Angle Sum of a Triangle

Statement

The angle sum of a triangle says that the three interior angles of any triangle add to \(180^{\circ}\). If the angles are labelled \(A\), \(B\) and \(C\), then

\[ A + B + C = 180^{\circ}. \]

Why it’s true (short reason)

• Draw any triangle \(ABC\). Through \(C\), draw a line parallel to the base \(AB\). Alternate and corresponding angle facts show that the two angles at \(A\) and \(B\) are “copied” onto that straight line at \(C\). A straight line makes \(180^{\circ}\), so the three interior angles add to \(180^{\circ}\).
• Equivalently, the exterior angle at a vertex equals the sum of the two opposite interior angles. Summing these facts around a triangle also leads to \(180^{\circ}\).

Recipe (how to use it)

1. Identify the three interior angles of the triangle. If one is missing, label it with a variable such as \(x\).
2. Form the equation \(A + B + C = 180^{\circ}\). Substitute any algebraic expressions given for angles.
3. Solve for the unknown(s). If the triangle has extra structure (isosceles, right-angled, exterior angle, or angles on a straight line), use those alongside the sum.
4. Check the result: the three angles should be positive and add to \(180^{\circ}\).

Spotting it

Look for:

• Two angles given inside a triangle with the third missing.
• Algebraic expressions on angles inside a triangle, e.g. \((2x+15)^{\circ}\), \((x-7)^{\circ}\).
• Right-angled triangles (one angle is \(90^{\circ}\), so the other two sum to \(90^{\circ}\)).
• Isosceles triangles (two equal angles) which, combined with the sum, quickly gives the third angle.
• Exterior angles created by extending one side; the exterior angle equals the sum of the two opposite interior angles, which is a fast route to the missing interior angle.

Common pairings

• Angles on a straight line: adjacent angles along a line add to \(180^{\circ}\). Often an exterior angle sits on a straight line with an interior one.
• Isosceles triangle facts: base angles equal, or equal sides imply equal base angles. Combine with the triangle sum to solve for unknowns.
• Right angles: knowing one angle is \(90^{\circ}\) reduces the problem to a simple two-angle sum.
• Parallel lines: alternate and corresponding angles can transfer information between triangles, then the triangle sum finishes it off.

Mini examples

1. Given: angles \(40^{\circ}\) and \(65^{\circ}\). Find: the third angle. Answer: \(180-40-65=75^{\circ}\).
2. Given: right-angled triangle with one acute angle \(x^{\circ}\) and the other \(3x^{\circ}\). Find: \(x\). Answer: \(x+3x=90 \Rightarrow x=22.5^{\circ}\).
3. Given: isosceles triangle with vertex angle \(38^{\circ}\). Find: each base angle. Answer: \((180-38)/2=71^{\circ}\).

Pitfalls

• Mixing interior and exterior angles: if a side is extended, the angle outside the triangle is an exterior angle. Do not add that to two interior angles and set equal to \(180^{\circ}\) unless you are using the straight-line fact.
• Forgetting degree symbols: keep \(^{\circ}\) in working and final answers where appropriate.
• Dropping brackets in algebra: if an angle is \((2x+15)^{\circ}\), include the brackets when adding or subtracting in equations.
• Isosceles mislabelling: equal sides are opposite equal angles. Double-check which angles are equal before setting them equal.

Exam strategy

• Mark known angles clearly on the diagram; write algebraic angles near their vertices.
• Decide which relation is fastest: triangle sum, straight line, exterior angle equals sum of opposite interior angles, or an isosceles/right-angle shortcut.
• Form a single clean equation and solve; if there are two unknowns, look for a second relation (e.g., equal angles in isosceles).
• State the final angle with units and, if asked, justify briefly (e.g., “triangle angles sum to \(180^{\circ}\)”).

Worked idea (exterior angle)

Suppose side \(BC\) is extended to \(D\). The exterior angle at \(C\) is \(\angle ACD\). Because \(\angle ACB + \angle ACD = 180^{\circ}\) (straight line) and \(A+B+C=180^{\circ}\), it follows that

\[ \angle ACD = \angle A + \angle B. \]

This is often quicker than finding the interior angle at \(C\) first.

Worked idea (isosceles)

If \(AB=AC\), then \(\angle B = \angle C\). With \(A+B+C=180^{\circ}\), we have \(A+2B=180^{\circ}\), so \(B=\frac{180^{\circ}-A}{2}\). This pattern appears frequently in GCSE problems: a vertex angle and equal sides lead to halving the remainder.

Summary

The interior angles of every triangle total \(180^{\circ}\). Combine this fact with straight lines, exterior angles, right angles and isosceles properties to solve unknowns efficiently. Keep equations tidy, track degree symbols, and check your three angles add to \(180^{\circ}\) at the end.