Skip Navigation Links
   
Geometry: Similarity and Congruence
To enroll in any of our courses, click here
 
   
In this lesson you'll learn how to determine that two figures are similar or congruent. Further on you'll know the important and related definitions, postulates, and theorems related to similar or congruent figures. The following approach may be taken: A. Analyze the properties of geometric figures. B. Identify and/or verify congruent and similar figures. Then apply equality or proportionality of their corresponding parts.

Main properties to remember and use:

Reflexive PropertyA quantity is congruent (equal) to itself.a = a         
Symmetric PropertyIf a = b, then b = a.
Transitive PropertyIf a = b and b = c, then a = c.
People who saw this lesson also found the
following lessons useful:
Circumcircles
Complementary Angles
Spherical Shell
Instalment Buying
Median
Learn and remember:

Important Theorems

Figure Reflection Theorem If a figure is determined by certain points, then its reflection image is the corresponding figure determined by the reflection images of those points 
Perpendicular Bisector TheoremIf a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment
Segment Congruence TheoremTwo segments are congruent if and only if they have the same length.
Betweeness TheoremIf B is between A and C, then AB+BC=AC
Angle Congruence TheoremTwo angles are congruent if and only if they have equal measures
CPCF Theorem i.e. Congruent Parts of Congruent Figures are congruentIf two figures are congruent, then any pair of corresponding parts is congruent
SSS Congruence TheoremIf three sides of one triangle are congruent to three sides of the other, then the triangles are congruent
SAS Congruence TheoremIf, two sides and the included angle of one triangle, are congruent to two sides and the included angle of the other, then the triangles are congruent
ASA Congruence TheoremIf, two angles and the included side of one triangle, are congruent to two sides and the included angle of the other, then the triangles are congruent
AAS Congruence TheoremIf, two angles and a non-included side of one triangle, are congruent respectively to two angles and a non-included side of the other then the two triangles are congruent
Hyp-Leg Congruence TheoremIf, the hypotenuse and leg of one triangle are congruent to the hypotenuse and leg of the other, then the two triangles are congruent
SSA Congruence TheoremIf, two sides and the angle opposite the longer of the two sides in one triangle are congruent respectively to two sides and the angle opposite the longer of the two sides in the other, then the two triangles are congruent.
Right Triangle Altitude TheoremIn a right triangle- The altitude to the hypotenuse is the geometric mean of the segments into which it divides the hypotenuse.
Each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.
Properties of a Parallelogram TheoremIn any parallelogram each diagonal forms two congruent triangles, opposite sides are congruent, and the diagonals intersect at their midpoints
SAS Inequality TheoremIf two sides of a triangle are congruent to two sides of a second triangle, and the measure of the included angle of the first triangle is less than the measure of the included angle of the second, then the third side of the first triangle is shorter than the third side of the second
Figure Size Change TheoremIf a figure is defined by certain points, then its changed size image is the corresponding figure determined by the size change images of those points. Notice the properties.
Extremes PropertyIf a/b = c/d, then ad = bc
Exchange Property If a/b = c/d, then a/c = b/d
Reciprocal PropertyIf a/b = c/d, then b/a = d/c
Similar Figures TheoremIf two figures are similar, then corresponding angles are congruent and corresponding lengths are proportional
Fundamental Theorem of SimilarityIf P ~ P' and r is the ratio of similitude, then:
Perimeter (P') = r times Perimeter (P)
Area (P') = r(sq) times Area(P)
Volume (P') = r(cu) times Volume (P)
SSS Similarity TheoremIf the three sides of one triangle are proportional to three sides of a second triangle, then the triangles are similar.
AA Similarity TheoremIf two triangles have two angles of one congruent to two angles of the other, then the triangles are similar.
SAS Similarity TheoremIn two triangles, if the ratios of two pairs of corresponding sides are equal and the included angles are congruent, then the triangles are similar.
