Geometry Mastery: A Review Guide
Geometry Mastery: A Review GuideMany students struggle in Geometry for two reasons:1) Students are not familiar or comfortable with this style of learning. Proofs in Geometry are rooted in logical reasoning, and it takes hard work, practice, and timefor many students to get the hang of it.2) Students do not study for Geometry enough. Not only must students learn to use logical reasoning to solve proofs in Geometry, but they must be able to recallmany theorems and postulates to complete their proof. In order to recall the theorems, they need to recognize which to use based on the information provided andthe figure, and they must have the information stored in memory to actually retrieve it.This guide lists the theorems you will need to master in order to succeed in your Geometry class. This does not list every theorem proven in Geometry, but it shouldcover the content you will see in your Geometry class. Follow the below tips to ensure you are well- prepared on your Geometry tests.1.Highlight your theorems. Read each definition, theorem, postulate, or property in this guide, then find it in your notes. Highlight the theorem and itscorresponding example or definition2.Create pictures to help you recall your theorems. Draw a picture next to each theorem that confuses you in this guide. The picture should representliterally what the theorem states. Ask your teacher or a tutor for help if you have trouble determining what picture to draw.Here is an example of a picture you should draw next to the theorem and its definition:3.Vertical Angles TheoremIf two angles are vertical, then they are congruentACA C4.Create flashcards. If you’re still having trouble remembering these concepts, create a flashcard for each theorem with its definition and example picture.You will use this to memorize each theorem. On one side of the flashcard you should write the theorem’s name. On the other side you should write thedefinition on the left and its picture on the right:TheoremVertical Angles TheoremDefinitionPictureIf two angles are vertical,then they are congruentCA C5.Refer to this guide when you are working on practice proofs, and use your flashcards each night to help you memorize the theorems. As you learn moretheorems in class, you will memorize more theorems, so this process is continuous.6.Finally, practice! Many students realize that the homework that his or her teacher assigns is not enough to completely master the concepts. You mayneed to request extra work from your teacher or find additional problems elsewhere to give you plenty of practice.7.If you are serious about you class and you are determined to make a good grade, then preparation, practice, and memorization is essential. Using thisguide or creating one for yourself is the first step to organizing your theorems and effectively learning them.Action Potential Learning Houston’s Math & Science Education Specialists www.aplearning.comPrivate Tutoring Education Tools Camps
Basic GeometryPostulatesRuler PostulateEach point on a number line can be paired to a unique real numberSegment addition postulateWhere B is between points A and C on a line: AB BC ACAngle addition postulateWhere point P is inside ABC, ABP PBC ABCLinear pair postulateIf two angles forma linear pair (on a line), they are supplementaryTheoremsSegment Congruence TheoremTwo segments are congruent if and only if they have the same lengthCommon Segments TheoremFor collinear points A, B, C, and D: If AB CD, then AC BD.Angle Congruence TheoremTwo angles are congruent if and only if they have the same measureRight Angle Congruence TheoremIf two angles are right angles, then they are congruentVertical Angles TheoremsIf two angles are vertical, then they are congruentCongruent Complements TheoremIf two angles are complementary to the same angles, then they are congruentCongruent Supplements TheoremIf two angles are supplementary to the same angle, then they are congruentLinear Pair TheoremIf two angles form a linear pair, then they are supplementary.Basic PropertiesAddition property of equalityIf a b, then a c b cSubtraction property of equalityIf a b, then a-c b-cMultiplication property of equalityIf a b, then ac bcDivision property of equalityIf a b, then 𝑎 𝑐 𝑏 𝑐Symmetric propertyIf a b, then b cTransitive propertyIf a b and b c, then a cSubstitution propertyIf a b and b 25, then a 25Distributive propertyIf c(a b) ca cbDefinitionsMidpointWhen P bisects segment AB, AP is congruent to PBAngle BisectorWhen segment BP bisects angle ABC, ABP is congruent to PBCCongruent SegmentsLine segments that have the same lengthCongruent AngleAngles that have the same measurePerpendicular LinesTwo lines that form congruent adjacent angles that measure 90 degreesComplementary AnglesTwo adjacent angles whose sum measure 90 degreesSupplementary AnglesTwo adjacent angles whose sum measure 180 degreesAction Potential Learning Houston’s Math & Science Education Specialists www.aplearning.comPrivate Tutoring Education Tools Camps
