LESSON 1 Geometry ReviewLESSON 1 Geometry Review

1.1 1.1 points, lines, and rayspoints, lines, and rays

Some fundamental mathematical terms are Some fundamental mathematical terms are impossible to define exactly. We call these terms impossible to define exactly. We call these terms primitive terms or undefined terms. We define primitive terms or undefined terms. We define these terms as best we can and then use them to these terms as best we can and then use them to define other terms. The words point, curve, line, define other terms. The words point, curve, line, and plane are primitive terms.and plane are primitive terms.

A point is a location. When we put a dot on a A point is a location. When we put a dot on a piece of paper to mark a location, the dot is not piece of paper to mark a location, the dot is not the point because a mathematical point has no the point because a mathematical point has no size and the dot does have size. We say that the size and the dot does have size. We say that the dot is the graph of the mathematical point and dot is the graph of the mathematical point and marks the location of the point. A curve is an marks the location of the point. A curve is an unbroken connection of points. Since points have unbroken connection of points. Since points have no size, they cannot really be connected. Thus, no size, they cannot really be connected. Thus, we prefer to say that a curve defines the path we prefer to say that a curve defines the path traveled by a moving point. We can use a pencil traveled by a moving point. We can use a pencil to graph a curve. These figures are curves.to graph a curve. These figures are curves.

A mathematical line is a straight curve that has no A mathematical line is a straight curve that has no ends. Only one mathematical line can be drawn that ends. Only one mathematical line can be drawn that passes through two designated points. Since a line passes through two designated points. Since a line defines the path of a moving point that has no width, defines the path of a moving point that has no width, a line has no width. The pencil line that we draw a line has no width. The pencil line that we draw marks the location of the mathematical line. When marks the location of the mathematical line. When we use a pencil to draw the graph of a mathematical we use a pencil to draw the graph of a mathematical line, we often put arrowheads on the ends of the line, we often put arrowheads on the ends of the pencil line to emphasize that the mathematical line pencil line to emphasize that the mathematical line has no ends.has no ends.

We can name a line by using a single letter (in this We can name a line by using a single letter (in this case, p) or by naming any two points on the line in case, p) or by naming any two points on the line in any order. The line above can be called line AB, any order. The line above can be called line AB, line BA, line AM, line MA, line BM, or line MB. line BA, line AM, line MA, line BM, or line MB. Instead of writing the word line, a commonly used Instead of writing the word line, a commonly used method is to write the letters for any two points method is to write the letters for any two points on the line in any order and to use an overbar with on the line in any order and to use an overbar with two arrowheads to indicate that the line continues two arrowheads to indicate that the line continues without end in both directions. All of the following without end in both directions. All of the following notations name the line shown above. These notations name the line shown above. These notations are read as "line AB," "line BA," etc. notations are read as "line AB," "line BA," etc.

AB BA AM MA BM MB

We remember that a part of a line is called a line We remember that a part of a line is called a line segment or just a segment. A line segment segment or just a segment. A line segment contains the endpoints and all points between the contains the endpoints and all points between the endpoints. A segment can he named by naming endpoints. A segment can he named by naming the two endpoints in any order. The following the two endpoints in any order. The following segment can be called segment AB or segment segment can be called segment AB or segment BA.BA.

Instead of writing the word segment, we can use Instead of writing the word segment, we can use two letters in any order and an overbar with no two letters in any order and an overbar with no arrowheads to name the line segment whose arrowheads to name the line segment whose endpoints are the two given points. Therefore, AB endpoints are the two given points. Therefore, AB means "segment AB" and BA means "segment means "segment AB" and BA means "segment BA." Thus, we can use eitherBA." Thus, we can use either

or or AB BA

to name a line segment whose endpoints are A and to name a line segment whose endpoints are A and B.B.

The length of a line segment is designated by using The length of a line segment is designated by using letters without the overbar. Therefore, AB designates letters without the overbar. Therefore, AB designates the length of segment AB and BA designates the the length of segment AB and BA designates the length of segment BA. Thus, we can use eitherlength of segment BA. Thus, we can use either

AB or BAAB or BA

to designate the length of the line segment to designate the length of the line segment shown below whose endpoints are A and B.shown below whose endpoints are A and B.

