For AI agents: use /llms.txt for the Nakafa content index.
Property:
Both vertices and lie on the same side of chord . The angles and therefore subtend the same arc from the same segment and have equal measures.
The central angle is twice the inscribed angle that subtends the same arc.
Property:
Every inscribed angle that subtends a diameter of the circle is always a right angle ().
Property: If is a diameter, then
This is known as Thales' Theorem.
A cyclic quadrilateral is a quadrilateral whose four vertices lie on a circle. The sum of opposite angles is .
Property: and
In a cyclic quadrilateral, an exterior angle at a vertex equals the interior angle at the opposite vertex.
Property: (exterior angle at equals interior angle at )
In a circle with center , the central angle is . Find the measure of inscribed angle !
Solution:
Using the property of central and inscribed angles:
In cyclic quadrilateral , given . Find the measure of !
Solution:
Using the property of cyclic quadrilaterals:
Point lies on a circle with as the diameter. Find the measure of !
Solution:
Since is a diameter and lies on the circle, by Thales' theorem:
In a circle with center , central angle intercepts the minor arc . Points and lie on the major arc . Find:
In cyclic quadrilateral , given:
Points , , and lie on a circle. If is a diameter and , find the measure of !
In a circle, inscribed angle . Find the measure of central angle !
In cyclic quadrilateral , the exterior angle at vertex is . Find the measure of the interior angle at vertex !
Finding inscribed angles subtending the same arc
Solution:
Using the relationship between central and inscribed angles:
Since inscribed angles subtending the same arc have equal measures:
Finding angles in a cyclic quadrilateral
Solution:
In a cyclic quadrilateral,
For angle (opposite to ):
Finding angle in an isosceles triangle with diameter
Finding central angle from inscribed angle
Solution:
Using the relationship between central and inscribed angles:
Finding interior angle from exterior angle in cyclic quadrilateral
Solution:
In a cyclic quadrilateral, an exterior angle at a vertex equals the interior angle at the opposite vertex.
If the exterior angle at is , then:
Published: . Updated: .
Find the measures of and !
For angle (opposite to ):
Solution:
Since is a diameter, by Thales' Theorem:
Since , triangle is a right isosceles triangle.
In a right isosceles triangle, both base angles are equal:
Since (base angles of isosceles triangle):
This is because vertices and are opposite in cyclic quadrilateral .
