# Reflexive example: AB = AB Symmetric example: AB = BA

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2.1 Properties of Segment Congruence: Reflexive, symmetric, and transitive
Reflexive example: AB = AB Symmetric example: AB = BA Transitive example: AB = CD; CD= BC therefore AB = BC

2.2 Properties of Angle Congruence: Angle congruence is reflexive, symmetric, and transitive.
Reflexive example: m<A = m< A Symmetric example: m<A = m< B; then m<B = m<A Transitive example: m<A = m<B; m<B = m<C; therefore m<A = m<C

2.3 Right Angle Congruence Theorem: All right angles are congruent
Ex:

2.4 Congruent Supplements Theorem: If 2 angles are supplementary to the same angle (or to congruent angles) then they are congruent. EX: m<1 + m<2 = 180 m<2 + m<3 =180 Therefore… m<1= m<3

2.5 Congruent Complements Theorem: If 2 angles are complementary to the same angle (or to congruent angles) then the 2 angles are congruent. EX: m<4 + m<5 = 90 M<5 + m<6 = 90 Therefore… m<4 = m< 6

2.6 Vertical angles theorem: Vertical angles are congruent.
EX: m<1 = m<3 M<2 = m<4

3.1 If 2 lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. EX:

3.2 If 2 sides of 2 adjacent acute angles are perpendicular, then they are complementary.
EX:

3.3 If 2 sides of 2 adjacent acute angles are perpendicular, then they intersect to form 4 right angles. EX:

3.4 Alternate Interior Angles: If 2 parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. EX: m<3 = m< 5 m<4 = m< 6

3.5 Consecutive Interior Angles: if 2 parallel lines are cut by a transversal, the n the pairs of consecutive interior angles are supplementary. EX: m<3 + m<6 =180 m<4 + m<5 =180

3.6 Alternate Exterior Angles: If 2 parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. EX: m<1 = m< 7 m<2 = m<8

3.7 Perpendicular Transversal: If a transversal is perpendicular to 1 of 2 parallel lines, then it is perpendicular to the others. EX:

3.8 Alternate Interior Angles Converse: If 2 lines are cut by a transversal so that the alternate interior angles are congruent, then the lines are parallel EX: If m<3 = m<5 or m<4 = m<6 then r ll s

3.9 Consecutive Interior Angles Converse: If 2 lines are cut by a transversal so that the consecutive interior angles are supplementary, then the lines are parallel. EX: m<3 + m<6 = 180 m<4 + m<5 = 180 Then r ll s

3.10 Alternate Exterior Angles Converse: If 2 lines are cut by a transversal so that the alternate exterior angles are supplementary, then the lines are parallel EX: m<1 = m<7 or m<2 = m< 8 then r ll s

3.11 If 2 lines are parallel to the same line, then they are parallel to each other.
EX: If p ll q and q ll r Then p ll r

3.12 In a plane, if 2 lines are perpendicular to the same line, then they are parallel to each other. EX:

4.1 Triangle Sum Theorem: the sum of the measures of the interior angles of a triangle is 180 degrees. EX: m<A + m<B + m<C = 180

4.2 Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the 2 non adjacent interior angles. EX: m<1 = m<A + m<B

4.3 Third Angle Theorem: If 2 angles of one triangle are congruent to the 2 angles of another triangle, then the third angles are also congruent. EX: If m<A = m<D and A D m<B = m<E then m<C = m<F B C E F

4.4 Reflexive Property of Congruent Triangles: Every triangle is congruent to itself Symmetric Property of congruent triangles examples Transitive property of congruent triangles examples EX: ABC = ABC ABC = DEF, then DEF = ABC ABC = DEF, and DEF= JKL, then ABC = JKL

4.5 Angle-Angle-Side (AAS): Congruence Theorem: If 2 angles and a nonincluded side of 1 triangle are congruent to 2 angles and the corresponding nonincluded side of a second triangle, then the 2 triangles are congruent. Angle m<A = m<D B E Angle m<B = m< E Side BC = EF A C D F

4.6 Base Angles Theorem: If 2 sides of a triangle are congruent, then the angles opposite them are congruent. If AB = CB, then m<A = m<C B A C

4.7 Converse of the Base Angles Theorem: If 2 angles of a triangle are congruent, then the sides opposite them are congruent. If m<A = m< C, then AB = CB B A C

4.8 Hypotenuse-Leg (HL) congruence theorem: If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second triangle, then the 2 triangles are congruent. If BC = EF, and AC = DF, then ABC = DEF A D ***2 RIGHT TRIANGLES*** B C E F

5.1 Perpendicular Bisector Theorem: If a point is on a perpendicular bisector of a segment, then it is equidistant from the end points of the segment. If CP is a perpendicular bisector of AB, then CA = CB

5.2 Converse of the Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. If DA = DB, then D lies on the perpendicular bisector of AB.