The words equal to, greater than, and less than are The words equal to, greater than, and less than are used only to compare numbers. Thus, when we say used only to compare numbers. Thus, when we say that the measure of one line segment is equal to that the measure of one line segment is equal to the measure of another line segment, we mean that the measure of another line segment, we mean that the number that describes the length of one line the number that describes the length of one line segment is equal to the number that describes the segment is equal to the number that describes the length of the other line segment. Mathematicians length of the other line segment. Mathematicians use the word congruent to indicate that designated use the word congruent to indicate that designated geometric qualities are equal. In the case of line geometric qualities are equal. In the case of line segments, the designated quality is understood to segments, the designated quality is understood to be the length. Thus, if the segments shown herebe the length. Thus, if the segments shown here

are of equal length, we could so state by writing are of equal length, we could so state by writing that segment PQ is congruent to segment RS or that segment PQ is congruent to segment RS or by writing that the length of PQ equals the length by writing that the length of PQ equals the length of RS. We use an equals sign topped by a wavy of RS. We use an equals sign topped by a wavy line () to indicate congruence.line () to indicate congruence.

CONGRUENCE OFCONGRUENCE OF EQUALITY OFEQUALITY OF LINE SEGMENTSLINE SEGMENTS LENGTHSLENGTHS

or or PQ = RSPQ = RS

Sometimes we will use the word congruent and at Sometimes we will use the word congruent and at other times we will speak of line segments whose other times we will speak of line segments whose measures are equal.measures are equal.

PQ RS

A ray is sometimes called a half line. A ray is part A ray is sometimes called a half line. A ray is part of a line with one endpoint—the beginning point, of a line with one endpoint—the beginning point, called the origin—and extends indefinitely in one called the origin—and extends indefinitely in one direction. The ray shown here begins at point T, direction. The ray shown here begins at point T, goes through points U and X, and continues goes through points U and X, and continues without end.without end.

When we name a ray, we must name the origin first When we name a ray, we must name the origin first and then name any other point on the ray. Thus we and then name any other point on the ray. Thus we can name the ray above by writing either "ray TU" or can name the ray above by writing either "ray TU" or "ray TX." Instead of writing the word ray, we can use "ray TX." Instead of writing the word ray, we can use two letters and a single-arrowhead overbar. The first two letters and a single-arrowhead overbar. The first letter must be the endpoint or origin, and the other letter must be the endpoint or origin, and the other letter can be any other point on the ray. Thus, we letter can be any other point on the ray. Thus, we can name the ray shown above by writing eithercan name the ray shown above by writing either

These notations are read as "ray TU" and "ray TX."These notations are read as "ray TU" and "ray TX."

TU TX or

Two rays of opposite directions that lie on the Two rays of opposite directions that lie on the same line (rays that are collinear) and that share same line (rays that are collinear) and that share a common endpoint are called opposite rays. a common endpoint are called opposite rays. Thus, rays XM and XP are opposite rays, and they Thus, rays XM and XP are opposite rays, and they are both collinear with line MX.are both collinear with line MX.

If two geometric figures have points in common, If two geometric figures have points in common, we say that these points are points of intersection we say that these points are points of intersection of the figures. We say that the figures intersect of the figures. We say that the figures intersect each other at these points. If two different lines each other at these points. If two different lines lie in the same plane and are not parallel, then lie in the same plane and are not parallel, then they intersect in exactly one point. Here we show they intersect in exactly one point. Here we show lines h and e that intersect at point Z.lines h and e that intersect at point Z.

1.2 Planes1.2 Planes

A mathematical line has no width and continues A mathematical line has no width and continues without end in both directions. A mathematical without end in both directions. A mathematical plane can be thought of as a flat surface like a plane can be thought of as a flat surface like a tabletop that has no thickness and that continues tabletop that has no thickness and that continues without limit in the two dimensions that define without limit in the two dimensions that define the plane. Although a plane has no edges, we the plane. Although a plane has no edges, we often picture a plane by using a four-sided figure. often picture a plane by using a four-sided figure. The figures below are typical of how we draw The figures below are typical of how we draw planes. We label and refer to them as plane P and planes. We label and refer to them as plane P and plane Q, respectively.plane Q, respectively.

Just as two points determine a line, three noncollinear Just as two points determine a line, three noncollinear points determine a plane. As three noncollinear points points determine a plane. As three noncollinear points also determine two intersecting straight lines, we can also determine two intersecting straight lines, we can see that two lines that intersect at one point also see that two lines that intersect at one point also determine a plane.determine a plane.

On the right, we see that two parallel lines also On the right, we see that two parallel lines also determine a plane. We say that lines that lie in the determine a plane. We say that lines that lie in the same plane are coplanar.same plane are coplanar.

