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Chapter 1

1-2 A point is the smallest thing in geometry. It has no size. A set of points that is straight and continuous without gaps that extends indefinitely in each direction is called a line. A line has length but no width. A plane is a flat surface that extends in all directions indefinitely. A segment consists of two points of a line and all the points between them. A ray, or half line, consists of one point of a line and all points of the line in one direction.

1-3 The precision of a ruler is the smallest unit of measure marked off. An angle is formed by two rays with a common endpoint, called a vertex. Angles can be classified by their angle measurement, usually degrees. Acute angles measure less than 90°, right angles measure 90°, obtuse angles measure between 90° and 180°, and straight angles measure 180°.

1-4 The ancient Greeks were particularly interested in drawings that could be made with only a compass and a straightedge. They called these drawings constructions.

1-5 TransIations, reflections, and rotations are transformations that move figures to new positions.

1-6 Inductive reasoning is reaching a conclusion on the basis of a series of examples. An example showing that a statement is not always true is called a counterexample.

Chapter 2

2-1 Deductive reasoning allows us to reach a conclusion based on given assumptions and rules of logic.

2-2 We can use the intuitive understanding of points, lines, and planes to give definitions for other terms. Space is the set of all points. Collinear points are points on the same line. Noncollinear points are points that are not all on the same line. Coplanar points are points that are in the same plane. Noncoplanar points are points that are not all in the
same plane. The intersection of two figures is the set of points that they have in common.

2-3 We begin the study of geometry with certain statements that we assume to be true. We call these statements postulates.
Postulate 1 Given any two points, there is exactly one line containing the two points.
Postulate 2 Given any three noncollinear points, there is exactly one plane containing them.
Postulate 3 Any line contains at least two points. Any plane contains at least three noncollinear points. Space contains at least four noncoplanar points.
Postulate 4 If two points lie in a plane, then the line containing them is in the plane.
Postulate 5 If two planes intersect, then their intersection is a line.

2-4 When definitions and postulates are used to prove a statement that statement is called a theorem.
Theorem 2.1 If two lines intersect, then they intersect in exactly one point.
Theorem 2.2 If a line and a plane intersect and the line is not in the plane, then they intersect in a point.
Theorem 2.3 If l is a line and P is a point not on the line, then l and P are contained in exactly one plane.

2-5 “If-then” sentences are called conditional sentences. The “if” part of the sentence is
called the hypothesis. The “then” part of the sentence is called the conclusion.

If we interchange the two parts of a conditional sentence, we obtain a new conditional sentence called its converse. If a conditional sentence is true, the converse might not be true.

Chapter 3

3-1 The number that corresponds to a point on number line is called the coordinate. The point with coordinate zero is called the origin. The distance between two points is the absolute value of the difference of the coordinates.

3-2 A line segment, AB, is the set of points of a line containing A, B, and all points between A and B. A and B are called the endpoints of the segment. A ray, AB, is the subset of a line AB that contains A and all points on the same side of the line as B. A point C between the points A and B on the line AB determines the opposite rays CA and CB with C the endpoint of each ray. The midpoint M is said to bisect segment AB when AM = MB.

3-3 A point on a plane can be named by a pair of numbers called its coordinates. The x-axis and y-axis intersect at right angles at a point called the origin. The x-axis and the y-axis determine the x-y plane.

3-4 An angle consists of two rays with a common endpoint, called a vertex.

3-5 We measure angles with a protractor. The unit of angle measure is called a degree.

Chapter 4

4-1 Congruent segments are segments that have the same length. Congruent angles are angles that have the same measure.

4-2 Two angles are complementary whenever the sum of their measures is 90°. Each angle is a complement of the other.
Two angles are supplementary whenever the sum of their measures is 180°. Each angle is a supplement of the other.

4-3 Two angles that have a common side and whose interiors do not intersect are called adjacent. Two adjacent angles are called a linear pair when their noncommon sides are opposite rays.

4-4 Two nonstraight angles are vertical angles whenever their sides form two pairs of opposite rays. Vertical angles are congruent.

