Proposition III.22 — The opposite angles of quadrilaterals in circles are equal t…

Proof

Let

A B C D

be a cyclic quadrilateral. Join

A C

and

B D

. By III.21,

∠ B A C = ∠ B D C

(both subtend arc

B C

from the opposite side), and

∠ C A D = ∠ C B D

(both subtend arc

C D

). So

∠ B A D = ∠ B A C + ∠ C A D = ∠ B D C + ∠ C B D

. In

△ B C D

, by I.32 the three angles sum to two right angles:

∠ B D C + ∠ C B D + ∠ B C D =

two right angles. Substituting

∠ B D C + ∠ C B D = ∠ B A D

∠ B A D + ∠ B C D =

two right angles.

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Full neighborhood

Depends on (3)

III.21Proposition III.21In a circle the angles in the same segment are equal to one another.
I.32Proposition I.32In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles,…
2Common notion 2If equals be added to equals, the wholes are equal.

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