior angle equals the sum of the two interior opposite angles, and forms a linear pair with the third — each check confirms the other.
- SAS needs the angle between the two sides; ASA needs the side between the two angles, and those two angles must total under .
- An altitude is the perpendicular from a vertex to the opposite side; every triangle has three.
- In an obtuse-angled triangle two altitudes fall outside, so the opposite side must be extended to draw them.
You will remember all of this far better after answering five questions on it than after reading it twice.
- An exterior angle equals the sum of the two interior opposite angles, and forms a linear pair with the third — each check confirms the other.
- SAS needs the angle between the two sides; ASA needs the side between the two angles, and those two angles must total under .
- An altitude is the perpendicular from a vertex to the opposite side; every triangle has three.
- In an obtuse-angled triangle two altitudes fall outside, so the opposite side must be extended to draw them.
You will remember all of this far better after answering five questions on it than after reading it twice.