A line not in a plane is parallel to the plane if the A line not in a plane is parallel to the plane if the line does not intersect the plane. If a line is not line does not intersect the plane. If a line is not parallel to a plane, the line will intersect the plane parallel to a plane, the line will intersect the plane and will do so at only one point. Here we show and will do so at only one point. Here we show plane M and line k that lies in the plane. We also plane M and line k that lies in the plane. We also show line c that is parallel to the plane and line f show line c that is parallel to the plane and line f that intersects the plane at point P.that intersects the plane at point P.

Skew lines are lines that are not in the same Skew lines are lines that are not in the same plane. Skew lines are never parallel, and they do plane. Skew lines are never parallel, and they do not intersect. However, saying this is not not intersect. However, saying this is not necessary because if lines are parallel or necessary because if lines are parallel or intersect, they are in the same plane. Thus, lines intersect, they are in the same plane. Thus, lines k and f in the diagram above are skew lines k and f in the diagram above are skew lines because they are not both in plane M, and they because they are not both in plane M, and they do not form another plane because they are not do not form another plane because they are not parallel and they do not intersect.parallel and they do not intersect.

1.3 Angles1.3 Angles

There is more than one way to define an angle. An There is more than one way to define an angle. An angle can be defined to be the geometric figure angle can be defined to be the geometric figure formed by two rays that have a common endpoint. formed by two rays that have a common endpoint. This definition says that the angle is the set of points This definition says that the angle is the set of points that forms the rays, and that the measure of the that forms the rays, and that the measure of the angle is the measure of the opening between the angle is the measure of the opening between the rays. A second definition is that the angle is the rays. A second definition is that the angle is the region hounded by two radii and the arc of a circle. In region hounded by two radii and the arc of a circle. In this definition, the measure of the angle is the ratio of this definition, the measure of the angle is the ratio of the length of the arc to the length of the radius.the length of the arc to the length of the radius.

Using the first definition and the left-hand figure Using the first definition and the left-hand figure on the preceding page, we say that the angle is on the preceding page, we say that the angle is the set of points that forms the rays AP and AX. the set of points that forms the rays AP and AX. Using the second definition and the right-hand Using the second definition and the right-hand figure on the preceding page, we say that the figure on the preceding page, we say that the angle is the set of points that constitutes the angle is the set of points that constitutes the shaded region, and that the measure of the angle shaded region, and that the measure of the angle is S over R. is S over R.

A third definition is that an angle is the difference A third definition is that an angle is the difference in direction of two intersecting lines. A fourth in direction of two intersecting lines. A fourth definition says that an angle is the rotation of a definition says that an angle is the rotation of a ray about its endpoint. This definition is useful in ray about its endpoint. This definition is useful in trigonometry. Here we show two angles.trigonometry. Here we show two angles.

The angle on the left is a 300 angle because the The angle on the left is a 300 angle because the ray was rotated through 30° to get to its terminal ray was rotated through 30° to get to its terminal position. On the right, the terminal position is the position. On the right, the terminal position is the same, but the angle is a 390° angle because the same, but the angle is a 390° angle because the rotation was 390°. Because both angles have the rotation was 390°. Because both angles have the same initial and terminal sides, we say that the same initial and terminal sides, we say that the angles are coterminal. We say the measure of the angles are coterminal. We say the measure of the amount of rotation is the measure of the angle. amount of rotation is the measure of the angle.

We can name an angle by using a single letter or We can name an angle by using a single letter or by using three letters. When we use three letters, by using three letters. When we use three letters, the first and last letters designate points on the the first and last letters designate points on the rays and the center letter designates the common rays and the center letter designates the common endpoint of the rays, which is called the vertex of endpoint of the rays, which is called the vertex of the angle. The angle shown here could be named the angle. The angle shown here could be named by using any of the notations shown on the right.by using any of the notations shown on the right.

We use the word congruent to mean "geometrically We use the word congruent to mean "geometrically identical." Line segments have only one geometric identical." Line segments have only one geometric quality, which is their length. Thus, when we say that quality, which is their length. Thus, when we say that two line segments are congruent, we are saying that two line segments are congruent, we are saying that their lengths are equal. When we say that two angles their lengths are equal. When we say that two angles are congruent, we are saying that the angles have equal are congruent, we are saying that the angles have equal measures. If the measure of angle A equals the measures. If the measure of angle A equals the measure of angle B, we may writemeasure of angle B, we may write

We will use both of these notations.We will use both of these notations.

If two rays have a common endpoint and point in If two rays have a common endpoint and point in opposite directions, the angle formed is called a opposite directions, the angle formed is called a straight angle. A straight angle has a measure of straight angle. A straight angle has a measure of 180 degrees, which can also be written as 180°. If 180 degrees, which can also be written as 180°. If two rays meet and form a "square corner," we two rays meet and form a "square corner," we say that the rays are perpendicular and that they say that the rays are perpendicular and that they form a right angle. form a right angle.

