Geometry Review Notes: Logic
LOGIC
Statements and Truth Values
1. Truth values: True and False
2. Statement: A sentence that can be determined to be either True of False.
3. Negation: Changing a statement to have the opposite truth value (usually by adding “not”).
Connectives
4. Conjunction: AND. True only if both parts are true: T and T = T, all others false.
5. Disjunction: OR. True if either part is true: T or T = T, T or F = T, F or T = T but F or F = F.
6. Conditional: IF . . . THEN . . . . Only false if T then F: T then F = F, all others true.
Ex: If a full circle has 360(, then a straight angle has 180(. T then T is TRUE
Ex: If the sum of the interior angles of a quadrilateral is 360(, then the sum of the interior angles of an
octagon is 720(. T then F is FALSE
Ex: If the sum of the interior angles of an octagon is 720(, then the sum of the interior angles of a
quadrilateral is 360(. F then T is TRUE.
Ex: If the diagonals of a trapezoid bisect each other, then the trapezoid is equilateral. F then F is TRUE.
Ex: If the diagonals of a quadrilateral bisect each other, then opposite sides of the quadrilateral are
congruent. Must be TRUE since we can never have T then F. There is no quadrilateral where the
diagonals bisect each other but the opposite sides are not congruent.
7. Biconditional: . . . IF AND ONLY IF . . . True if both parts have same truth value: T iff T = T and
F iff F = T but T iff F = F and F iff T = F.
A biconditional is a conditional that works “both ways:” p iff q means “If p then q AND if q then p.”
A good definition is a biconditional.
Ex: A triangle is isosceles if and only if it has at least two congruent sides. This is TRUE. It is the
definition of an isosceles triangle. It works “both ways:”
If a ( has 2 ( sides then it is iso. AND if a ( is iso. then it has 2 ( sides.
Ex: In (ABC, a2 + b2 = c2 if and only if (C is a right angle. TRUE. It works both ways:
If (C is a right angle, then a2 + b2 = c2 AND if a2 + b2 = c2 then (C is a right angle.
Ex: The diagonals of a quadrilateral are perpendicular if and only if the quadrilateral is a rhombus.
FALSE. Does not work both ways. “If a quadrilateral is a rhombus then its diagonals are
perpendicular” is TRUE but “if a quadrilateral has perpendicular diagonals then it is a rhombus”
is FALSE.
Related Conditional Statements
8. Conditional: If a quadrilateral is a rectangle, then its diagonals are congruent. TRUE
9. Converse: “Switch” the hypothesis and the conclusion.
If a quadrilateral has congruent diagonals, then it is a rectangle. FALSE
10. Inverse: Negate both parts.
If a quadrilateral is not a rectangle, then its diagonals are not congruent. FLASE
11. Contrapositive: “Switch” and negate.
If a quadrilateral does not have congruent diagonals, then it is not a rectangle. TRUE
12. A conditional and its contrapositive are always logically equivalent: they always have the same truth value.
Laws of Logic (You do NOT need to know the names)
13. Law of Detachment
If a conditional is true and its hypothesis (first part) is true, then its conclusion (second part) is also true.
Ex: Given: If a cow is brown, then she gives chocolate milk.
Elsa the cow is brown.
Conclusion: Elsa gives chocolate milk
Ex: Given: If a cow is brown, then she gives chocolate milk.
Bessie the cow is not brown.
Conclusion: No conclusion.
Ex: Given: If a cow is brown, then she gives chocolate milk.
Mrs. O’Leary’s cow gives chocolate milk.
Conclusion: No conclusion.
13. Law of the Contrapositive
If a conditional is true, then its contrapositive is also true (they are logically equivalent).
Ex: Given: If Jack knows logic, then he passes his test.
Conclusion: If Jack does not pass his test, then he does not know logic.
14. Law of Modus Tollens
If a conditional is true and its conclusion is false, then it hypothesis must also be false.
Ex: Given: If Jack knows logic, then he passes his test.
Jack did not pass his test.
Conclusion: Jack does not know logic.
Ex: Given: If Jill knows logic, then she passes her test.
Jill passed her test.
Conclusion: No conclusion.
Ex: Given: If Jane knows logic, then she passes her test.
Jane does not know logic.
Conclusion: No conclusion.
Ex: Given: If John knows logic, then he passes his test.
John knows logic.
Conclusion: John passes his test (by Law of Detachment)
15. Law of Disjuctive Inference
If a disjunction (OR) is true and either part is false, then the other part must be true.
Ex: Given: Dragons are dreadful or griffins are gruesome.
Griffins are not gruesome.
Conclusion: Dragons are dreadful
Ex: Given: Fish swim or titmice fly
Fish swim.
Conclusion: No conclusion.
