Postulates, Theorems, and Corollaries. Chapter 1

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1 Chapter 1 Post Through any two points there is exactly one line. Post Through any three noncollinear points there is exactly one plane containing them. Post If two points lie in a plane, then the line containing those points lies in the plane. Post If two lines intersect, then they intersect in exactly one point. Post If two planes intersect, then they intersect in exactly one line. Post Ruler Postulate The points on a line can be put into a one-to-one correspondence with the real numbers. Post Segment Addition Postulate If B is between A and C, then AB + BC = AC. Post Protractor Postulate Given AB and a point O on AB, all rays that can be drawn from O can be put into a one-to-one correspondence with the real numbers from 0 to 180. Post Angle Addition Postulate If S is in the interior of PQR, then m PQS + m SQR = m PQR. Thm Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. PT2

2 Chapter 2 Thm Linear Pair Theorem If two angles form a linear pair, then they are supplementary. Thm Congruent Supplements Theorem If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent. Thm Right Angle Congruence Theorem All right angles are congruent. Thm Congruent Complements Theorem If two angles are complementary to the same angle (or to two congruent angles), then the two angles are congruent. Thm Common Segments Theorem Given collinear points A, B, C, and D arranged as shown, if AB CD, then AC BD. A B C D Thm Vertical Angles Theorem Vertical angles are congruent. Thm If two congruent angles are supplementary, then each angle is a right angle. PT3

3 Chapter 3 Post Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Thm Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Thm Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of alternate exterior angles are congruent. Thm Same-Side Interior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are supplementary. Post Converse of the Corresponding Angles Postulate If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel. Post Parallel Postulate Through a point P not on line l, there is exactly one line parallel to l. Thm Converse of the Alternate Interior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel. Thm Converse of the Alternate Exterior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel. Thm Converse of the Same-Side Interior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the two lines are parallel. Thm If two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular. Thm Perpendicular Transversal Theorem In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line. Thm If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other. Thm Parallel Lines Theorem In a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope. Any two vertical lines are parallel. Thm Perpendicular Lines Theorem In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. Vertical and horizontal lines are perpendicular. PT4

4 Chapter 4 Thm Triangle Sum Theorem The sum of the angle measures of a triangle is 180. Cor The acute angles of a right triangle are complementary. Cor The measure of each angle of an equiangular triangle is 60. Thm Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. Thm Third Angles Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third pair of angles are congruent. Post Side-Side-Side (SSS) Congruence Postulate If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. Post Side-Angle-Side (SAS) Congruence Postulate 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. Post Angle-Side-Angle (ASA) Congruence Postulate 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. Thm Angle-Angle-Side (AAS) Congruence Theorem If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent. Thm Hypotenuse-Leg (HL) Congruence Theorem If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. Thm Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite the sides are congruent. Thm Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. Cor If a triangle is equilateral, then it is equiangular. Cor If a triangle is equiangular, then it is equilateral. PT5

5 Chapter 5 Thm Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Thm 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. Thm Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. Thm Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. Thm Circumcenter Theorem The circumcenter of a triangle is equidistant from the vertices of the triangle. Thm Incenter Theorem The incenter of a triangle is equidistant from the sides of the triangle. Thm Centroid Theorem The centroid of a triangle is located 2 of the distance from each vertex to 3 the midpoint of the opposite side. Thm Triangle Midsegment Theorem A midsegment of a triangle is parallel to a side of a triangle, and its length is half the length of that side. Thm If two sides of a triangle are not congruent, then the larger angle is opposite the longer side. Thm If two angles of a triangle are not congruent, then the longer side is opposite the larger angle. Thm Triangle Inequality Theorem The sum of any two side lengths of a triangle is greater than the third side length. Thm Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the longer third side is across from the larger included angle. Thm Converse of the Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle and the third sides are not congruent, then the larger included angle is across from the longer third side. Thm Converse of the Pythagorean Theorem If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. Thm Pythagorean Inequalities Theorem In ABC, c is the length of the longest side. If c 2 > a 2 + b 2, then ABC is an obtuse triangle. If c 2 < a 2 + b 2, then ABC is an acute triangle. Thm Triangle Theorem In a triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times 2. Thm Triangle Theorem In a triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times 3. PT6

6 Chapter 6 Thm Polygon Angle Sum Theorem The sum of the interior angle measures of a convex polygon with n sides is (n - 2) 180. Thm Polygon Exterior Angle Sum Theorem The sum of the exterior angle measures, one angle at each vertex, of a convex polygon is 360. Thm If a quadrilateral is a parallelogram, then its opposite sides are congruent. Thm If a quadrilateral is a parallelogram, then its opposite angles are congruent. Thm If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. Thm If a quadrilateral is a parallelogram, then its diagonals bisect each other. Thm If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram. Thm If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Thm If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Thm If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. Thm If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Thm If a quadrilateral is a rectangle, then it is a parallelogram. Thm If a parallelogram is a rectangle, then its diagonals are congruent. Thm If a quadrilateral is a rhombus, then it is a parallelogram. Thm If a parallelogram is a rhombus, then its diagonals are perpendicular. Thm If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. Thm If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. Thm If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. Thm If one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. Thm If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. Thm If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus. Thm If a quadrilateral is a kite, then its diagonals are perpendicular. Thm If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. Thm If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent. Thm If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles. Thm A trapezoid is isosceles if and only if its diagonals are congruent. Thm Trapezoid Midsegment Theorem The midsegment of a trapezoid is parallel to each base, and its length is one half the sum of the lengths of the bases. PT7

