Angles in Inscribed Quadrilateral Theorem

Use the Outside Angle Theorem. x=125°−27°/2 x=98°/2 x=49°

Find the measure of x.

x=120°−32°/2 x=88°/2 x=44°

Find the value of x

m⌒AB=360-250=110 x=½m⌒AB=½(110) x=55

Find the value of x if m⌒ACB=250.

(7x+1)°+105°= 180 7x+106°= 7x=180° x =10.57 (4y+14)°+(7y+1)°=180° 11y = 15° =180° 11Y = 165 Y=15

Find x and y in the picture

1) m∠1=½m⌒AC=½(110)= 55° 2) Since m⌒BC= 180-110=70, m∠2=½m⌒BC=½(70)= 35° 3) m∠3=½m⌒BC=½(70)= 35° 4) m∠4=m⌒AC= 110° 5) m∠5=½m⌒AC=½(110)= 55° 6) m∠6=½m⌒AB=½(180)= 90° **∠1 and ∠2 are formed by a chord and a tangent. ∠4 is a central angle. ∠3, ∠5, and ∠6 are inscribed angles.**

In Circle O, m⌒AC=110. Find the measure of each of the numbered angles.

Chord-Tangent Angle Theorem

m∠ABC=½x

Outside Angle Theorem: The measure of an angle formed by two secants, two tangents, or a secant and a tangent from a point outside the circle is half the difference of the measures of the intercepted arcs.

m∠D=m⌒EF−m⌒GH/2 m∠L=m⌒MPNˆ−m⌒MN/2 m∠Q=m⌒RS−m⌒RT/2

Inscribed angles that intercept the same arc are congruent. This is called the Congruent Inscribed Angles Theorem and is shown in the diagram.

∠ADB and ∠ACB intercept ⌒AB, so m∠ADB=m∠ACB. Similarly, ∠DAC and ∠DBC intercept ⌒DC, so m∠DAC=m∠DBC.

Square

A quadrilateral with FOUR CONGRUENT sides and FOUR right angles; Opposite sides that are parallel; TWO diagonals that BISECT at right angles The only regular quadrilateral.

Rhombus

A quadrilateral with FOUR equal sides, opposite sides are parallel, congruent sides and diagonals bisect at right angles.

Adjacent Angle

An angle immediately next to another angle.

Inscribed Quadrilateral

Any four sided figure whose vertices all lie on a circle

m∠AEC=½(x+y)

Chord-Chord Angle Theorem

Parallelogram

Has opposite sides that are parallel and of equal length and opposite angles that are equal.

Inscribed Quadrilateral Theorem: A quadrilateral can be inscribed in a circle if and only if the opposite angles are supplementary.

If ABCD is inscribed in ⨀E, then m∠A+m∠C=180° and m∠B+m∠D=180°. Conversely, If m∠A+m∠C=180° and m∠B+m∠D=180°, then ABCD is inscribed in ⨀E.

Inscribed Quadrilateral Theorem

If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

Adjacent

Next to

m∠P=½(x-y)

Secant-Secant, Secant-Tangent, Tangent-Tangent Theorem

Intercepted Arc

The arc that is formed when segments intersect portions of a circle and create arcs.

Central Angle (At Center)

The measure of the intercepted arc.

Bisect , Bisector

To divide into two equal sections, cut in half. A line , point or plane that bisects something.

Supplementary

Two angles whose sum is 180º

m∠1=m∠2=½(m⌒AC+m⌒BD) m∠3=m∠4=½(m⌒AD+m⌒BC)

Vertical Angles

Inscribed Angle (On Circle)

½ the measure of the intercepted arc.

Chord-Chord Angle (Interior of Circle)

½ the sum of the measures of the intercepted arcs.

S-S, S-T, T-T Angles (Exterior of Circle)

½ the difference of the measures of the intercepted arcs.

Chord-Tangent Angle (On Circle)

½ the measure of the intercepted arc.

x=72°−22°/2 x=50°/2 x=25°

Find the measure of x.

40° is not the intercepted arc. The intercepted arc is 120°, (360°−200°−40°) . x= 200°−120°/2 x=80°/2 x=40°

Find the measure of x. Use the Outside Angle Theorem.

First note that the missing arc by angle x measures 32° because the complete circle must make 360°. Then, x=141°−32°/2 x=109°/2 x=54.5°

Find the value of x