Download Geometry - Semester 2

Geometry - Semester 2
Mrs. Day-Blattner
2/2/2016
Agenda 2/2/2016
1) Circles, Chords, Diameters and their relationships - finish
up Exercises 1-6
2) Homework - correct and grade
3) Lesson Summary and Graphic organizer
4) Homework
5) Exit ticket
Learning Log (important things to remember)
We can only find the circumcenter of 3 points that are not
colinear - if we have three points that are colinear there could
not be a location equidistant from all three points .
(We could find a point closer to one or 2, but not all 3.)
A.
B.
C.
1. Prove the theorem: If a diameter of a circle
bisects a chord, then it must be perpendicular to
the chord.
Given: Circle C with diameter DE, chord AB, and
AF = BF
Prove DE is perpendicular to AB
Proof version 1.
Statements
1.
2.
3.
4.
5.
6.
7.
AF = BF
FC = FC
AC = BC
triangle AFC is congruent to triangle
BFC
measure of angle AFC is equal to
measure of angle BFC
angles AFC and BFC are right
angles
Line segment DE is perpendicular
to line segment AB
Reasons.
1.
2.
3.
4.
5.
6.
7.
Given
Reflexive property
radii of same circle are equal in
measure
Side-side-side congruency
postulate
corresponding angles of congruent
triangles are equal in measure
equal angles that form a linear pair
each measure 90 degrees
Definition of perpendicular lines
Proof version 2.
Statements
1.
2.
3.
4.
5.
6.
7.
AF = BF
AC = BC
measure of angle FAC is equal to
measure of angle FBC
triangles AFC and BFC are
congruent
measure angle AFC = measure of
angle BFC
angles AFC and BFC are right
angles
Line segment DE is perpendicular
to line segment AB
Reasons.
1.
2.
3.
4.
5.
6.
7.
Given
radii of same circle are equal in
measure
base angles of an isosceles triangle
are equal in measure
SAS
Corresponding angles of congruent
triangles are equal in measure
equal angles that form a linear pair
each measure 90 degrees
Definition of perpendicular lines
2. Prove the theorem: If a diameter of a circle is
perpendicular to a chord, then it must bisect the
chord.
Given: Circle C with diameter DE, chord AB, and
DE is perpendicular to AB
Prove: DE bisects AB
2.Proof .
Statements
1. Line segment DE is
perpendicular to line segment
AB
2. angles AFC and BFC are right
angles
3. angle AFC is congruent to
angle BFC
4. AC = BC
5. measure of angle FAC is equal
to measure of angle FBC
Reasons.
1. Given
2. Definition of perpendicular
lines
3. all right angles are
congruent
4. radii of the same circle are
equal in measure
5. base angles of isosceles
triangles are congruent
2.Proof .
Statements
1.
2.
3.
4.
5.
Line segment DE is perpendicular to line segment AB
angles AFC and BFC are right angles
angle AFC is congruent to angle BFC
AC = BC
measure of angle FAC is equal to measure of angle FBC
6. measure of angle ACF is
equal to measure of angle
BCF
7. triangles AFC and BFC are
congruent
8. AF = BF
9. Line segment DE bisects line
segment AB
1.
2.
3.
4.
5.
Reasons.
Given
Definition of perpendicular lines
all right angles are congruent
radii of the same circle are equal in measure
base angles of isosceles triangles are congruent
6. two angles of triangle are
equal in measure, so third
angles are equal
7. ASA
8. corresponding sides of
congruent triangles are
equal in length
9. Definition of segment
bisector.
Lesson Summary
Theorems about chords and diameters in a circle
and their converses:
● If a diameter of a circle bisects a chord, then it
must be perpendicular to the chord.
● If a diameter of a circle is perpendicular to a
chord, then it bisects the chord.
Lesson Summary cont.
● If two chords are congruent, then the center is
equidistant from the two chords.
●
● If the center is equidistant from two chords, then
the two chords are congruent.
Lesson Summary cont.
● Congruent chords define central angles equal in
measure.
● If two chords define central angles equal in
measure, then they are congruent.
Use these theorems to complete the graphic
organizer sheet for circles.
Homework
Practice questions sheet for quiz next
week on Feb 10th.