Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Multilateration wikipedia , lookup
Line (geometry) wikipedia , lookup
Perceived visual angle wikipedia , lookup
Integer triangle wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Rational trigonometry wikipedia , lookup
Area of a circle wikipedia , lookup
Trigonometric functions wikipedia , lookup
Euler angles wikipedia , lookup
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.