7 Chapter 7 Post Angle-Angle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Thm Side-Side-Side (SSS) Similarity Theorem If the three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar. Thm Side-Angle-Side (SAS) Similarity Theorem If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. Thm Triangle Proportionality Theorem If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally. Thm Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. Cor Two-Transversal Proportionality Corollary If three or more parallel lines intersect two transversals, then they divide the transversals proportionally. Thm Triangle Angle Bisector Theorem An angle bisector of a triangle divides the opposite side into two segments whose lengths are proportional to the lengths of the other two sides. Thm Proportional Perimeters and Areas Theorem If the similarity ratio of two similar figures is a 2, then the ratio of their perimeters is a, and the ratio of their areas is a b b or b ( a 2 b) 2. PT8

8 Chapter 8 Thm The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle. Cor Geometric Means Corollary The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse. Cor Geometric Means Corollary The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Thm The Law of Sines For any ABC with side lengths a, b, and c, sin A a = sin B = sin C b c. Thm The Law of Cosines For any ABC with sides a, b, and c, a 2 = b 2 + c 2-2bc cos A, b 2 = a 2 + c 2-2ac cos B, and c 2 = a 2 + b 2-2ab cos C. PT9

9 Chapter 9 Thm A composition of two isometries is an isometry. Thm The composition of two reflections across two parallel lines is equivalent to a translation. The translation vector is perpendicular to the lines. The length of the translation vector is twice the distance between the lines. The composition of two reflections across two intersecting lines is equivalent to a rotation. The center of rotation is the intersection of the lines. The angle of rotation is twice the measure of the angle formed by the lines. Thm Any translation or rotation is equivalent to a composition of two reflections. PT10

10 Chapter 10 Post Area Addition Postulate The area of a region is equal to the sum of the areas of its nonoverlapping parts. PT11

11 Chapter 12 Thm If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. Thm If a line is perpendicular to a radius of a circle at a point on the circle, then the line is tangent to the circle. Thm If two segments are tangent to a circle from the same external point, then the segments are congruent. Post Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. Thm In a circle or congruent circles: (1) congruent central angles have congruent chords, (2) congruent chords have congruent arcs, and (3) congruent arcs have congruent central angles. Thm In a circle, if a radius (or diameter) is perpendicular to a chord, then it bisects the chord and its arc. Thm In a circle, the perpendicular bisector of a chord is a radius (or diameter). Thm Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. Cor If inscribed angles of a circle intercept the same arc or are subtended by the same chord or arc, then the angles are congruent. Thm An inscribed angle subtends a semicircle if and only if the angle is a right angle. Thm If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. Thm If a tangent and a secant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc. Thm If two secants or chords intersect in the interior of a circle, then the measure of each angle formed is half the sum of the measures of its intercepted arcs. Thm 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 half the difference of the measures of its intercepted arcs. Thm Chord-Chord Product Theorem If two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal. Thm Secant-Secant Product Theorem If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. Thm Secant-Tangent Product Theorem If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared. Thm Equation of a Circle The equation of a circle with center (h, k) and radius r is (x - h) 2 + (y - k) 2 = r 2. PT12

12 Constructions Angle Bisector p. 23 Center of a Circle p. 822 Centroid of a Triangle p. 326 Circle Circumscribed About a Triangle pp. 325, 826 Circle Inscribed in a Triangle p. 325 Circle Through Three Noncollinear Points p. 809 Circumcenter of a Triangle pp. 319, 325 Congruent Angles p. 22 Congruent Segments p. 14 Congruent Triangles Using ASA p. 261 Congruent Triangles Using SAS p. 251 Congruent Triangles Using SSS p. 256 Dilation pp. 650, 656 Equilateral Triangle p. 228 Incenter of a Triangle p. 325 Irrational Numbers p. 375 Kite p. 447 Midpoint p. 16 Midsegment of a Triangle p. 339 Orthocenter of a Triangle p. 332 Parallel Lines pp. 163, 170, 171, 179, 416 Parallelogram p. 416 Perpendicular Bisector of a Segment p. 172 Perpendicular Lines p. 179 Rectangle p. 436 Reflection pp. 604, 609 Regular Decagon p. 393 Regular Dodecagon p. 392 Regular Hexagon p. 392 Regular Octagon p. 392 Regular Pentagon p. 393 Rhombus pp. 427, 436 Right Triangle p. 266 Rotation pp. 619, 624 Segment Bisector p. 16 Square pp. 392, 436 Tangent to a Circle at a Point p. 794 Tangent to a Circle from an Exterior Point p. 827 Translation pp. 611, 616 Triangle Circumscribed About a Circle p. 827 PT13

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