Proposition 14

Theorem

If at a point (B) in a line (AB ), two other lines (CB , BD ) on opposite sides make the adjacent angles (∠CBA, ∠ABD) together equal to two right angles, then these two lines form one continuous line.

---

Commentary

Let AB  be a given line segment, and let BC  and BD  be two lines that are on the opposite sides of AB . Two adjacent angles, ∠CBA and ∠ABD, are formed by these three lines.

If the two angles ∠CBA and ∠ABD add up to two right angles, then BC  and BD  form a straight line.

These two adjacent angles are called supplementary angles, meaning angles that add up to two right angles.

This proposition is known as the linear pair of angles and is the converse of the previous proposition, Book 1 Proposition 13.

---

---

Computable version

In[1]:=

GeometricScene[{{A.,B.,C.,O.,D.,E.,F.,G.},{}},​​{GeometricStep[{CircleThrough[{A.,B.,C.},O.],Line[{A.,O.,C.}],Line[{D.,E.}],EuclideanDistance[D.,E.]≤EuclideanDistance[A.,C.]}],​​GeometricStep[{F.∈Line[{A.,C.}],EuclideanDistance[A.,F.]EuclideanDistance[D.,E.]}],​​GeometricStep[{GeometricAssertion[{CircleThrough[{F.},A.],CircleThrough[{A.,B.,C.},O.]},{"Concurrent",G.}]}],​​GeometricStep[{Line[{A.,G.}]}]},​​{EuclideanDistance[A.,G.]EuclideanDistance[D.,E.]}​​]//RandomInstance

Out[]=

---

Additional instances

• 
• 
• 
• 
• 

---

Dependency graphs