PLANE GEOMETRY

Plane geometry has to do with measurement and relationships among lines, planes, volume, angles, circles and polygons. Polygons are figures formed by line segments such as triangles and squares.

NOTATION AND SYMBOLS

A, B Points often designated with capital letters

Line segment with endpoints at A and B

^ Perpendicular (Line 1 ^ Line 2)

Ða or ÐABC Angle a or angle ABC

ANGLES

Angle measurement is based on the circle. There are 360 degrees in a circle.

The angle measure of a line is 180 degrees.

Right Angles

A right angle is a 90 degree angle. Often a small box at the vertex is used to indicate that the degree measure of an angle is 90 degrees. In the illustration below, angle ABC is 90 degrees.

Acute Angles, Obtuse Angles, and Supplementary Angels

An angle that is less than 90 degrees is an acute angle and an angle that is greater than 90 degrees is an obtuse angle. In the illustration below, Ðb is an acute angle and Ða is an obtuse angle. We call angles a and b "supplementary angles" because the sum of their measures is 180 degrees.

Vertical Angles

Four angles are formed by the intersection of two lines. The angles that are opposite each other are called vertical angles.

Important fact: Vertical angles are equal. In the illustration below, Ða=Ða' and Ðb=Ðb'.

LINES

Lines are assumed to have infinite length.

Parallel lines do not intersect.

Perpendicular lines intersect at a 90 degree angle.

Important fact:

When a line intersects two parallel lines, corresponding angles are equal. In the illustration below: Ða=Ða', Ðb=Ðb', Ðc=Ðc', and Ðd=Ðd'. Since vertical angles are equal it is also true that

Ða=Ða'=Ðd=Ðd' and Ðb=Ðb'=Ðc=c'.

Example: Find the measure of angle E if BC is parallel to DE.

Solution: To help visualize the solution extend AE, BC and DE. Angle ACE and angle E are corresponding angles formed by the intersections of a line crossing two parallel lines. Therefore, angle E = 48 degrees.

TRIANGLES

The sum of the three angles is 180 degrees.

The perimeter of a triangle is the sum of the lengths of the sides.

The area is half the base multiplied by the height: A=(1/2)bh

The height is the distance from a vertex to the line containing the opposite side.

In each of the triangles below, the base is 3 and the height is 4.

In each triangle A=(1/2)bh=(1/2)(3)(4)=6.

A right triangle has a right angle. The Pythagorean Theorem applies to right triangles. It is one of the most important theorems to know, so we'll save our discussion of the Pythagorean Theorem for the next section which is devoted entirely to this theorem.

An equilateral triangle is a triangle which has three equal sides. The angles are also equal and are each 60 degrees.

Extra: Area of an equilateral triangle:

It is unlikely but possible that you will need to know the area of an equilateral triangle without being given the height.

An isosceles triangle has two equal sides. The angles opposite the two equal sides are also equal. In the triangle below, AB = AC and ÐB=ÐC.

Exercise: What is the measure of the third angle?

Answer: Since triangles have 180 degrees and two of the angles are 70 degrees, the third angle is 180-(70+70)=40.

Important Triangle Fact: "Similar triangles" have the same shape. The angles in the two triangles will have the same degree measures but the sides may not have the same length. However, the sides will be proportional to each other.

For example, triangle ABC is similar to triangle A'B'C'. The angles are equal (angle A = angle A', etc.) and the sides are proportional (3:6::5:10 etc.)

Important Triangle Fact: A line that intersects two sides of a triangle and that is parallel to the third side will form two similar triangles. In the figure below, triangle ABC is similar the triangle AB'C'. Angle ABC = angle AB'C' etc.

Important Triangle Fact: The sum of the lengths of two sides of a triangle is always longer than the length of the third side.

Important Triangle Fact: The largest angle in a triangle is opposite the longest side and the smallest angle is opposite the smallest side.

Example: In triangle ABC, if AB = 4 and AC = 7, what can be said about BC?