A right angle has a measure of 90°. Thus, the A right angle has a measure of 90°. Thus, the words right angle, 90° angle, and perpendicular words right angle, 90° angle, and perpendicular all have the same meaning. As we show below, a all have the same meaning. As we show below, a small square at the intersection of two lines or small square at the intersection of two lines or rays indicates that the lines or rays are rays indicates that the lines or rays are perpendicular.perpendicular.

Acute angles have measures that are greater' Acute angles have measures that are greater' than 0° and less than 90°. Obtuse angles have than 0° and less than 90°. Obtuse angles have measures that are greater than 90° and less than measures that are greater than 90° and less than 180°. If the sum of the measures of two angles is 180°. If the sum of the measures of two angles is 180°, the two angles are called supplementary 180°, the two angles are called supplementary angles. If the sum of the measures of two angles angles. If the sum of the measures of two angles is 900, the angles are called complementary is 900, the angles are called complementary angles. On the left we show two adjacent angles angles. On the left we show two adjacent angles whose measures sum to 180°. We can use this whose measures sum to 180°. We can use this fact to write an equation to solve for x.fact to write an equation to solve for x.

In the center we show two adjacent angles whose In the center we show two adjacent angles whose measures sum to 90°. We can use this fact to measures sum to 90°. We can use this fact to write an equation and solve for y. Two pairs of write an equation and solve for y. Two pairs of vertical angles are formed by intersecting lines. vertical angles are formed by intersecting lines. Vertical angles have equal measures. On the right Vertical angles have equal measures. On the right we show two vertical angles, write an equation, we show two vertical angles, write an equation, and solve for z.and solve for z.

1.4 betweenness, tick marks, and assumptions1.4 betweenness, tick marks, and assumptions

One point is said to lie between two other points One point is said to lie between two other points only if the points lie on the same line (are only if the points lie on the same line (are collinear). When three points are collinear, one collinear). When three points are collinear, one and only one of the points is between the other and only one of the points is between the other two.two.

Thus, we can say that point C is between points A Thus, we can say that point C is between points A and B because point C belongs to the line and B because point C belongs to the line segment determined by the two points A and B segment determined by the two points A and B and is not an endpoint of this segment. Point X is and is not an endpoint of this segment. Point X is not between points A and B because it is not on not between points A and B because it is not on the same line (is not collinear) that contains the same line (is not collinear) that contains points A and B.points A and B.

We will use tick marks on the figures to designate We will use tick marks on the figures to designate segments of equal length and angles of equal measure.segments of equal length and angles of equal measure.

Here we have indicated the following equality of Here we have indicated the following equality of segment lengths: AB = CD, EF = GH, and KL = MN. The segment lengths: AB = CD, EF = GH, and KL = MN. The equality of angle measures indicated is mLA = nrLB, equality of angle measures indicated is mLA = nrLB, mLC = mLD, and mLE = mLF.mLC = mLD, and mLE = mLF.

In this book we will consider the formal proofs of In this book we will consider the formal proofs of geometry. We will use geometric figures in these geometry. We will use geometric figures in these proofs. When we do, some assumptions about the proofs. When we do, some assumptions about the figures are permitted and others are not permitted. It figures are permitted and others are not permitted. It is permitted to assume that a line that appears to be a is permitted to assume that a line that appears to be a straight line is a straight line, but it is not permitted to straight line is a straight line, but it is not permitted to assume that lines that appear to be perpendicular are assume that lines that appear to be perpendicular are perpendicular. Further, it is not permitted to assume perpendicular. Further, it is not permitted to assume that the measure of one angle or line segment is equal that the measure of one angle or line segment is equal to, greater than, or less than the measure of another to, greater than, or less than the measure of another angle or line segment. We list some permissible and angle or line segment. We list some permissible and impermissible assumptions on the following page.impermissible assumptions on the following page.