5.3 Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. If m<BAD = m<CAD, then DB = DC

5.4 Converse of the Angle Bisector Theorem: If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle. If DB = DC, then m<BAD = m<CAD

5.5 Concurrency if the Perpendicular Bisectors of a Triangle: The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. PA = PB = PC B A C

5.6 Concurrency of Angle Bisectors of a Triangle: The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. PD = PE = PF B D F A P E C

5.7 Concurrency of the Medians of a Triangle: The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side If P is the centroid of triangle ABC, then AP= 2/3 AD, BP= 2/3 BF,and CE= 2/3 CP

5.8 Concurrency of Altitudes of a Triangle: The lies containing the altitudes of a triangle are congruent. If AE, BF, and CF are the altitudes of triangle ABC, then the lines AE, BF, and CD intersect at some point.

5.9 Midsegment Theorem: The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. DE ll AB, and DE = ½ AB

5.10 If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. m<A > m<C B A C

5.11 If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. EF > DF

5.12 Exterior Angle Inequality: The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles. m<1 > m<A and m<1 > m<B

5.13 Triangle Inequality: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. AB + BC > AC A AC + BC > AB AB + AC > BC C B

5.14 Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second RT > VX R V 100 S T W X

5.15 Converse of the Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second. m<A > m<D B E A D C F

6.1 Interior Angles of Quadrilateral: The sum of the measures of the interior angles of a quadrilateral is 360. m<1 + m<2 + m<3 + m<4 = 360 degrees

6.2 If a quadrilateral is a parallelogram, then its opposite sides are congruent
PQ = RS and SP = QR Q R P S

6.3 If a quadrilateral is a parallelogram, then its opposite angles are congruent.
m<P = m<R and m<Q = m<S Q R P S

6.4 If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
m<P + m<Q = 180, m<Q + m<R = 180 M<R + m<S = 180, m<S + m<P = 180 Q R P S

6.5 If a quadrilateral is a parallelogram, then its diagonals bisect each other
QM = SM and PM = RM Q R P S

6.6. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is parallelogram. ABCD is a parallelogram A B C D

6.7 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. ABCD is a parallelogram A B C D

6.8 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. ABCD is a parallelogram A B C D

6.9 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is parallelogram.
ABCD is a parallelogram A B C D

ABCD is a parallelogram A B C D
6.10 If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. Rhombus Corollary: A quadrilateral is a rhombus if and only if it has four congruent sides. Rectangle Corollary: A quadrilateral is a rectangle if and only if it has four right angles. Square Corollary: A quadrilateral is a square if and only if it is a rhombus and a rectangle. ABCD is a parallelogram A B C D

6.11 A parallelogram is a rhombus if and only if its diagonals are perpendicular.
ABCD is a rhombus iff AC perpendicular to BD B C A D

6.12 A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.
ABCD is a rhombus iff AC bisects <DAB and <BCD and BD bisects <ADC and <CBA B C A D

6.13 A parallelogram is a rectangle if and only if its diagonals are congruent.
ABCD is a rectangle iff AC = BD A B C D

6.14 If a trapezoid is isosceles, then each pair of base angles is congruent.
m<A = m<B, m<C =m<D A B C D

6.15 If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid
ABCD is an isosceles trapezoid A B C D

6.16 A trapezoid is isosceles if and only if its diagonals are congruent.
ABCD is isosceles iff AC = BD A B C D

6.17 Midsegment Theorem for Trapezoids: The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. MN ll AD, MN ll BC, MN = ½ (AD + BC) B C M N A D

6.18 If quadrilateral is a kite, then its diagonals are perpendicular.
AC perpendicular BD C B D A

6.19 If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.
m<A = m<C, m<B is not equal m<D C B D A

6.20 Area of a Rectangle: The area of a rectangle is the product of its base and height.
A = bh h b

6.21 Area of a Parallelogram: The area of a parallelogram is the product of a base and its corresponding height. A = bh h b

6.22 Area of a Triangle: The area of a triangle is one half the product of a base and its corresponding height. A = ½ bh h b

6.23 Area of a Trapezoid: The area of a trapezoid is one half the product of the height and the sum of the bases. A = ½ h(b 1 + b 2 ) b1 h b2

6.24 Area of a Kite: the area of a kite is one half the product of the lengths of its diagonals.
A = ½ d1 *d2 d1 d2

6.25 Area of a Rhombus: the area of a rhombus is equal to one half the product of the lengths of diagonals. A = ½ d1 *d2 d1 d2

7.1 Reflection Theorem: A reflection is an isometry
Isometry is a transformation that preserves lengths line of reflection

7.2 Rotation Theorem: A rotation is an isomentry.
Point of rotation 180 degree rotation

7.3 Rotational theorem using lines and coordinate planes
Rotational point

7.4 Translation Theorem: A translation is an isometry.
P’ P Q’ Q

7.5 Reflections k m Q Q’ Q’’ P P’ P”

7.6 Composition Theorem: The composition of two (or more) isometrics is an isometry
When 2 or more transformations are combined to produce a single transformation EX: reflection then translation

8.1 If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths. Ratio perimeter = ratio of sides K L P Q N M S R

8.2 Side-Side-Side (SSS) Similarity Theorem: If the corresponding sides of two triangles are proportional, then the triangles are similar. IF AB = BC = CA PQ QR RP’ Then ABC ~ PQR

8.3 Side-Angle-Side (SAS) Similarity Theorem: If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. If m<X = m<M and ZX = XY PM MN’ Then XYZ ~ MNP

8.4 Triangle Proportionally Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. If TU ll QS, then RT = RU TQ US

8.5 Converse of the Triangle Proportionally Theorem: If a line divides two sides of a triangle proportionally, then it is parallel to the third side. IF RT = RU, then TU ll QS TQ US

8.6 If three parallel lines intersect two transversals, then they divide the transversals proportionally. If r ll s and s ll t, and l and m intersect r, s, and t, then UW = VX WY XZ

8.7 If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. If CD bisects <ACB, then

9.1 If an altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.

9.2 In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse divides they hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments.

