### My Scribe

__Hello to everyone! __Today’s class started off with two investigations. Investigation number nine and investigation number ten, which had to do with circle geometry.

Investigation 9:

A *cyclic quadrilateral *is a quadrilateral all of whose vertices lie in the circumference of a circle.

First we had to construct a polygon with all four points on the circumference. After that, for that quadrilateral we measured the sum of the opposite angles and found out that they were equal to 180°. We then did those same steps two more times to confirm that it wasn’t just for that one type of quadrilateral. The last step for this investigation was to articulate a general rule. We found out that “If two angles are opposite each other in a cyclic quadrilateral, then the angles are supplementary.”

Investigation 10:

The diagonals of convex polygons are drawn at a single vertex. Complete the table of values showing the relation between the number of sides in the polygon and the number of triangles formed.

In conclusion, a convex polygon with n sides can be divided into n - 2 triangles.

You have learned that the sum of the interior angles of a triangle is 180°. Complete the following table.

Refer to the preceding table. Describe in words the relation between the number of sides in a polygon and the sum of its interior angles.

The sum of the interior angles is found by the number of sides (n) subtract two, then

Multiplied by 180°.

Again, referring to the table, write a formula that describes this relation.

The sum of the interior angles of a polygon with n sides = ( n - 2 )( 180° )

After all of that we then went into our dictionaries to take some notes. Here they are:__Perpendicular Bisector Theorem:__**Part I **If a line bisects a chord and passes through the center of a circle then it is perpendicular to the chord.**Part II **If a line is perpendicular to a chord and passes through the center of a circle then it bisects the chord.__Inscribed-central Angle Theorem:__**Part I **If a central angle and an inscribed angle are subtended by the same arc then the central angle is twice the inscribed angle.**Part II **If an inscribed angle and a central angle are subtended by the same arc then the inscribed angle is half the central angle.__A.K.A. “The star trek theorem”____Special Case:__

Inscribed-diameter (Thales) theorem

An inscribed angle subtended by a diameter is always 90°.__Inscribed Angle Theorem:__

If two (or more) inscribed angles are subtended by the same arc then they are congruent.__Parallel Chord Theorem:__

The arcs between parallel chords are congruent.__Congruent Chords Theorem:__

Congruent chords subtend congruent arcs.

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