A Sum of 360° — Key Facts (Grade 7)
This short note covers four related angle facts that all come from the sums 180° and 360°.
1. Angles on a straight line
When two or more angles lie on a straight line, their measures add up to 180° (a straight angle).
If two angles are adjacent on a line, we often write:
\[
\alpha + \beta = 180^\circ
\]
Example 1.1 Find \( \beta \) if \( \alpha = 65^\circ \) and \( \alpha + \beta = 180^\circ \).
\( \beta = 180^\circ - 65^\circ = 115^\circ \).
2. Angles around a point
All angles that meet at a single point and make a full turn add to 360°.
\[
\alpha + \beta + \gamma + \dots = 360^\circ
\]
Example 2.1 Three angles around a point are \(80^\circ\), \(120^\circ\) and \(x\). Find \(x\).
\(x = 360^\circ - (80^\circ+120^\circ) = 360^\circ - 200^\circ = 160^\circ\).
3. Angles in a triangle
The three interior angles of any triangle add to 180°:
\[
A + B + C = 180^\circ
\]
Example 3.1 Triangle with angles \(A=50^\circ\), \(B=60^\circ\). Find \(C\).
\(C = 180^\circ - (50^\circ+60^\circ) = 70^\circ\).
Example 3.2 (Isosceles) In an isosceles triangle two equal angles are \(x\). If the third angle is \(40^\circ\), find \(x\).
\(2x + 40^\circ = 180^\circ \Rightarrow 2x = 140^\circ \Rightarrow x = 70^\circ\).
4. Angles in a quadrilateral
The four interior angles of a quadrilateral add to 360° (because two triangles make a quadrilateral):
\[
A + B + C + D = 360^\circ
\]
Example 4.1 A quadrilateral has three angles \(95^\circ\), \(85^\circ\) and \(110^\circ\). Find the fourth angle.
Fourth \(= 360^\circ - (95^\circ+85^\circ+110^\circ) = 360^\circ - 290^\circ = 70^\circ\).
Quick practice (try before checking)
- Two angles on a line are \(x\) and \(3x\). Find \(x\).
- Four angles around a point are in ratio \(1:2:3:4\). Find them.
- In a triangle angles are \(2y\), \(3y\) and \(4y\). Find \(y\).
- A quadrilateral has angles \(90^\circ\), \(2z\), \(z\), and \(60^\circ\). Find \(z\).
Answers
- \(x + 3x = 180^\circ \Rightarrow 4x=180^\circ \Rightarrow x=45^\circ\).
- Total parts \(1+2+3+4=10\). Each part is \(360^\circ/10=36^\circ\). Angles: \(36^\circ,72^\circ,108^\circ,144^\circ\).
- \(2y+3y+4y=180^\circ \Rightarrow 9y=180^\circ \Rightarrow y=20^\circ\).
- \(90^\circ+2z+z+60^\circ=360^\circ \Rightarrow 3z+150^\circ=360^\circ \Rightarrow 3z=210^\circ \Rightarrow z=70^\circ\).
Tip: Many problems mix these facts — look for straight lines or points where angles meet, split shapes into triangles, or use symmetry (isosceles, regular polygons) to find equal angles.