9.3 In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. Each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

9.4 Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

9.5 Converse of the Pythagorean Theorem: If the square of the length of the longest sides a triangle is equal to the sum of the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

9.6 If the square of the length of the longest side of triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute.

9.7 If the square of the length of the longest side of a triangle is greater than the sum of the squares of the length of the other two sides, then the triangle is obtuse.

9.8 45° -45 ° -90° Triangle Theorem: In a 45° -45 ° -90° triangle, the hypotenuse is √2 times as long as each leg.

9.9 30° -60° -90° Triangle Theorem: In a 30° -60° -90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg √3 times as long as the shorter leg.

10.1 If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

10.2 In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

10.3 If two segments from the same exterior point are tangent to a circle, then they are congruent.

10.4 In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

10.5 If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

10.6 If one chord is a perpendicular bisector of another chord, then the first chord is a diameter.

10.7 In the same circle or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

10.8 If an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc.

10.9 If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

10.10 If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one of an inscribed triangle is a diameter of the circle, then the triangle is right triangle and the angle opposite the diameter is the right angle.

10.11 A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

10.12 If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

10.13 If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

10.14 If a tangent and a secant, two tangents, or two secants intersect in the exterior of a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

10.15 If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

10.16 If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segments equals the product of the length of the other secant segment and the length of its external segment.

10.17 If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.

11.1 Polygon Interior Angles Theorem: The sum of the measures of the interior angles of a convex n-gon is (n-2) ∙ 180°.

11.2 Polygon Exterior Angles Theorem: The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360°. Corollary: The measure of each exterior angle of regular n-gon is 1/n ∙ 360°, or 360°/n

11.3 Area of an Equilateral Triangle: The area of an equilateral triangle is one fourth the square of the length of the side times √3.

11.4 Area of a Regular Polygon: The area of a regular n-gon with side length s is half the product of the apothem a and the perimeter P, so A = ½ aP, or A = ½a ∙ ns.

11.5 Areas of Similar Polygons: If two polygons are similar with the lengths of corresponding sides in the ratio of a:b, then the ratio of their areas is a2:b2.

11.6 Circumference of a Circle: The circumference C of a circle is C = pd or C = 2pr, where d is the diameter of the circle and r is the radius of the circle. Arc Length Corollary: In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360°.

11.7 Area of a Circle: The area of a circle is p times the square of the radius, or A = pr2.

11.8 Area of a Sector: The ratio of the area A of a sector of a circle to the area of the circle is equal to the ratio of the measure of the intercepted arc to 360°.

12.1 Euler’s Theorem: The number of faces (F), vertices (V), and the edges of a polyhedron are related by the formula F + V= E+2.

12.2 Surface Area of a Right Prism: The surface area S of a right prism can be found using the formula S = 2B+Ph where B is the area of a base, P is the perimeter of a base, and h is the height.

12.3 Surface Area of Right Cylinder: The surface area S of a right cylinder is S = 2B + Ch = 2pr2 + 2prh2, where B is the area of a base, C is the circumference of a base, r is the radius of a base and h is the height.

12.4 Surface Area of a Regular Pyramid: The surface area S of a regular pyramid is S of a regular pyramid is S = B + ½ P l, where B is the area of a base, P is the perimeter of the base, and l is the slant height.

12.5 Surface Area of a Right Cone: The surface area S of a right cone S =pr2 +prl, where r is the radius of the base and l is the slant height.

12.6 Cavalieri’s Principle: IF two solids have the same height and the same cross- sectional area at every level then they have the same volume.

12.7 Volume of a Prism: The volume V of a prism is V =Bh, where B is the area of a base and h is the height.

12.8 Volume of a Cylinder: The volume of V of a cylinder is V = Bh = pr2h, where B is the area of a base, and h is the height, and r is the radius of a base.

12.9 Volume of a Pyramid: The volume V of a pyramid is V = 1/3Bh, where B is the area of a base, h is the height.

12.10 Volume of a Cone: The volume V of a cone is V = 1/3Bh = 1/3pr2h, where B is the area of the base, h is the height, and r is the radius of the base.

12.11 Surface Area of a Sphere: The surface area S of a sphere with radius r is S = 4pr2.

12.12 Volume of a Sphere: The volume V of a sphere with the radius r is V = 4/3pr3.

12.13 Similar Solids Theorem: If two similar solids have a scale factor of a:b, then corresponding areas have a ratio of a2:b2, and corresponding volumes have a ratio of a3:b3.

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