A. BC>7

B. BC<11

C. BC=11

D. BC>11

E. without more information it is impossible to tell anything about BC

Answer: B. The sum of the lengths of two sides of a triangle is always longer than the length of the third side. Answer choice A is incorrect because while a visual estimate might lead us to conclude that BC is longer than AC, we can't be sure whether this is true.

QUADRILATERALS

Quadrilaterals are four sided figures. The sum of the measures of the angles is 360 degrees. To see that this is true, draw a diagonal in a quadrilateral to create two triangles, each of which will have 180 degrees.

Squares: A square is a quadrilateral with four equal sides and four right angles.

All four sides are equal (indicated by the marks on the sides).

Opposite sides are parallel.

All four angles are 90 degrees.

The diagonals are equal.

The diagonals are perpendicular to each other.

Perimeter is the sum of the sides (P=4s)

Area is the square of the length of a side (A=s²)

Rectangles: A rectangle is a quadrilateral which has parallel and equal opposite sides and four right angles. (Note: If all the sides of a rectangle are equal to then it is also a square, so here we will examine rectangles that are not squares.)

Opposite sides are equal and parallel.

All four angles are 90 degrees.

The diagonals are equal.

The diagonals are not be perpendicular to each other.

Perimeter is the sum of the sides (P=2l+2w)

Area is the length multiplied by the width (A=lw)

Parallelograms: A parallelogram is a quadrilateral with parallel and equal opposite sides. (Note: If a parallelogram has 90 degree angles then it is also a rectangle, so we we'll just examine parallelograms that are neither rectangles nor squares.)

Opposite sides are equal and parallel.

Opposite angles are equal.

The diagonals are not equal.

The diagonals are not be perpendicular to each other.

Perimeter is the sum of the lengths of the sides.

Area is the base multiplied by the height (A=bh). The height is the perpendicular distance between two opposite sides.

Trapezoids: A trapezoid is a quadrilateral with at least one pair of parallel sides. (Note: If a trapezoid had two pairs of parallel sides it also a parallelogram, so we'll examine trapezoids that are not parallelograms.)

One pair of opposite sides are parallel.

Opposite angles are not equal.

The diagonals may or may not be equal.

The diagonals may or may not be perpendicular to each other.

Perimeter is the sum of the lengths of the sides.

Area is the average length of the bases multiplied by the height.

A=(1/2)(b₁+b₂)h or A=(h/2)( b₁+b₂). The height is the perpendicular distance between the two parallel sides. (It not too likely that you will need to know the area of a trapezoid.)

OTHER POLYGONS

The number of degrees in a polygon is (n-2)180 where n is the number of sides of the polygon. As we did with quadrilaterals, we can also find the number of degrees by dividing the polygon into the fewest number of triangles possible. See the figure below: there are six sides and four triangles. The number of degrees is (6-2)180=720.

The perimeter can be found by adding the lengths of the sides.

The area of some polygons can be found by breaking the area into sections.

Example: Find the area of the figure below. Assume that all angles are right angles.

Solution: Since all angles are right angles we can divide the shape into rectangles. Since the height is 6, the height of the small rectangle is 2 (=6-2-2) and the sum of the areas of the rectangles is 18+4+18=40.

Alternate solution: Sometimes it is easier to "fill in" a shape and then subtract. The area of the large rectangle is the base (3+2+3) multiplied by the height (6). The area of each shaded section is 4 (=2x2). The area of the original shape is therefore 48-4-4=40.

CIRCLES

Radius: The radius is the distance from the center of a circle to a point on the circle.

Chord: A chord is a line segment with both ends on a circle.

Diameter: The longest chord in a circle is the diameter, which goes through the center of the circle.

Circumference: The circumference is the distance around a circle. The circumference can be found with the formula C=2pr where p is approximately 3.14 and r is the radius. Another formula for the circumference is pD where D is the diameter.

Area: The area of a circle is given by the formula A=pr², where p is approximately 3.14 and r is the radius.

Arc: An arc is part of the circumference.

Central angle: A central angle is an angle formed by the center and 2 points on a circle. The sides of the angle are radii.

Inscribed angle: An inscribed angle is an angle formed by three points on a circle. The sides of the angle are chords.