PERMISSIBLE PERMISSIBLE NOT PERMISSIBLE NOT PERMISSIBLE Straight lines are straight angles Straight lines are straight angles Right anglesRight angles Collinearity of points Collinearity of points Equal lengths Equal lengths Relative location of points Relative location of points Equal angles Equal angles Betweenness of points Betweenness of points Relative size Relative size

of segments and anglesof segments and angles Perpendicular or parallel linesPerpendicular or parallel lines

In the figure above we may assume the following:In the figure above we may assume the following:

1. That the four line segments shown are 1. That the four line segments shown are segments of straight linessegments of straight lines

2. That point C lies on line BD and on line AE2. That point C lies on line BD and on line AE 3. That point C lies between points B and D and 3. That point C lies between points B and D and

points A and Epoints A and E

We may not assume:We may not assume:

1. That AB and DE are of equal length1. That AB and DE are of equal length 2. That BC and CD are of equal length2. That BC and CD are of equal length 3. That LD is a right angle3. That LD is a right angle 4. That mLA equals mLE4. That mLA equals mLE 5. That AB and DE are parallel5. That AB and DE are parallel

1.51.5 TrianglesTriangles

Triangles have three sides and three angles. The Triangles have three sides and three angles. The sum of the measures of the angles in any triangle sum of the measures of the angles in any triangle is 1800. Triangles can be classified according to is 1800. Triangles can be classified according to the measures of their angles or according to the the measures of their angles or according to the lengths of their sides. If the measures of all lengths of their sides. If the measures of all angles are equal, the triangle is equiangular. angles are equal, the triangle is equiangular.

The Greek prefix iso- means "equal" and the The Greek prefix iso- means "equal" and the Greek word gonia means "angle." We put them Greek word gonia means "angle." We put them together to form isogonic, which means "equal together to form isogonic, which means "equal angles." An isogonic triangle is a triangle in which angles." An isogonic triangle is a triangle in which the measures of at least two angles are equal. If the measures of at least two angles are equal. If one angle is a right angle, the triangle is a right one angle is a right angle, the triangle is a right triangle. If all angles have a measure less than triangle. If all angles have a measure less than 90°, the triangle is an acute triangle. 90°, the triangle is an acute triangle.

If one angle has a measure greater than 90°, the If one angle has a measure greater than 90°, the triangle is an obtuse triangle. An oblique triangle triangle is an obtuse triangle. An oblique triangle is a triangle that is not a right triangle. Thus, is a triangle that is not a right triangle. Thus, acute triangles and obtuse triangles are also acute triangles and obtuse triangles are also oblique triangles. oblique triangles.

Triangles are also classified according to the Triangles are also classified according to the relative lengths of their sides. The Latin prefix relative lengths of their sides. The Latin prefix equi- means "equal" and the Latin word latus equi- means "equal" and the Latin word latus means "side." We put them together to form means "side." We put them together to form equilateral, which means "equal sides." An equilateral, which means "equal sides." An equilateral triangle is a triangle in which the equilateral triangle is a triangle in which the lengths of all sides are equal. lengths of all sides are equal.

Since the Greek prefix iso- means "equal" and the Since the Greek prefix iso- means "equal" and the Greek word skelos means "leg," we can put them Greek word skelos means "leg," we can put them together to form isosceles, which means "equal together to form isosceles, which means "equal legs." An isosceles triangle is a triangle that has legs." An isosceles triangle is a triangle that has at least two sides of equal length. If all the sides at least two sides of equal length. If all the sides of a triangle have different lengths, the triangle is of a triangle have different lengths, the triangle is called a scalene triangle.called a scalene triangle.

The lengths of sides of a triangle and the The lengths of sides of a triangle and the measures of the angles opposite these sides are measures of the angles opposite these sides are related. If the lengths of the sides are equal, then related. If the lengths of the sides are equal, then the measures of angles opposite these sides are the measures of angles opposite these sides are also equal. This means that every isogonic also equal. This means that every isogonic triangle is also an isosceles triangle and that triangle is also an isosceles triangle and that every isosceles triangle is also an isogonic every isosceles triangle is also an isogonic triangle. triangle.

Every equilateral triangle is also an equiangular Every equilateral triangle is also an equiangular triangle and every equiangular triangle is also an triangle and every equiangular triangle is also an equilateral triangle. If the measure of an angle in equilateral triangle. If the measure of an angle in a triangle is greater than the measure of a a triangle is greater than the measure of a second angle, the length of the side opposite the second angle, the length of the side opposite the angle is greater than the length of the side angle is greater than the length of the side opposite the second angle. The sum of the opposite the second angle. The sum of the measures of the angles of any triangle is 180°.measures of the angles of any triangle is 180°.

