SME
Grade 10 Math · Problem type
Angles from arcs
Use this for “Two chords cross inside a circle, cutting off arcs of 70° and 110°. Find the angle.”
Diagram
Two chords crossing inside a circle: the angle is half the sum of the arcs it and its vertical angle stand on (blue and yellow).
x
=
p + q
2
=
70° + 110°
2
=
90°.
This is an example. Type your own values to start a new problem.
x
°
=
(
p
°
q
°)
÷
2
Formulas
x
=
p
±
q
2
90
=
70 + 110
2
Assumptions
•
Two chords crossing inside the circle (+): the angle is half the sum of its arc and its vertical angle’s arc.
•
Two secants meeting outside (−): the angle is half the far arc minus the near arc.
•
p is arc AC inside, or the far arc outside; q is the other arc.
Step-by-step
WE KNOW
p = 70°, ±: +, q = 110°
FIND
the angle (x)
Step 1 · Find the angle (x)
Use x = (p ± q) ÷ 2
Inside the circle: half the sum of the two arcs.
x
=
p
+
q
2
x
=
70 + 110
2
x
=
180
2
x
=
90°
Check
Put the numbers back in. Both sides should match.
90
=
70 + 110
2
✓
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Circles: arcs, chords, tangents and inscribed angles
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