4-5 Two lines are called perpendicular when they intersect to form right angles. Four right angles are formed. The perpendicular bisector of a segment is a line perpendicular to the segment at the segment's midpoint

Chapter 5

5-1 A triangle consists of three segments determined by three noncollinear points. The segments are the sides and the points the vertices. In ABC, side BC and angle A are opposite each other. AB is called the included side of angle A and angle B. Likewise, angle C is called the included angle of AC and BC. Triangles may be classified by their angles as acute, equiangular, right, or obtuse. Triangles may also be classified by their sides as equilateral, isosceles,or scalene.

5-2 A figure has reflectional symmetry if the reflected image of each point on one side line dividing the figure is also a point of the figure.The line is called the line of symmetry.

5-3 Two triangles are congruent whenever their vertices can be matched so that the corresponding sides and the corresponding angles are congruent. There are six corresponding parts of two congruent triangles: three sides and three angles. These are called corresponding parts of congruent triangles. Corresponding parts of congruent triangles are congruent.

5-4 Marks are added to drawings to show right angles and to indicate the congruent parts in figure. Segments with equal number of marks are congruent. Congruent angles are indicated in the same way.

5-5 Triangles can be shown congruent by several postulates. SSS (Side-Side-Side) Postulate two triangles are congruent if three sides of one triangle are congruent to three sides of the other triangle. SAS (Side-Angle-Side) Postulate two triangles are congruent if two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle. ASA (Angle-Side-Angle) Postulate two triangles are congruent if two angles and the included side of one triangle are congruent to two angles and the included side of the other triangle.

5-6 There are many ways to make a convincing argument that two triangles are congruent. One way is to make a drawing, mark the congruent parts, and decide whether the triangles are congruent by SSS.SAS, or ASA.

5-7 Sometimes we can conclude that angles and segments are congruent by first showing that triangles are congruent.

5-8 AAS (Angle-Angle-Side) Postulate two triangles are congruent if two angles and a non-included side of one triangle are congruent to two angles and the same non-included side of the other triangle. HL (Hypotenuse-Leg) Postulate two right triangles are congruent if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of the other right triangle.

Chapter 6

6-1 Isosceles triangles have at least two congruent sides, called legs. The third side is called the base. The angles opposite the congruent sides are called the base angles. The angle opposite the base is called the vertex angle.
Theorem 6.1 The Isosceles Triangle Theorem Two sides of a triangle are congruent whenever the angles opposite them are congruent. Two angles of a triangle are congruent whenever the sides opposite them are congruent

6-2 An angle that forms a linear pair with an angle of a triangle is called an exterior angle of the triangle. The angles of the triangle that are not adjacent to a specific exterior angle are called remote interior angles. There are two exterior angles at each vertex. These are each associated with the same pair of remote interior angles.
Theorem 6.2 The Exterior Angle Theorem The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles.

6-3 Theorem 6.3 The Opposite Parts Theorem In any ABC, if CA > CB, then measure of angle B > measure of
angle A.
Theorem 6.4 In any ABC, if measure of angle B > measure of angle A, then CA > CB.

6-4 Theorem 6.5 The Triangle Inequality The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

6-5 The Hinge theorem and its converse tell us about inequalities in two triangles. Theorem 6.6 The Hinge Theorem In ABC and DEF, if AB = DE and AC = DP, and measure of angle A < measure of angle D, then BC < EF.
Theorem 6.7 In ABC and DEF, if AB = DE and AC = DP, and BC < EF, then measure of angle A < measure of
angle D.

6-6 Lines are concurrent if they intersect in a single point.
The point of intersection is called the point of concurrency. Segments and rays are concurrent if they are contained in concurrent lines. (Theorem 6.8) The angle bisectors of a triangle are concurrent in a point that is equidistant from the sides of a triangle, called the incenter. While the (Theorem 6.9) perpendicular bisectors of the sides of a triangle are concurrent in a point that is equidistant from the vertices, called the circumcenter.

Chapter 7

7-1 Two lines are parallel whenever they are in the same plane and do not intersect. We write k || m to say k is parallel to m .Two segments or rays are parallel whenever the lines containing them are parallel. Two lines that do not lie in the same plane are called skew lines. Two segments contained in two skew lines are called skew segments.

7-2 A transversal is a line that intersects two or more coplanar lines in different points. When a transversal intersects a pair of lines, eight angles are formed. Certain pairs of these have special names: interior angles, corresponding angles, alternate interior angles, and alternate exterior angles.