1.6 Transversals; alternate and corresponding 1.6 Transversals; alternate and corresponding anglesangles

We remember that two lines in a plane are called We remember that two lines in a plane are called parallel lines if they never intersect. A transversal parallel lines if they never intersect. A transversal is a line that cuts or intersects one or more other is a line that cuts or intersects one or more other lines in the same plane. When two parallel lines lines in the same plane. When two parallel lines are cut by a transversal, two groups of four angles are cut by a transversal, two groups of four angles whose measures are equal are formed. The four whose measures are equal are formed. The four small (acute) angles have equal measures, and the small (acute) angles have equal measures, and the four large (obtuse) angles have equal measures. If four large (obtuse) angles have equal measures. If the transversal is perpendicular to the parallel the transversal is perpendicular to the parallel lines, all eight angles are right angles. lines, all eight angles are right angles.

In the figures above, note the use of arrowheads In the figures above, note the use of arrowheads to indicate that lines are parallel. In the left-hand to indicate that lines are parallel. In the left-hand figure, we have named the small angles B and figure, we have named the small angles B and the large angles A. In the center figure, we show the large angles A. In the center figure, we show a specific example where the small angles are a specific example where the small angles are 50° angles and the large angles are 1300 angles. 50° angles and the large angles are 1300 angles.

Also, note that in each case, the sum of the Also, note that in each case, the sum of the measures of a small angle and a large angle is measures of a small angle and a large angle is 180° because together the two angles always 180° because together the two angles always form a straight line. The angles have special form a straight line. The angles have special names that are useful. The four angles between names that are useful. The four angles between the parallel lines are called interior angles, and the parallel lines are called interior angles, and the four angles outside the parallel lines are the four angles outside the parallel lines are called exterior angles. called exterior angles.

Angles on opposite sides of the transversal are Angles on opposite sides of the transversal are called alternate angles. There are two pairs of called alternate angles. There are two pairs of alternate interior angles and two pairs of alternate interior angles and two pairs of alternate exterior angles, as shown below. It is alternate exterior angles, as shown below. It is important to note that if two parallel lines are cut important to note that if two parallel lines are cut by a transversal, then each pair of alternate by a transversal, then each pair of alternate interior angles has equal measures and each pair interior angles has equal measures and each pair of alternate exterior angles also has equal of alternate exterior angles also has equal measures.measures.

Corresponding angles are angles that have Corresponding angles are angles that have corresponding positions. There are four pairs of corresponding positions. There are four pairs of corresponding angles, as shown below. It is corresponding angles, as shown below. It is important to note that if two parallel lines are cut important to note that if two parallel lines are cut by a transversal, then each pair of corresponding by a transversal, then each pair of corresponding angles has equal measures. angles has equal measures.

We summarize below some properties of parallel We summarize below some properties of parallel lines. In the statements below, remember that lines. In the statements below, remember that saying two angles are congruent is the same as saying two angles are congruent is the same as saying that the angles have equal measures. saying that the angles have equal measures.

We also state below the conditions for two lines We also state below the conditions for two lines to be parallel.to be parallel.

Some geometry textbooks will postulate, or in Some geometry textbooks will postulate, or in other words assume true without proof, one of the other words assume true without proof, one of the statements in each of the boxes and then prove statements in each of the boxes and then prove the other statements as theorems. We decide to the other statements as theorems. We decide to postulate all the statements in the boxes and use postulate all the statements in the boxes and use these statements to prove other geometric facts these statements to prove other geometric facts later.later.

When two transversals intersect three parallel When two transversals intersect three parallel lines, the parallel lines cut the transversals into lines, the parallel lines cut the transversals into line segments whose lengths are proportional, as line segments whose lengths are proportional, as we will show in Lesson 8. This fact will allow us to we will show in Lesson 8. This fact will allow us to find the length of segment x in the diagram on find the length of segment x in the diagram on the left. These segments are proportional. This the left. These segments are proportional. This means that the ratios of the lengths are equal.means that the ratios of the lengths are equal.

We decided to put the lengths of the segments on We decided to put the lengths of the segments on the left on the top and the lengths of the the left on the top and the lengths of the segments on the right on the bottom.segments on the right on the bottom.

1.7 Area and Sectors of Circles1.7 Area and Sectors of Circles

The area of a rectangle equals the product of the The area of a rectangle equals the product of the length and the width. The altitude, or height, of a length and the width. The altitude, or height, of a triangle is the perpendicular distance from either triangle is the perpendicular distance from either the base of the triangle or an extension of the the base of the triangle or an extension of the base to the opposite vertex. Any one of the three base to the opposite vertex. Any one of the three sides can be designated as the base. The altitude sides can be designated as the base. The altitude can (a) he one of the sides of the triangle, (b) fall can (a) he one of the sides of the triangle, (b) fall inside the triangle, or (c) fall outside the triangle. inside the triangle, or (c) fall outside the triangle.