Important fact: Let AOB be a central angle where A and B are points on a circle and O is the center. Let AXB be an inscribed angle on the same circle. Then the measure of angle AOB is twice the measure of angle AXB.

VOLUME AND SURFACE AREA

Cube: The volume of a cube is given by the formula V=s³, where s is the length of the edge of the cube.

Example: Find the volume of a cube if one of the edges is 4 centimeters.

Solution: V=s³=(4 cm.)³=64 cm³. Note that the volume is in cubic centimeters.

The surface area is the sum of the areas of the faces. The area of each face is s² (A=s²) and there are 6 faces. Therefore, S.A.=6s².

Example: If the volume of a cube is 8 cubic feet, how much wrapping paper is needed to completely cover the cube? Assume there is no overlap.

Solution: If the volume is 8 cubic feet then one edge is 2 feet. (Note that V=s³=8. Therefore, the length of a side is the cube root of 8.) The surface area is then 6s²=6(2)²=24 square feet.

Rectangular Solid: A rectangular solid looks like a box. In other words, it has six faces that are rectangles.

The volume is the length multiplied by the width multiplied by the height (V=lwh).

Formula for the surface area: S.A.=2lw+2wh+2lh where l is the length, w is the width, and h is the height. (It is unlikely that you will need to know the surface area of a rectangular solid.)

Cylinder: A cylinder looks like a can. In other words, the top and bottom are circles.

Volume of a cylinder: V= pr²h. Surface Area: S.A.=2pr²+ 2prh. (It is highly unlikely that you will need to know the surface area of a cylinder.)

Example: A tin box (rectangular solid) has length 3 inches, width 4 inches, and height 10 inches. The base of a tin can (cylinder) is 12 square inches and the height is 10 inches. If gold dust is poured into both of them, which one will be worth more?

Solution: The volume of the box is 120 cubic inches (V=lwh=3x4x10). The base of the can is 12 square inches. The area of a circle is pr², therefore, pr²=12. Now V= pr²h=12h=12(10)=120 cubic inches. Since the box and the can have equal volume, they will hold the same quantity of gold dust and therefore they have equal value.

LINES, AREAS, AND VOLUMES

There is an important relationship between linear measure, area measure, and volume measure which we will illustrate with examples. Let S be the side of a square as well as the edge of a cube. Then the area of the square is S² and the volume of the cube is S³. Now if the length of S is multiplied by 2, the area of the square is multiplied by 4, and the volume of the cube is multiplied by 8.

Example: Let S be 3. Then the area of a square with side S is 3²=9 and the volume of a cube with side S is 3³=27. Now if we double S we have 2S or 6. The area of the square will be (2S)²=6²=36 and the volume of the cube will be (2S)³=6³=216. The new area is 4 times as large as the old area (36=4x9), and the volume is 8 times as large as the old volume (216=8x27).

The same idea applies to rectangular solids. Let L and W be the length and width of a rectangle. If we double the length and width, the area is quadrupled. If we let H be the height of a rectangular solid with length L and width W, then if we double all three dimensions the new volume will be 8 times the original volume.

Algebraically we can see these relationships in the chart below.

Linear Measure	Area Measure	Volume Measure
S	S^2	S^3
2S	(2S)^2=2^2S^2=4S^2	(2S)^3=2^3S^3=8S^3
L, W, H	A=LW	A=LWH
2L, 2W, 2H	A=(2L)(2W)=4LW	V=(2L)(2W)(2H)=8LWH

Note also that if we triple the side of a cube, the area of one of the faces will be 9 (=3²) times as large and the volume will be 27 (=3³) times as large.

The relationship between linear, area, and volume measures apply to other shapes as well. If we double the base and height of a triangle the area will be 4 times as large as the area of the original triangle, and if we double the radius of a circle the new area will again be 4 times as large as the area of the original circle.

SUMMARY

The important formulas for plane geometry are summarized in the table below.