7-3 When a transversal intersects two lines, we can look at the related angles that are formed and determine if the lines are parallel.
Theorem 7.1 If two lines and a transversal form congruent alternate interior angles, then the lines are parallel.
Theorem 7.2 If two lines and a transversal form congruent corresponding angles, then the lines are parallel.
Theorem 7.3 If two lines and a transversal form congruent alternate exterior angles, then the lines are parallel.
Theorem 7.4 If two lines and a transversal form supplementary interior angles on the same side of the transversal, then the lines are parallel.

7-4 Postulate 15 The Parallel Postulate Given a line k and a point P not on k there is at most one line that contains P and is parallel to k .
Theorem 7.5 If a transversal intersects two parallel lines, then the alternate interior angles are congruent.
Theorem 7.6 If a transversal intersects two parallel lines, then the corresponding angles are congruent.
Theorem 7.7 If a transversal intersects two parallel lines, then the alternate exterior angles are congruent.
Theorem 7.8 If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary .

7-5 There are many theorems involving the angles of a triangle.
Theorem 7.9 The Angle Sum Theorem The sum of the measures of the angles of a triangle is 180°.
Theorem 7.l0 If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
Theorem 7.11 Acute angles of a right triangle are complementary.
Theorem 7.12 The measures of an exterior angle of a triangle is the sum of the measures of the two remote interior angles.

Chapter 8

8-1 A quadrilateral consists of four coplanar segments that intersect only at their endpoints. Each endpoint belongs to exactly two segments. The segments are called sides, the endpoints are the vertices. A diagonal of a quadrilateral is a segment that joins two opposite vertices. The sum of the measures of the angles of a quadrilateral is 360°. Some special types of quadrilaterals are defined by their sides and angles. Among these classifications are parallelogram, two pairs of parallel sides; rectangle, a parallelogram with four right angles; rhombus, a parallelogram with four congruent sides; square, a rectangle with four congruent sides or a rhombus with four right angles; trapezoid, exactly one pair of parallel sides; and kite, exactly two pairs of congruent consecutive sides.

8-2 Parallelograms have many properties, among them:
Theorem 8.2 A diagonal of a parallelogram determines two congruent triangles.
Theorem 8.3 The opposite angles of a parallelogram are congruent.
Theorem 8.4 The opposite sides of a parallelogram are congruent.
Theorem 8.5 Consecutive angles of a parallelogram are supplementary.
Theorem 8.6 The diagonals of a parallelogram bisect each other.
The perimeter of a quadrilateral is the sum of the lengths of its four sides.

8-3 Converses of several of the properties of a parallelogram can be used to determine whether a quadrilateral is a parallelogram.
Theorem 8.7 If the opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Theorem 8.8 If the opposite sides of a quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Theorem 8.9 If the diagonals of a quadrilateral bisect each other; then the quadrilateral is a parallelogram.
Theorem 8.10 If two sides of a quadrilateral are both parallel and congruent then the
quadrilateral is a parallelogram.

8-4 Rhombuses, rectangles, and squares are special kinds of parallelograms. They have two pairs of parallel sides along with other properties. Here are some properties involving diagonals:
Theorem 8.11 A parallelogram is a rhombus if its diagonals are perpendicular.
Theorem 8.12 A parallelogram is a rectangle if its diagonals are congruent.
Theorem 8.13 A parallelogram is a square if its diagonals are perpendicular and congruent.

8-5 A trapezoid is a quadrilateral that has exactly one pair of parallel sides. The bases of a trapezoid are the pair of parallel sides. The legs of a trapezoid are the other two sides. The median of a trapezoid is the segment determined by the midpoints of the legs.
Theorem 8.14 A median of a trapezoid is parallel to the bases and is half as long as the sum of the lengths of the two bases.
An isosceles trapezoid is a trapezoid with two congruent, non-parallel sides.
Theorem 8.15 In an isosceles trapezoid, each pair of base angles is congruent.
Theorem 8.16 The diagonals of an isosceles trapezoid are congruent.

Chapter 9

9-1 A comparison of two numbers by division is called a ratio. The ratio of a to b is the quotient, a/b, which may also be written as a ÷b or a : b . When writing ratios it is important that the numbers being compared are expressed in the same units.

9-2 A proportion is a statement that two ratios are equal. We say the two ratios are proportional. The following theorem provides a convenient way to check whether two ratios form a proportion.
Theorem 9.1 If a/b = c/d , then ad = bc .