When the altitude falls outside the triangle, we When the altitude falls outside the triangle, we have to extend the base so that the altitude can be have to extend the base so that the altitude can be drawn. This extension of the base is not part of the drawn. This extension of the base is not part of the length of the base. The area of any triangle equals length of the base. The area of any triangle equals one half the product of the base and the altitude.one half the product of the base and the altitude.

The area of a circle equals , where r represents The area of a circle equals , where r represents the radius of the circle. The perimeter of a circle the radius of the circle. The perimeter of a circle is called the circumference of the circle and is called the circumference of the circle and equals the product of and the diameter of the equals the product of and the diameter of the circle. The length of the diameter equals twice circle. The length of the diameter equals twice the length of the radius.the length of the radius.

An arc is two points on a circle and all the points on An arc is two points on a circle and all the points on the circle between them. If we draw two radii to the circle between them. If we draw two radii to connect the endpoints of the arc to the center of the connect the endpoints of the arc to the center of the circle, the area enclosed is called a sector of the circle, the area enclosed is called a sector of the circle. circle.

We define the degree measure of the arc to be We define the degree measure of the arc to be the same as the degree measure of the central the same as the degree measure of the central angle formed by the two radii. There are 3600 in angle formed by the two radii. There are 3600 in a circle. One degree of arc is of the a circle. One degree of arc is of the circumference of the circle. Eighteen degrees of circumference of the circle. Eighteen degrees of arc is , of the circumference of the circle. arc is , of the circumference of the circle.

The sector designated by 18° of arc is of the area The sector designated by 18° of arc is of the area of the circle. The length of an 18° arc is of the of the circle. The length of an 18° arc is of the circumference of the circle. The radius of the circumference of the circle. The radius of the circle shown below is 5 cm, so the area of the 18° circle shown below is 5 cm, so the area of the 18° sector is times the area of the whole circle. The sector is times the area of the whole circle. The length of an 18° arc is times the circumference length of an 18° arc is times the circumference of the whole circle.of the whole circle.

example 1.1 example 1.1

m. Find the area of the shaded region of the circle.m. Find the area of the shaded region of the circle.

SolutionSolution

Since the full measure of a circle is 3600, the Since the full measure of a circle is 3600, the measure of the central angle of the shaded sector measure of the central angle of the shaded sector of the circle is 360° minus 126°, or 234°. The of the circle is 360° minus 126°, or 234°. The area of the shaded sector equals the product of area of the shaded sector equals the product of the fraction of the central angle that is shaded the fraction of the central angle that is shaded and the area of the whole circle.and the area of the whole circle.

Area of 234° sector = 360 x (2 m)2 = 13m2Area of 234° sector = 360 x (2 m)2 = 13m2

1.8 Concept Review Problems1.8 Concept Review Problems

Virtually all the problems in the problem sets are Virtually all the problems in the problem sets are designed to afford the student practice with designed to afford the student practice with concepts or skills. Often, a problem may be concepts or skills. Often, a problem may be contrived so that it requires the use of a contrived so that it requires the use of a particular technique or the application of a particular technique or the application of a particular concept. However, there are some particular concept. However, there are some problems that defy simple classification. These problems that defy simple classification. These may be problems that appear very difficult but may be problems that appear very difficult but can be easily solved with clever reasoning or a can be easily solved with clever reasoning or a "trick." "trick."

These may be problems that can be solved using These may be problems that can be solved using either very long and tedious calculations or may either very long and tedious calculations or may be very easily solved through some "shortcut" be very easily solved through some "shortcut" requiring a deep understanding of the underlying requiring a deep understanding of the underlying concepts. Knowing what "shortcuts" and "tricks" concepts. Knowing what "shortcuts" and "tricks" to use requires experience. We do not believe to use requires experience. We do not believe one can become a good problem solver by one can become a good problem solver by reading about the philosophy of problem solving. reading about the philosophy of problem solving.

One learns the art of problem solving by solving One learns the art of problem solving by solving problems. What we will do is provide conceptually problems. What we will do is provide conceptually oriented review problems at the end of many oriented review problems at the end of many problem sets to permit exposure to a wide problem sets to permit exposure to a wide spectrum of problems, many of which appear in spectrum of problems, many of which appear in some similar form on standardized exams such as some similar form on standardized exams such as the ACT and SAT. Through time and practice, the ACT and SAT. Through time and practice, students gain confidence, experience, and students gain confidence, experience, and competence solving these types of problems.competence solving these types of problems.

