Geometric Proofs properties and Circle Properties Study Guide

definition of midpoint

the point that divides a segment into 2 congruent angles

definition of angle bisector

ray that divides an angle into 2 congruent angles

vertical angles are congruent

vertical angles are congruent

midpoint theorem

if M is the midpoint then the segments are 1/2 the total measure

reflexive property

ab=ab (duh property)

transitive property

if a=b and b=c then a=c

definition of right angle

an angle whose measure is 90

definition of perpendicular lines

2 lines that intersect to form a right angle

2 lines = perpendicular = congruent adjacent angles

the lines are perpendicular... the conclusion is that they form 90 degree angles, or congruent adjacent angles

definition of complementary angles

2 angles whose sum is 90

definition of supplementary angles

2 angles whose sum is 180

Same Side Interior Angles

Angles inside two parallel lines on the same side of the transversal are supplementary

If Two Angles are Congruent and Supplementary

then each is a right angle

All Right Angles

are congruent to each other

Alternate Interior Angles Theorem

Angles inside two parallel lines on opposite sides of the transversal are congruent

Corresponding Angles Theorem

The two angles are in the same position at each parallel line. They are congruent.

Alternate Exterior Angles Theorem

Angles outside two parallel lines on opposite sides of the transversal are congruent

CPCTC

corresponding parts of congruent triangles are congruent

5 Triangle congruency theorems

ASA

SAS

AAS

SSS

HL

definition of parallelogram

A quadrilateral with two pairs of parallel sides

definition of isosceles

A triangle is isosceles if it has at least two equal sides and a triangle having 2 equal angles

Proving right triangles

prove the right angles first because right angles form right triangles

All radii are congruent

all radii are congruent

Congruent angle subtraction

If two congruent angles are subtracted from two other congruent angles, then the differences are congruent.

Congruent segment - congruent segment = congruent segment

EX:

congruent angle addition

If an angle is added to two congruent angles, then the sums are congruent

Definition of Bisector

a line or segment that intersects a segment or angle and divides it into two congruent halves

Definition of midpoint

A point that divides a segment into two congruent segments

Definition of median

line segment drawn from any vertex of a triangle to the midpoint of the opposite side (bisecting the segment)

If 2 angles of a triangle are congruent than the third pair will be congruent too

If 2 angles of a triangle are congruent than the third pair will be congruent too

In a triangle sides opposite of congruent angles are congruent

In a triangle sides opposite of congruent angles are congruent

An angle bisector divides an angle into two equal parts

An angle bisector divides an angle into two equal parts

angles across from congruent sides are congruent

angles across from congruent sides are congruent

Chord of Central Angles

If two chords in a circle are congruent, then the two central angles are congruent

Chord Arcs

If two chords in a circle are congruent, then their arcs are congruent

Chord distance to center

two congruent chords in a circle are equidistance from the center

Perpendicular to a chord

the perpendicular bisector of a chord passes through the center of the circle. (perpendicular --> forms right angle, Bisector---> splits into 2 equal segments)

Inscribed angles

the measure of an angle inscribed in a circle is 1/2 the central angle (1/2 the arc bc central angle=arc)

Angle in semi-circle

Angles inscribed in a semicircle are right angles

Angles intercepting arcs

inscribed angles that intercept the same arc are congruent

Inscribed quadrilateral

for an inscribed quadrilateral, the opposite angles are supplementary (add up to 180)

Parallel Lines intercepted Arcs

Parallel lines intercept congruent arcs in a circle

Tangent to a Chord

Perpendicular to the center and If 2 tangent lines meet at a point; they are congruent.