Boarding School and Day School
Twice the Angle
- Circle Theorems 3: Angle at the Centre Theorem Definitions An arc of a circle is a contiguous (i.e. no gaps) portion of the circumference. An arc which is half of a circle is called a semi-circle. An arc which is shorter than a semi-circle is called a minor arc. An arc which is greater than a semi-circle is called a major arc. Clearly, for every minor arc there is a corresponding major arc.
A segment of a circle is a figure bounded by an arc and its chord. If the arc is a minor arc then the segment is a minor segment. If the arc is a major arc then the segment is a major segment. Clearly, for every minor segment there is a corresponding major segment. A sector of a circle is a figure bounded by two radii and the included arc. If the arc is a minor arc then the sector is a minor sector. If the arc is a major arc then the sector is a major sector. Clearly, for every minor sector there is a corresponding major sector.
The word subtend means to hold up or support. For example, the minor arc BC subtends an angle
θ (∠BDC) at a point D on the major arc. ∠BDC at the point D.
We could also have said that the chord BC subtends Similarly, the chord or minor arc BC subtends
∠BAC at the centre A. The ∠BAC is
called a central angle and is sometimes measured by the length of the minor arc.
Copyright 2007, Hartley Hyde
Page 1 of 6
Twice the Angle
Prior Learning Theorem: The exterior angle of any triangle is equal to the sum of the interior opposite angles. Given: To Prove: any ABC with BC extended to D
m∠ACD = m∠BAC + m∠ABC
Construction: Draw CE // BA Proof: m∠ACE = m∠BAC = m∠ECD = m∠ABC = adding
α
(alternate ∠s: CE // BA)
β (corresponding ∠s: CE // BA) α+β
QED
⇒
m∠ACD = m∠BAC + m∠ABC =
Corollary:
The sum of the angles of any triangle is 180° ABC is any triangle Since
α + β + γ measure adjacent angles on the line BCD α + β + γ = 180°.
ABC is also
It follows that
But the sum of the angles of the
α+β+γ
⇒
Purpose
The sum of the angles of any
- Circle Theorems 3: Angle at the Centre Theorem Definitions An arc of a circle is a contiguous (i.e. no gaps) portion of the circumference. An arc which is half of a circle is called a semi-circle. An arc which is shorter than a semi-circle is called a minor arc. An arc which is greater than a semi-circle is called a major arc. Clearly, for every minor arc there is a corresponding major arc.
A segment of a circle is a figure bounded by an arc and its chord. If the arc is a minor arc then the segment is a minor segment. If the arc is a major arc then the segment is a major segment. Clearly, for every minor segment there is a corresponding major segment. A sector of a circle is a figure bounded by two radii and the included arc. If the arc is a minor arc then the sector is a minor sector. If the arc is a major arc then the sector is a major sector. Clearly, for every minor sector there is a corresponding major sector.
The word subtend means to hold up or support. For example, the minor arc BC subtends an angle
θ (∠BDC) at a point D on the major arc. ∠BDC at the point D.
We could also have said that the chord BC subtends Similarly, the chord or minor arc BC subtends
∠BAC at the centre A. The ∠BAC is
called a central angle and is sometimes measured by the length of the minor arc.
Copyright 2007, Hartley Hyde
Page 1 of 6
Twice the Angle
Prior Learning Theorem: The exterior angle of any triangle is equal to the sum of the interior opposite angles. Given: To Prove: any ABC with BC extended to D
m∠ACD = m∠BAC + m∠ABC
Construction: Draw CE // BA Proof: m∠ACE = m∠BAC = m∠ECD = m∠ABC = adding
α
(alternate ∠s: CE // BA)
β (corresponding ∠s: CE // BA) α+β
QED
⇒
m∠ACD = m∠BAC + m∠ABC =
Corollary:
The sum of the angles of any triangle is 180° ABC is any triangle Since
α + β + γ measure adjacent angles on the line BCD α + β + γ = 180°.
ABC is also
It follows that
But the sum of the angles of the
α+β+γ
⇒
Purpose
The sum of the angles of any