Side-Splitting TheoremIf a line intersects ray OP and ray OQ in distinct points X and Y such that OX/XP = OY/YQ, then line XY is parallel to line PQ.
Radius-Tangent TheoremA line is tangent to a circle; if and only if it is perpendicular to the radius at endpoint of the radius of circle.
Exterior Angle TheoremIn a triangle, the measure of an exterior angle is equal to the sum of the measures of the two non adjacent interior angles.
Exterior Angle InequalityIn a triangle, the measure of an exterior angle is equal to the sum of the measures of the two nonadjacent interior angles.
Unequal Sides TheoremIf two sides of a triangle are not congruent, then the angles opposite to them are not congruent, and the largest angle is opposite the longer side.
Unequal Angles TheoremIf two angles of a triangle are not congruent, then the sides opposite to them are not congruent, and the longer side is opposite the larger angle
Isosceles Right Triangle TheoremIn an isosceles right triangle, if a leg is x then the hypotenuse is x times the 2.
30-60-90 Triangle TheoremIn a 30-60-90 right triangle, if the short leg is x then the longer leg is x times 3 and the hypotenuse is 2x.
Isoperimetric TheoremOf all solids with the same volume, the sphere has the least surface area.
Isoperimetric TheoremOf all solids with the same surface area, the sphere has the max volume
Isoperimetric TheoremOf all plane figures with the same area, circle has the least perimeter
Isoperimetric InequalityIf a plane figure has area A and perimeter P, then A is greater then or equal to P2/4.
Tangent Square TheoremThe power of point P for circle O is square of the length of a segment tangent to circle O.
Secant Length TheoremSuppose one secant intersects a circle at A and B, and a second secant intersects the circle at C and D. If the secants intersect at P, then AP times BP equals CP times DP.
Tangent-Segment TheoremThe measure of an angle between two tangents, or between a tangent and a secant, is half the distance of the intercepted arcs
Tangent-Chord TheoremThe measure of an angle formed by a tangent and a chord measures half of the measures of intercepted arc.
Angle-Secant TheoremThe measure of an angle formed by two secants intersecting outside a circle is half the difference of the arcs intercepted by it.
Angle-Chord TheoremThe measure of an angle formed by two intersecting chords is one-half the sum of the measures of the arcs intercepted by it and its vertical angle.
Inscribed Angle Theorem In a circle the measure of an inscribed angle is one-half the measure of its intercepted arc.
Arc-Chord Congruence TheoremIn a circle or in congruent circles:
(1) If two arcs have the same measure, they are congruent and their chords are congruent
(2) If two chords have the same length, their minor arcs have the same measure.
Chord-Center Theorem(1) The line containing the center of a circle perpendicular to a chord bisects the chord.
(2) The line containing the center of a circle and the midpoint of a chord bisects the central angle determined by the chord.
(3) The bisector of the central angle of a chord is perpendicular to the chord and bisects the chord.
(4) The perpendicular bisector of a chord of a circle contains the center of the circle
Vector Addition TheoremThe sum of the vectors (a,b) and (c,d) is the vector (a + c, b + d). Notice the following properties:
A.Vector addition is commutative.
B.Vector addition is associative
C.(0,0) is an identity for vector addition
D.Every vector (a,b) has an additive inverse (-a,-b)
 
   
As many of you know, Winpossible's online courses use a unique teaching method where an instructor explains the concepts in any given area to you in his/her own voice and handwriting, just like you see your teacher explain things to you on a blackboard in your classroom. All our courses include teacher's instruction, practice questions as well as end-of-lesson quizzes for practice. You can enroll in any of our online courses by clicking here.

The format of Winpossible's online courses is also very suitable for teachers who are using an interactive whiteboard such as Smartboard on Promethean in their classrooms, because the course lessons can be easily displayed on such interactive whiteboards. Volume pricing is available for schools interested in our online courses. For more information, please contact us at educators@winpossible.com.

 
       
     
 Copyright © Winpossible, 2010 - 2011
Best viewed in 1024x768 & IE 5.0 or later version