9-3 Two figures are similar if their vertices can be matched so that the corresponding sides are proportional and the corresponding angles are congruent. The ratio of any two corresponding sides gives the scale factor that transforms one figure into its image. Similar figures can be thought of as size transformations of each other.

9-4 Two triangles are similar whenever their vertices can be matched so that corresponding angles are congruent and the lengths of corresponding sides are proportional. We can use proportions and the fact that triangles are similar to find missing lengths.

9-5 Postulate 16 The AAA Similarity Postulate For any two triangles, if the corresponding angles are congruent, then the triangles are similar.
Theorem 9.2 The AA Similarity Theorem For any two triangles, if two pairs of corresponding angles are congruent then the triangles are similar.

9-6 In addition to the AA Similarity Theorem, there are two other ways we can determine if triangles are similar. They are summarized in the following theorems:
Theorem 9.3 The SAS Similarity Theorem For any two triangles, if one pair of corresponding angles is congruent and the sides that include this angle are proportional, then the triangles are similar.
Theorem 9.4 The SSS Similarity Theorem For any two triangles, if all three pairs of corresponding sides are proportional, then the triangles are similar.

Chapter 10

10-1 In the figure to the right, triangle PQR is a right triangle with the right angle at Q . QS is an altitude, the length of which is the height of triangle PQR.
Theorem 10.1 In a right triangle, the altitude to the hypotenuse
forms two triangles, with both triangles similar to the right
triangle and each one similar to the other.
The positive number x is called the geometric mean of
the positive numbers a and b whenever a/x = x/b .
Theorem 10.2 In a right triangle, the length of the altitude to the hypotenuse is the geometric mean of the lengths of the segments on the hypotenuse. a/h = h/b

10-2 One of the most important theorems in all of geometry is call the Pythagorean Theorem . Theorem 10.3 The Pythagorean Theorem In a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.

10-3 We can use the converse of the Pythagorean theorem to determine whether a triangle is a right triangle.
Theorem 10.4 The Converse of the Pythagorean Theorem If the lengths of the sides of a triangle are a, b, and c, and

then the triangle is a right triangle with right angle opposite the longest side, whose length is c.

10-4 A special right triangle is the isosceles right triangle. Theorem 10.5 The Isosceles Right Triangle Theorem
For any isosceles right triangle, the hypotenuse is

times as long as either leg. Since the angles in an isosceles right triangle are 45°, 45°, and 90° the Isosceles Right Triangle Theorem is sometimes called the 45°- 45°- 90° Triangle Theorem.

10-5 Another special right triangle is the 30°- 60° right triangle. Theorem 10.6 The 30°- 60° Right Triangle Theorem
For any 30°- 60° right triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is

times as long as the shorter leg. Since the angles of an equilateral triangle are each 60°, and the angle formed by the altitude is
30°, we can use the 30°- 60° Right Triangle Theorem to find the height of an equilateral triangle.

Chapter 11

11-1 A polygon is a figure formed by the line segments connecting three or more coplanar points. The segments are called sides and the endpoints are called vertices. Each side intersects exactly two other sides, one at each vertex. No three consecutive vertices can be collinear. A polygon is convex whenever no line containing a side of the polygon intersects the interior of the polygon. If a polygon is not convex, then it is concave.

11-2 A diagonal of a polygon is a segment that joins two vertices of a polygon but is not a side. Theorem11.1 For any polygon with n sides, he number of diagonals, D, is
1/2 n(n - 3).

11-3 The perimeter of a polygon is the sum of the lengths of its sides.

11-4 Theorem 11.2 The sum, S, of the angle measures of any convex polygon with n sides is given by the formula
S = (n- 2) 180. An angle is an exterior angle of a convex polygon whenever it forms a linear pair with an angle of the polygon. Theorem 11.3 For a convex polygon, the sum of the measures of the exterior angles, one at each vertex, is 360°.

11-5 Two polygons are similar whenever their vertices can be matched so that the corresponding angles are congruent and the lengths of the corresponding sides are proportional. Similarity can be used to find the lengths of sides of polygons.