Polygon	Perimeter	Area
Triangle	Sum of lengths of sides	A=(1/2)bh
Equilateral Triangle	Sum of lengths of sides
Square	P = 4s	A= s^2
Rectangle	P = 2(l + w) or 2l + 2w	A=lw
Parallelogram	Sum of lengths of sides	A=bh
Trapezoid	Sum of lengths of sides	A=(1/2)(b1+b2) h
Pentagon, Hexagon etc.	Sum of lengths of sides
Circle	C = 2pr or pD	A=pr^2
Shape	Volume	Surface Area
Cube	V = s^3	S.A.=6 x s^2
Rectangular solid	V = lwh	S.A.=2hw + 2wl + 2 hl
Cylinder	V = pr^2h	S.A.=2pr^2+ 2prh

Example:

ABCD is a rectangle. Find the ratio of the shaded area to the area of the rectangle.

Solution: The area of a rectangle can be found by multiplying the base times the height. The area of a triangle is 1/2 the base times the height.

Area of triangle CXD = (1/2)CDxAC = (1/2)x area ABCD.

Example:

(A) Which is greater: The volume of a rectangular solid with dimensions 3x3x4 or the volume of a rectangular solid with dimensions 2x2x9?

(B) Which is greater: The surface area of a rectangular solid with dimensions 3x3x4 or the surface area of a rectangular solid with dimensions 2x2x9?

Answer: The volumes are equal but the surface area of the rectangular solid with dimensions 2x2x3 is greater.

Volume of rectangular solid with dimensions 3x3x4:

V=lwh

= 3x3x4

= 36

Volume of rectangular solid with dimensions 2x2x9:

V=lwh

= 2x2x9

= 36

Surface area of rectangular solid with dimensions 3x3x4:

S. A.=2(lw)+2(lh)+2(wh)

=(2x3x3)+(2x3x4)+(2x3x4)

=18+24+24

=66

Surface area of rectangular solid with dimensions 2x2x9:

S. A.=2(lw)+2(lh)+2(wh)

=(2x2x2)+(2x2x9)+(2x2x9)

= 8+36+36

= 80

Example:

Point X is located somewhere inside rectangle ABCD. AXB and DXC are triangles with common vertex at X. Find the ratio of the triangles (shaded area) to the rectangle.

A. 1/2

B. /4

C. 2/3

D. 3/4

E. not enough information

Answer: A

Construct a line through X that is parallel to AB. Now there are two rectangles, ABFE and EFCD, the sum of whose areas is equal to the area of ABCD.

Area of ABFE = bh = EFxAE.

Area of triangle AXB = bh = (0.5)EFxAE = (0.5) Area of ABFE.

Area of EFCD = bh = DCxED

Area of triangle DXC = (0.5)bh = (0.5)DCxED = (0.5) EFCD.

Therefore, the sum of the shaded areas is equal to (0.5)ABFE + (0.5)EFCD = (0.5)ABCD.

Example:

In triangle ABC, ÐE = 90°, AB is parallel to DE, AB = 5, DE = 3, and BC =4. Find the shaded area.

Answer: 6.4

One way to find the shaded area is to find the area of ABC and then subtract the area of DEC. Since AB and DE are parallel, ÐB = ÐE = 90°. Therefore, the heights of ABC and DEC are CB and CE respectively. So first we must find the length of CE.

Triangles ABC and DEC are similar since two angles are congruent (ÐB = ÐE and ÐC = ÐC). Therefore, the sides are in proportion to each other.

Let CE = x.

Then x/3= 4/5

so that x = (3)(4)/5=12/5= 2.4

Area ABC = 0.5bh

= 0.5(5)(4) = 10

Area DEC = 0.5(3)(2.4) = 3.6

Finally, the shaded area is the difference: 10 - 3.6 = 6.4

Special note: Recall that if the dimensions of a figure, such as a triangle is doubled, the area is quadrupled. Look at triangles DEC and ABC. Note that

5 = 3(5/3) and 4 = (12/5)(5/3). That is, the base and the height were multiplied by 5/3 (=1.67). Then the area of the smaller triangle is multiplied by (5/3)² (=1.67)². Therefore, the area of the larger triangle is 3.6x(1.67)²=3.6x2.79=10.008 which is approximately equal to 10.