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Honors Geometry Theorems Name_______________ Date_______________ Possible Algebraic Reasons Reason Meaning If a statement is listed in the given information of the problem. Given For every number a, a a. Ref. Example for segments: AB AB Sym. Trans. For all numbers a and b , if a b , then b a. Example for segments: If AB CD, then CD AB a, b , and c, For all numbers , and b c , then a c. if a b Example for segments: and CD EF , then AB EF . If AB CD numbers a, b , and c, For all if a b , then a c b c. Sub. a, b , and c, For all numbers , then a c b c. if a b Mult. a, b , and c, For all numbers , then a c b c . if a b Add. Div. Subst. Reference.doc a, b , and c, For all numbers and c 0, then a/c b/c. if a b For all numbers a and b , if a b , then a may be replaced by b in any equation or expression. Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Reason Meaning Dist. For all numbers a, b , and c, a(b + c) = a ³ b + a ³ c . Comm. For all numbers a, b , and c, a + b = b + a. Reference.doc Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Possible Reasons for Segments Reason Meaning Def. Two segments having the same measure are congruent. SAP If B is between A and C, then AB BC AC. Def. midpt. Def. bis. Reference.doc of PQ The midpoint is a point between P M and Q such that PM MQ. Any segment, ray or line that passes through of a segment, bisectsthe the midpoint segment. Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Possible Reasons for Angles Reason Meaning If R is in the interior of PQS , then mPQR mRQS mPQS. AAP Def. comp. Two angles with measures that have a sum of 90. Def. supp. Two angles with measures that have a sum of 180. Def. lin. pr. Def. rt. Def. bis. VAT A pair of adjacent angles whose noncommon sides are opposite rays. An angle with a degree measure of 90. A ray that divides an angle into two congruent parts. If two angles are vertical angles, then they are congruent. ST Supplements of congruent angles are congruent. CT Complements of congruent angles are congruent. Reference.doc Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Possible Reasons for Isosceles Triangles Reason Meaning ITT If two sides of a triangle are congruent, then the angles opposite those sides are congruent. CITT Reference.doc If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Possible Reasons for Angles in Parallel Lines Reason Meaning PAI If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent. PAE If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent. PCA If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent. PCIS If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary. PCES If two parallel lines are cut by a transversal, then each pair of consecutive exterior angles is supplementary. 2PT Reference.doc In a plane, if a line (transversal) is perpendicular to one of two parallel lines, then it is perpendicular to the other. Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Possible Reasons for Proving Parallel Lines Reason Meaning AIP If two lines are cut by a transversal and a pair of alternate interior angles is congruent, then the lines are parallel. AEP If two lines are cut by a transversal and a pair of alternate exterior angles is congruent, then the lines are parallel. CAP If two lines are cut by a transversal and a pair of corresponding angles is congruent, then the lines are parallel. CISP If two lines are cut by a transversal and a pair of consecutive interior angles is supplementary, then the lines are parallel. CESP If two lines are cut by a transversal and a pair of consecutive exterior angles is supplementary, then the lines are parallel 2PT Reference.doc If two lines in a plane are each perpendicular to the same transversal, then the lines are parallel. Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Possible Reasons for Angles #2 Reason Meaning AST The sum of the measures of the angles of a triangle is 180. TAT If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent. EAT The measure of an exterior angle of a triangle is equal to the sum of the two remote interior angles. Reference.doc Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Possible Reasons for Congruent Triangles Reason SSS SAS ASA AAS Meaning If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of a second triangle, then the triangles are congruent. CPCTC If two triangles are congruent, then corresponding parts of the congruent triangles are congruent. CPCTE If two triangles are congruent, then the measures of corresponding parts of the congruent triangles are equal. Reference.doc Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Possible Reasons for Triangle Inequalities Reason EAI BSBA BABS TI SASI (HINGE) SSSI (CONVERSE HINGE) Reference.doc Meaning If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles. If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. The sum of the length of any two sides of a triangle is greater than the length of the third side. If two sides of a triangle are congruent to two sides of another triangle and the included angle in one triangle has a greater measure than the included angle in the other, then the third side of the first triangle is longer than the third side of the second triangle. If two sides of a triangle are congruent to two sides of another triangle and the third side in one triangle is longer than the third side in the other, then the angle between the pair of congruent sides in the first triangle is greater than the corresponding angle in the second triangle. Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Possible Reasons for Right Triangles Reason Meaning LL If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent. HA LA HL Reference.doc If the hypotenuse and acute angle of one right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the triangles are congruent. If one leg and an acute angle of one right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the triangles are congruent. If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Properties of Parallelograms Picture Meaning B A D C J K L M K J L M K J L M J K M L P C D B A D Reference.doc Theorem: If a quadrilateral is a parallelogram, then its opposite sides are congruent. Theorem: If a quadrilateral is a parallelogram, then its opposite angles are congruent. Theorem: If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. Theorem: If a parallelogram has one right angle, then it has four right angles. B A Definition: If a quadrilateral is a parallelogram, then both pairs of opposite sides are parallel. C Theorem: If a quadrilateral is a parallelogram, then its diagonals bisect each other. Theorem: If a quadrilateral is a parallelogram, then each diagonal separates the parallelogram into two congruent triangles. Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Conditions for Parallelograms Picture A Meaning B Definition: If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram. B Theorem: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. B Theorem: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. B Theorem: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. B Theorem: If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. C D A C D A C D A C D A D Reference.doc C Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Properties of Rectangles Picture Meaning A B D C A B D C A B D C A B D C A B D C J K M L Reference.doc Definition: If a quadrilateral is a rectangle, then all four angles are right angles. Definition: If a quadrilateral is a rectangle, then its opposite sides are parallel and congruent. Definition: If a quadrilateral is a rectangle, then its opposite angles are congruent. Definition: If a quadrilateral is a rectangle, then its consecutive angles are supplementary. Definition: If a quadrilateral is a rectangle, then its diagonals bisect each other. Theorem: If a parallelogram is a rectangle, then its diagonals are congruent. Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Conditions for Rectangles Picture Meaning W X Z Y Reference.doc Theorem: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Properties of Rhombi and Squares Picture Meaning B A D C B A D C P N 5 3 1 6 4 7 2 8 Q R E F H G Reference.doc Definition: If a parallelogram is a rhombus, then its four sides are congruent. Theorem: If a parallelogram is a rhombus, then its diagonals are perpendicular. Theorem: If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. Definition: If a parallelogram is a square, then its four sides are congruent and its four angles are right angles. Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Conditions for Rhombi and Squares Picture Meaning B A D C 5 3 Theorem: If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. P N 1 6 7 4 2 8 Q R B A D C E F H G Reference.doc Theorem: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. Theorem: If one pair of consecutive sides of a parallelogram is congruent, the parallelogram is a rhombus. Theorem: If a quadrilateral is both a rectangle and a rhombus, then it is a square. Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Properties of and Conditions for Trapezoids Picture Meaning G H E F M N L K M N L K M N L K A B C Reference.doc F E D Definition: If a quadrilateral is a trapezoid, then it has exactly one pair of parallel sides. Theorem: If a trapezoid is isosceles, then each pair of base angles is congruent. Theorem: If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid. Theorem: A trapezoid is isosceles if and only if its diagonals are congruent. Theorem: The midsegment of a trapezoid is parallel to each base and its measure is one half the sum of the lengths of the bases. Miss Jo Ann Fricker Lower Moreland HS Honors Geometry Theorems Name_______________ Date_______________ Properties of Kites Picture Meaning P S Q Definition: If a quadrilateral is a kite, then it has exactly two pairs of consecutive congruent sides. Q Theorem: If a quadrilateral is a kite, then its diagonals are perpendicular. R P S R K J L Theorem: If a quadrilateral is a kite, then exactly one pair of opposite angles is congruent. M Reference.doc Miss Jo Ann Fricker Lower Moreland HS