Problems that compare the values of quantities Problems that compare the values of quantities come in many forms and can be used to provide come in many forms and can be used to provide practice in mathematical reasoning. In these practice in mathematical reasoning. In these problems, a statement will be made about two problems, a statement will be made about two quantities A and B. The correct answer is A if quantities A and B. The correct answer is A if quantity A is greater and is B if quantity B is quantity A is greater and is B if quantity B is greater. The correct answer is C if the quantities greater. The correct answer is C if the quantities are equal and is D if insufficient information is are equal and is D if insufficient information is provided to determine which quantity is greater.provided to determine which quantity is greater.

example 1.2 example 1.2

Let x and y be real numbers. If x > 0 and y < 0, Let x and y be real numbers. If x > 0 and y < 0, compare: A. x + y B. x - ycompare: A. x + y B. x - y

Solution Solution

We will subtract expression B from expression A. We will subtract expression B from expression A. If the result is a positive number, then A is If the result is a positive number, then A is greater. If the result is a negative number, then B greater. If the result is a negative number, then B is greater. If the result is zero, the expressions is greater. If the result is zero, the expressions have equal value and the answer is C. If have equal value and the answer is C. If insufficient information is provided to determine insufficient information is provided to determine which quantity is greater, the answer is D. which quantity is greater, the answer is D.

Therefore, we haveTherefore, we have (x+y)—(x—y)=2y(x+y)—(x—y)=2y We were told that y < 0, so 2y is a negative We were told that y < 0, so 2y is a negative

number. number.

Thus,Thus, (x + y) - (x - y) <0 expression(x + y) - (x - y) <0 expression x + y < x - y added x - y to both sidesx + y < x - y added x - y to both sides Therefore, quantity B is greater, so the answer is Therefore, quantity B is greater, so the answer is

B.B.

example 1.3 example 1.3

Let x and y be real numbers. If 1 < x < 6 and 1 < Let x and y be real numbers. If 1 < x < 6 and 1 < y < 6, compare: A. x - y B. y - xy < 6, compare: A. x - y B. y - x

Solution Solution

We are given that x is greater than 1 and less than 6 and We are given that x is greater than 1 and less than 6 and y is greater than 1 and less than 6. y is greater than 1 and less than 6.

There are three cases that we must consider:There are three cases that we must consider: 1. If x < y, then x - y is a negative number and y - x is a 1. If x < y, then x - y is a negative number and y - x is a

positive number.positive number. 2. If x > y, then x - y is a positive number and y - x is a 2. If x > y, then x - y is a positive number and y - x is a

negative number.negative number. 3. If x = y, then x - y = 0 and y - x = 0.3. If x = y, then x - y = 0 and y - x = 0. We can see that if 1 < x < 6 and 1 < y < 6, then all three We can see that if 1 < x < 6 and 1 < y < 6, then all three

cases are possible. Therefore, insufficient information is cases are possible. Therefore, insufficient information is provided to determine which quantity is greater, so the provided to determine which quantity is greater, so the answer is D.answer is D.

example 1.4 example 1.4

Given two intersecting lines with angles k, 1, m, Given two intersecting lines with angles k, 1, m, and n, as shown, compare:and n, as shown, compare:

A. (k + l + m)° B. (180 - n)°A. (k + l + m)° B. (180 - n)°

solution solution

We know that a straight angle has a measure of We know that a straight angle has a measure of 1800. Therefore, we have1800. Therefore, we have

k + I = 180k + I = 180 straight anglestraight angle m + n = 180m + n = 180 straight anglestraight angle k + l + m + n = 360k + l + m + n = 360 addedadded Thus,Thus, k+l+m = 360 – n = 180 + (180 - n)k+l+m = 360 – n = 180 + (180 - n) Thus,Thus, k + I + m > 180 – nk + I + m > 180 – n Therefore, quantity A is greater, so the answer is A.Therefore, quantity A is greater, so the answer is A.

Given and are parallel, as shown. Given and are parallel, as shown. Compare: Compare:

A. Area of AWXY A. Area of AWXY B. Area of AZXYB. Area of AZXY

solution solution

We know that the area of any triangle equals one We know that the area of any triangle equals one half the product of the base and the altitude. For half the product of the base and the altitude. For both triangles, XY is the base. Also, since ZW is both triangles, XY is the base. Also, since ZW is parallel to XY, the altitudes of the two triangles parallel to XY, the altitudes of the two triangles are of equal length. Thus, the areas of the two are of equal length. Thus, the areas of the two triangles AWXY and AZXY are equal. Therefore, triangles AWXY and AZXY are equal. Therefore, the quantities are equal, so the answer is C.the quantities are equal, so the answer is C.