11-6 A convex polygon is regular whenever it is both equilateral and equiangular. Theorem 11.4 The measure of each interior angle of a regular polygon with n sides is
(n - 2) 180/ n

Theorem 11.5 The measure of each exterior angle of a regular polygon with n sides is
360/ n

Chapter 12

12-1 A polygonal region is a polygon and its interior. The Area Postulate states that for every polygonal region, there corresponds one positive number called its area. This number is dependent upon the given unit. The area of a polygonal region is the sum of the areas of the non overlapping regions that it contains. Congruent regions have the same area.

12-2 We often call the adjacent sides of a rectangle the length and the width of the rectangle. We can also call them the base and the height. A rectangle can be rotated so that any side of the rectangle is the base and its adjacent side is the height.
Postulate 20 The area of a rectangle is the product of the lengths of any two adjacent sides. Area = bh, where A is the area, b is the base length, and h is the corresponding height.
Theorem 12.1 The area of a square is the square of the length of a side.

12-3 Any side of a parallelogram may be used as a base. The height, h, is the length of any segment that is
perpendicular to the lines containing the base and the side opposite the base. The perpendicular segment is called the altitude of the parallelogram.
Theorem 12.2 The area of a parallelogram is the product of a base and the corresponding height.

12-4 Any side of a triangle may be used as a base. The height, h, is the length of any segment that is perpendicular
to the base and contains the opposite vertex.
Theorem 12.3 The area of a triangle is one-half the product of a base and the corresponding height.

12-5 Recall that the bases of a trapezoid are the pair of parallel sides. The height, h, is the length of any segment that is perpendicular to the lines containing the bases. The perpendicular segment is called the altitude of the trapezoid.
Theorem 12.4 The area of a trapezoid is one-half the product of the sum of its bases and the its height.

Chapter 13

13-1 A circle is the locus of all points in a plane a given distance (r) from a given point (0) in the plane. The center is point 0, The radius is a segment whose endpoints are the center of the circle and a point on the circle. The length of this segment is also called the radius. A chord is a segment with both endpoints on a circle. A chord that contains the center is a diameter. All radii of a circle are congruent (Theorem 13.1) and the diameter, d, of a circle is twice the
length of the radius, r, of the circle, d = 2r (Theorem 13.2).

13-2 Chords of circles have some special properties summarized by these theorems:
Theorem 13.3 If a radius of a circle is perpendicular to a chord, then it bisects the chord.
Theorem 13.4 If a radius of a circle bisects a chord that is not a diameter, then it is perpendicular to the chord. Theorem 13.5 The perpendicular bisector of a chord of a circle contains the center of the circle.

13-3 A line, coplanar with a circle, is tangent to the circle if it intersects the circle in exactly one point. A secant is a line that intersects a circle in two points. A line is tangent to circle 0 at a point P whenever the line is perpendicular to the radius OP at P (Theorem 13.7). A tangent segment is a segment from a point to a circle that is contained in a tangent. The tangent segments from a point to a circle are congruent (Theorem 13.8).

13-4 A central angle is an angle whose vertex is the center of a circle. The part of the circle cut off by a central angle is called an arc. The measure of an arc is the same as the measure of its central angle. Arcs are classified by their degree measure: a minor arc has angle measure less than 180°, a semicircle has angle measure of 180°, and a major arc has angle measure greater than 180°.

13-6 An inscribed angle is an angle whose vertex is on the circle and whose sides each intersect the circle in one other point. Inscribed angles are related to their intercepted arc by the following theorems. The measure of an
inscribed angle is one-half the measure of its intercepted arc (Theorem 13.10).
Theorem 13.11 An angle inscribed in a semicircle is a right angle.
Theorem 13.12 Inscribed angles are congruent if they intercept the same arc or congruent arcs.

13-7 A polygon is inscribed in a circle if every vertex of the polygon lies on the circle.
13-8 The circle is said to be circumscribed about the polygon. The distance around a circle is its circumference. The ratio circumference÷diameter is the same number for all circles (Theorem 13.14) and we use the Greek letter π (pi) to represent this number.
13-9 The area of a circle with radius r is . A sector of a circle is a region bounded by two radii and an arc.
The area of a sector with radius r and arc (or central angle) measure n/360 •

Chapter 14 - this space is reserved for chapter 14 summary

Chapter 15 - this space is reserved for chapter 15 summary

Chapter 16 - this space is reserved for chapter 16 summary