Lines and anglesPostulatesParallel postulateIf there is a line and point somewhere not on that line, then there exists exactly one line through the point that isparallel to the other linePerpendicular postulateIf there is a line and one point somewhere not on that line, then there exists exactly one line through the pointthat is perpendicular to the other line.Slope of parallel linesTwo lines are parallel if and only if they have the same slopeSlope of perpendicular linesTwo lines are perpendicular if and only if the product of their slopes is equal to -1Corresponding angles postulateIf two parallel lines are cut by a transversal, then the corresponding angles are congruentCorresponding angles converseIf two coplanar lines cut by a transversal result in congruent corresponding angles, then the two lines are parallelTheoremsSame Side Interior Angles TheoremIf two parallel lines are cut by a transversal, then the two pairs of same side interior angles formed aresupplementary.Alternate Interior Angles TheoremIf two parallel lines are cut by a transversal, then the two pairs of alternate interior angles formed are congruent.Alternate Exterior Angles TheoremIf two parallel lines are cut by a transversal, then the two pairs of alternate exterior angles formed are congruent.Same side Interior Angles ConverseIf two coplanar lines cut by a transversal result in a pair of same side interior angles that are supplementary, thenthe two lines are parallelAlternate Interior Angles ConverseIf two coplanar lines cut by a transversal result in congruent alternate interior angles, then the two lines areparallelAlternate Exterior Angles ConverseIf two coplanar lines cut by a transversal result in congruent alternate exterior angles, then the two lines areparallelPerpendicular Transversal TheoremIf a transversal line is perpendicular to one of two parallel lines, then it is also perpendicular to the other parallelline.DefinitionsCoplanarPoints or lines existing in the same planeParallel LinesTwo coplanar lines that do not intersectTransversal lineA line that intersects two or more coplanar linesSkew LinesTwo lines that are not coplanar and do not intersectAction Potential Learning Houston’s Math & Science Education Specialists www.aplearning.comPrivate Tutoring Education Tools Camps
TrianglesPostulatesSide- Side- Side (SSS) Congruence PostulateIf all three sides of one triangle are congruent to the three sides of another triangle, then those triangles arecongruent.Side- Angle- Side (SAS) CongruencePostulateIf two sides and the included angle of one triangle are congruent to two sides and the included angle ofanother triangle, then those triangles are congruent.Angle- Side- Angle (ASA) CongruencePostulateIf two angles and the included side of one triangle are congruent to two angles and the included side ofanother triangle, then the triangles are congruent.Angle- Angle (AA) Similarity PostulateIf two angles of one triangle are congruent to two angles of another triangle then those two triangles aresimilar.TheoremsTriangle Sum TheoremThe sum of the angles of a triangle always equals 180Exterior Angle TheoremThe measure of an exterior angle of a triangle is equal to the sum of the two non-adjacent angles.Third Angle TheoremIf two angles of one triangle are congruent to two angles of another triangle, then the third angles arecongruent.Angle- Angle- Side Congruence TheoremIf two angles and a non-included side of one triangle are congruent to the corresponding two angles and anon- included side of another triangle, then the two triangles are congruent.Hypotenuse- Leg TheoremIf the hypotenuse and one leg of a triangle are congruent to the hypotenuse and leg of another triangle thenthe triangles are congruentIsosceles Triangle TheoremIf two sides of a triangle are congruent, then their opposite angles are also congruentPerpendicular Bisector TheoremIf a perpendicular line is drawn so that a point is on the bisector of a segment, then the point is equidistantfrom the endpoints of the segmentAngle Bisector TheoremIf a line is drawn so that a point is on the bisector of an angle, then the point is equidistant from the sides ofthe angle.Circumcenter TheoremThe circumcenter of a triangle is equidistant from that triangle’s vertices.Centroid TheoremThe centroid of a triangle occurs where the three lines created from each vertex to the midpoint of theopposite side intersectIncenter TheoremThe incenter of a triangle is equidistant from a triangles sides. It is created at the intersection of the threeangle bisectorsHinge TheoremIf two sides of one triangle are congruent to two sides of another triangle, but their included angles are notcongruent, then the third side across from the longer included angle is the larger side.Triangle Midsegment TheoremThe midsegment of a triangle is half the length of the side it is parallel to.Triangle Inequality TheoremThe sum of any two sides of a triangle is greater than the triangle’s third side.Pythagorean TheoremIf the triangle is a right triangle, then the sum of the squares of the lengths of the legs is equal to the squareof the length of the hypotenuse.Pythagorean Inequalities TheoremIf two sides of a triangle are congruent to two sides of another triangle; then when the included angle of thefirst triangle is greater than the included angle of the second triangle, the third side of the first triangle mustbe longer than the third side of the second triangle.45-45-90 Triangle TheoremIn a triangle with angle measures 45 , 45 , and 90 , the legs are congruent and the length of the hypotenuseis the length of the legs multiplied by 2.30-60-90 Triangle TheoremIn a triangle with angle measure 30, 60, and 90, the longer leg is the length of the shorter legs times 3 andthe length of the hypotenuse is 2 times the length of the shorter leg.Action Potential Learning Houston’s Math & Science Education Specialists www.aplearning.comPrivate Tutoring Education Tools Camps
