Postulates and Theorems - CompassLearning …

1 P o s tu l ate s an d Th e o r e m sP r o p e r ti e s an d P o s tu l ate sSegment Addition PostulatePoint B is a point on segment AC, B is between A and C, if and only if AB + BC = ACConstructionFrom a given point on (or not on) a line, one and only one perpendicular can be drawn to the points determine a straight PostulateThe whole is equal to the sum of its PostulateA quantity may be substituted for its equal in any PostulateIf equal quantities are divided by equal nonzero quantities, the quotients are PostulateIf equal quantities are multiplied by equal quantities, the products are equal. Subtraction PostulateIf equal quantities are subtracted from equal quantities, the differences are PostulateIf equal quantities are added to equal quantities, the sums are PropertyIf a = b and b = c, then a = PropertyA quantity is congruent (equal) to itself.

2 A = a Symmetric PropertyIf a = b, then b = n g l e sExterior AngleThe measure of an exterior angle of a triangle isequal to the sum of the measures of the twonon-adjacent interior Angle theorem (Isosceles Triangle)If two sides of a triangle are congruent, the anglesopposite these sides are Angle Converse(Isosceles Triangle)If two angles of a triangle are congruent, the sidesopposite these angles are SumThe sum of the interior angles of a triangle is 180 .Vertical AnglesVertical angles are PairIf two angles form a linear pair, they ComplementsComplements of the same angle are of congruent angles are SupplementsSupplements of the same angle are of congruent angles are AnglesAll straight angles are Addition PostulatePoint C lies in the interior of ABD , if and only if, m ABC + m CBD = m ABDR ight AnglesAll right angles are i an g l eIf an angle of one triangle is congruent to the corresponding angle ofanother triangle and the lengths of the sides including these angles arein proportion, the triangles are two triangles are similar.

3 The corresponding sides are in segment connecting the midpoints of two sides of a triangle isparallel to the third side and is half as the three sets of corresponding sides of two triangles are in proportion,the triangles are two angles of one triangle are congruent to two angles of anothertriangle, the triangles are parts of congruent triangles are leg of a right triangle is the mean proportional between thehypotenuse and the projection of the leg on the two sides of a triangle are congruent, the angles opposite these sidesare two angles of a triangle are congruent, the sides opposite these anglesare a triangle, the longest side is across from the largest a triangle, the largest angle is across from the longest sum of the lengths of any two sides of a triangle must be greaterthan the third sideThe altitude to the hypotenuse of a right triangle is the mean proportionalbetween the segments into which it divides the the hypotenuse and leg of one right triangle are congruent to thecorresponding parts of another right triangle, the two right triangles two angles and the non-included side of one triangle are congruent tothe corresponding parts of another triangle.

4 The triangles are two angles and the included side of one triangle are congruent to thecorresponding parts of another triangle, the triangles are three sides of one triangle are congruent to three sides of anothertriangle, then the triangles are for SimilaritySide ProportionalityMid-segment theorem (also called mid-line)SSS for SimilarityAngle-Angle (AA)Similarity CPCTCLeg RuleBase Angle theorem (Isosceles Triangle)Base Angle Converse(Isosceles Triangle)Longest SideSum of Two SidesAltitude RuleHypotenuse-Leg (HL)Congruence (right triangle)Angle-Angle-Side (AAS)CongruenceAngle-Side-Angle (ASA)Congruence Side-Side-Side (SSS)Congruence Side-Angle-Side (SAS)

5 Congruence If two sides and the included angle of one triangle are congruent to thecorresponding parts of another triangle, the triangles are al l e l sIf two lines are cut by a transversal and the interior angles on the sameside of the transversal are supplementary, the lines are two lines are cut by a transversal and the alternate exterior angles arecongruent, the lines are two lines are cut by a transversal and the alternate interior angles arecongruent, the lines are two parallel lines are cut by a transversal, the interior angles on thesame side of the transversal are supplementaryIf two lines are cut by a transversal and the alternate interior angles arecongruent, the lines are two parallel lines are cut by a transversal, then the alternate interiorangles are two parallel lines are cut by a transversal.

6 Then the pairs ofcorresponding angles are congruentInteriors on Same SideConverseAlternate Exterior AnglesConverseAlternate Interior Angles ConverseInteriors on Same SideAlternate Exterior AnglesAlternate Interior AnglesCorresponding AnglesCorresponding AnglesConverseIf two lines are cut by a transversal and the corresponding angles arecongruent, the lines are i r c l e sAn angle inscribed in a semi-circle is a right a circle, inscribed circles that intercept the same arc are opposite angles in a cyclic quadrilateral are a circle, or congruent circles, congruent central angles have congruent the same circle, or congruent circles, congruent central angles have congruentarcs.

7 (and converse)Tangent segments to a circle from the same external point are congruentIn a circle, a radius perpendicular to a chord bisects the chord and the a circle, a radius that bisects a chord is perpendicular to the a line is tangent to a circle, it is perpendicular to the radius drawn to the pointof a circle, or congruent circles, congruent chords are equidistant from the center.(and converse)In a circle, or congruent circles, congruent chords have congruent arcs.(and converse)In the same circle, or congruent circles, congruent central angles have congruentchords (and converse)In a circle, parallel chords intercept congruent arcs