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 Property A quantity is congruent (equal) to itself.a = a Symmetric Property If a = b, then b = a. Transitive Property If a = b and b = c, then a = c.
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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 Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment Segment Congruence Theorem Two segments are congruent if and only if they have the same length. Betweeness Theorem If B is between A and C, then AB+BC=AC Angle Congruence Theorem Two angles are congruent if and only if they have equal measures CPCF Theorem i.e. Congruent Parts of Congruent Figures are congruent If two figures are congruent, then any pair of corresponding parts is congruent SSS Congruence Theorem If three sides of one triangle are congruent to three sides of the other, then the triangles are congruent SAS Congruence Theorem If, 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 Theorem If, 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 Theorem If, 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 Theorem If, 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 Theorem If, 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 Theorem In 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 Theorem In any parallelogram each diagonal forms two congruent triangles, opposite sides are congruent, and the diagonals intersect at their midpoints SAS Inequality Theorem If 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 Theorem If 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 Property If a/b = c/d, then ad = bc Exchange Property If a/b = c/d, then a/c = b/d Reciprocal Property If a/b = c/d, then b/a = d/c Similar Figures Theorem If two figures are similar, then corresponding angles are congruent and corresponding lengths are proportional Fundamental Theorem of Similarity If 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 Theorem If the three sides of one triangle are proportional to three sides of a second triangle, then the triangles are similar. AA Similarity Theorem If two triangles have two angles of one congruent to two angles of the other, then the triangles are similar. SAS Similarity Theorem In 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 Theorem If 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 Theorem A 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 Theorem In 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 Inequality In a triangle, the measure of an exterior angle is equal to the sum of the measures of the two nonadjacent interior angles. Unequal Sides Theorem If 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 Theorem If 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 Theorem In an isosceles right triangle, if a leg is x then the hypotenuse is x times the 2. 30-60-90 Triangle Theorem In 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 Theorem Of all solids with the same volume, the sphere has the least surface area. Isoperimetric Theorem Of all solids with the same surface area, the sphere has the max volume Isoperimetric Theorem Of all plane figures with the same area, circle has the least perimeter Isoperimetric Inequality If a plane figure has area A and perimeter P, then A is greater then or equal to P2/4 . Tangent Square Theorem The power of point P for circle O is square of the length of a segment tangent to circle O. Secant Length Theorem Suppose 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 Theorem The measure of an angle between two tangents, or between a tangent and a secant, is half the distance of the intercepted arcs Tangent-Chord Theorem The measure of an angle formed by a tangent and a chord measures half of the measures of intercepted arc. Angle-Secant Theorem The measure of an angle formed by two secants intersecting outside a circle is half the difference of the arcs intercepted by it. Angle-Chord Theorem The 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 Theorem In 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 Theorem The 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)

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