Perpendicular Bisector ConverseThe point equidistant from the endpoints of a segment lies on the perpendicular bisector of that segment.Isosceles Triangle Theorem ConverseIf two angles of a triangle are congruent, then their opposite sides are also congruent.Angle Bisector ConverseIf a point located on the interior of an angle is equidistant from its sides, then it lies on the bisector of theangle.Hinge Theorem ConverseIf two sides of one triangle are congruent to two sides of another triangle, but their third sides are notcongruent, then the third angle across from the longer side is the larger angle.Pythagorean Theorem ConverseIf the sum of the squares of the length of the legs is equal to the square of the length of the hypotenuse in atriangle, then the triangle is a right triangle.Side- Side- Side (SSS) Similarity TheoremIf three sides of a triangle are proportional to three sides of another triangle, then the triangles are similar.Side- Angle- Side (SAS) Similarity TheoremIf two sides and their included angle of a triangle are proportional to two sides and their included angle ofanother triangle, then the triangles are similar.Triangle Proportionality TheoremIf a parallel line to one side of a triangle intersects the other two sides, then the sides are dividedproportionally.Triangle Proportionality Theorem ConverseIf a line intersects two sides of a triangle and those sides are divided proportionally, then the line is parallelto the third side.Triangle Angle Bisector TheoremAn angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional tothe other two sides of the triangleThe Law of Sines TheoremFor any triangle ΔABC with sides a, b, and c,The Law of Cosines TheoremFor any triangle ΔABC with sides a, b, and c:a2 b2 c2 - 2bcosA; b2 a2 c2 – 2bcosB; c2 a2 b2 – 2abcosAsin 𝐴𝑎 sin 𝐵𝑏 sin 𝐶𝑐The altitude of the hypotenuse of a right triangle forms two triangles that are similar to each other and aresimilar to the original triangleIf two sides of a triangle are not congruent to each other, then the largest angle is always opposite thelonger side.If two angles of a triangle are not congruent to each other, then the longer side is always opposite thelargest angle.Triangle PropertiesReflexiveSymmetricA triangle is congruent to itself.If ΔABC ΔDEF, then ΔDEF ΔABCIf ΔABC ΔDEF and ΔDEF ΔGHK, then ΔABC ΔGHKTransitiveDefinitionsIncluded angleThe angle between two linesIncluded sideThe side between two anglesTriangleA figure with three sides and three anglesEquilateral TriangleA triangle with three equal sides and each angle measures 60 Isosceles TriangleA triangle with two equal sides and two equal anglesScalene triangleA triangle with no equal sides and no equal anglesAction Potential Learning Houston’s Math & Science Education Specialists www.aplearning.comPrivate Tutoring Education Tools Camps
QuadrilateralsPostulatesArea Addition PostulateThe area of a region is the sum of the areas of each non- overlapping part.Area Congruence PostulateIf two polygons are congruent, then they have the same area.Area of a Square PostulateThe area of a square is the square of the length of a side.TheoremsInterior Angles of a QuadrilateralThe sum of the measures of the interior angles of a quadrilateral is 306 .Polygon Angle Sum TheoremThe sum of the measures of the interior angles of a polygon with n sides is(n-2) 180 .Polygon Exterior Angle SumThe sum of the measures of the exterior angles of a polygon is 360 .Trapezoid Midsegment TheoremThe midsegment of a trapezoid is parallel to each of the trapezoids’ bases, and its length is one half the sumof the lengths of the bases.Area of a ParallelogramThe area of a parallelogram is the product of a base and its corresponding height.Area of a RectangleThe area of a rectangle is the product of the base and its height.Area of a RhombusThe area of a rhombus is on half the product of the lengths of its diagonals.Area of a KiteThe area of a kite is one half the product of the length of its diagonals.Area of a TrapezoidThe area of a trapezoid is one half the product of the height and the sum of the base.Area of a TriangleThe area of a triangle is one half the product of the base and its corresponding height.Proportional Perimeters and AreasTheoremsIf the ratio of two similar figures is , then the ratio of their perimeters is and the ratio of their areas is𝑎𝑎𝑎2𝑏𝑏𝑏2.If a quadrilateral is a parallelogram, then its opposite sides are congruent.If a quadrilateral is a parallelogram, then its opposite angles are congruent.If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is aparallelogram.If a quadrilateral is a parallelogram, then its diagonals bisect each other.If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.If both pairs of the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.If both pairs of the opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.If a pair of opposite sides are parallel and congruent to each other, then the quadrilateral is a parallelogram.If a quadrilateral is a rectangle, then it is a parallelogram.If a parallelogram is a rectangle, then its diagonals are congruent.If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle.If a quadrilateral is a rhombus, then it is a parallelogram.If a parallelogram is a rhombus then its diagonals are perpendicular to each other.Action Potential Learning Houston’s Math & Science Education Specialists www.aplearning.comPrivate Tutoring Education Tools Camps
If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.If a parallelogram is a rhombus, then each of its diagonals bisects a pair of opposite angles.If a diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.If a pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.If a quadrilateral is a kite, then its diagonals are perpendicular.If a quadrilateral is a kite, then exactly one pair of its opposite angles are congruent.If a quadrilateral is an isosceles trapezoid, then each pair of its base angles are congruent.If a pair of base angles or a trapezoid are congruent, the trapezoid is isosceles.A trapezoid is isosceles if and only if its diagonals are congruent.DefinitionsPolygonA plane figure with at least three straight sides and angleParallelogramA quadrilateral with pairs of parallel opposite sides.RectangleA parallelogram with four congruent angles.RhombusA parallelogram with four congruent sides.SquareA parallelogram with four congruent angles and four congruent sides.KiteA quadrilateral with two pairs of consecutive congruent sides, with the opposite sides not congruent.TrapezoidA quadrilateral with exactly one pair of parallel sides.Equilateral PolygonAny polygon with all sides of the same length.Isosceles PolygonA polygon with two sides of the same length.Action Potential Learning Houston’s Math & Science Education Specialists www.aplearning.comPrivate Tutoring Education Tools Camps
CirclesPostulatesArc Addition PostulateThe measure of an arc that is formed by two adjacent arcs is equal to the sum of those two arcs.TheoremsInscribed Angle TheoremThe measure of an inscribed angle is half the measure of its central angle (or half the measure of itsintercepted arc)Chord- Chord Power TheoremIf two chords intersect in the interior of a circle, then the product of the segments of one chord is equal to theproduct of the segment of the other chord.Secant- Secant Power TheoremIf two secants intersect in the exterior of a circle, then the product of the measures of one secant segmentand its external segment is equal to the product of the measures of the other secant segment and its externalsegment.Secant- Tangent Power TheoremIf a secant segment and a tangent segment intersect on the exterior of a circle, then the product of themeasures of the secant segment and its external segment is equal to the square of the measure of thetangent segment.Equation of a CircleThe equation of a circle with center (a,b) and radius r is(x-a)2 (y-b)2 r2If a line is tangent to a circle, then it is perpendicular to the radius of the circle drawn to the tangent point.If a line is perpendicular to the radius of a circle at a point on the circle, then the line is tangent to the circle.If two segments are tangent to a circle from the same point external to the circle, then the segments arecongruent.Congruent central angles also have congruent cords; Congruent cords also have congruent arcs; Congruentarcs also have congruent central angles.If the radius of a circle is perpendicular to a chord, then it bisects both the chord and its arc.The perpendicular bisector of a chord is a radius or diameter of the circle.If inscribed angles of a circle intercept the same arc then the angles are congruent.An angle inscribed in a semicircle is always a right angleIf a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.If a tangent and a secant or chord intersect on a tangent point of a circle and the sides of the angles interceptarcs on the circle, then the measure of the angle formed is half the measure of the intercepted arc.If two secants or two chords intersect in the interior of a circle and the sides of the angles intercept arcs onthe circle, then the measure of each of the angles formed is half the sum of the measures of the interceptedarc.If two secants intersect in the exterior of a circle, and the sides of the angle intercept arcs on the circle, thenthe measure of the angle formed is equal to half the difference of the measures of its intercepted arcs.DefinitionsChordA segment whose endpoints both lie on a circleSecantA segment that intersects two points on a circleTangentA line that intersects the circle (or any curve) on a point where the slope of the line is equal to the slop of thecircleArcA closed segment on a circle (or any curve)Action Potential Learning Houston’s Math & Science Education Specialists www.aplearning.comPrivate Tutoring Education Tools Camps
Triangle Sum Theorem Exterior Angle Theorem Third Angle Theorem Angle - Angle- Side Congruence Theorem non Hypotenuse- Leg Theorem the triangles are congruent Isosceles Triangle Theorem Perpendicular Bisector Theorem If a perpendicular line is
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