BASIC GEOMETRY
Vocabulary
1. Collinear: All on the same line.
2. Congruent: Same measure. (Same length of segments; same number of degrees for angles)
3. Midpoint: Divides a segment into 2 congruent parts.
4. Bisect/bisector: Divides a segment (or an angle) into 2 congruent parts.
5. Parallel lines: Two lines (in the same plane) that never intersect.
6. Perpendicular: Form right angle(s)
7. Complementary angles: Two angles whose measures add to 90(
8. Supplementary angles: Two angles whose measures add to 180(
9. Adjacent angles: Two angles that share a ray (side) but neither is “inside” the other.
10. Vertical angles: Nonadjacent angles formed by the intersection of two lines.
11. Transversal: A line that intersects two other lines at different points.
12. Corresponding angles: (1 and (5, (2 and (6, (3 and (7, (4 and (8 in diagram.
13. Alternate interior angles: (3 and (5, (4 and (6 in diagram.
14. Same side interior angles: (3 and (6, (4 and (5 in diagram.
Theorems (Facts)
12. Vertical angles are congruent.
13a. When two parallel lines are cut by a transversal, corresponding angles are congruent.
13b. When two parallel lines are cut by a transversal, alternate interior angles are congruent.
13c. When two parallel lines are cut by a transversal, same side interior angles are supplementary.
14a. When two lines are cut by a transversal and corresponding angles are congruent, the lines are parallel.
14b. When two lines are cut by a transversal and alternate interior angles are congruent, the lines are parallel.
14c. When two lines are cut by a transversal and same side interior angles are supplementary, the lines are parallel.
TRIANGLES
Vocabulary
1. Scalene: No sides (or angles) congruent
2. Isosceles: (At least) two sides (and two angles) congruent
3. Equilateral: All three sides (and all three angles) congruent
4. Acute: All angles acute
5. Right: One right angle
6. Obtuse: One obtuse angle
Theorems (facts)
7. The sum of the interior angles in any triangle is 180(
8a. If two sides of a triangle are congruent, the angles
opposite those sides are congruent (base angles of
an isosceles triangle are congruent).
8b. If two angles of a triangle are congruent, the sides
opposite those angles are congruent.
9. The LONGEST side is opposite the LARGEST angle.
10. An exterior angle of a triangle is equal to the SUM of the two
nonadjacent interior angles. (Also, the exterior angle is larger
than either of the two nonadjacent interior angles.)
11a. If the lengths of two sides of a triangle are known, the length
of the third side must be BETWEEN the DIFFERENCE and
the SUM of the other two.
11b. The two shorter sides of a triangle must add up to MORE
THAN the longer side.
Concurrent segments
12. The medians of a triangle intersect at the centroid.
- The centroid is the center of gravity (balance point) of the (
- The centroid divides the medians in the ratio 2:1
- In coordinate geometry, the coordinates of the centroid are the
AVERAGE of the coordinates of the three vertices.
13. The angle bisectors of a triangle intersect at the incenter
- The incenter is the center of the inscribed circle
- The incenter is equidistant from all three sides of the (
14. The perpendicular bisectors of the sides of a triangle
intersect at the circumcenter
- The circumcenter is the center of the circumscribed circle
- The circumcenter is equidistant from all three vertices of the (
15. The altitudes of a triangle intersect at the orthocenter
- The orthocenter is the only “center” that can be outside the (
- It will only be outside the triangle in an obtuse (
- In a right triangle, it will be at the right angle.
Triangle congruence
16. Two triangles may be proved congruent by
- SSS
- SAS
- ASA
- AAS (SAA)
- RHL (Regents just uses HL)
17. CPCTC may only be used AFTER you prove two triangles congruent
QUADRILATERALS
Theorems (facts)
1. The sum of the interior angles in any quadrilateral is 360(
2. If a quadrilateral is inscribed in a circle,
opposite angles are supplementary.
Special Quadrilaterals
3. Trapezoid: Exactly one pair of opposite sides (bases) parallel
- Angles at the ends of the same leg are supplementary
4. Isosceles trapezoid: Trapezoid with congruent legs
- Base angles congruent
- Opposite angles supplementary
- Diagonals congruent
5a. Parallelogram: Both pairs of opposite sides ||
- Both pairs of opposite sides (
- Both pairs of opposite angles (
- All pairs of consecutive angles supplementary
- Diagonals bisect each other
5b. To prove a quadrilateral is a parallelogram prove:
- both pairs opposite sides || OR
- both pairs of opposite sides ( OR
- ONE pair opposite sides both || and (
- diagonals bisect each other
6a. Rectangle: Four right angles
- A rectangle IS a parallelogram
- diagonals are (
6b. To prove a quadrilateral is a rectangle, prove
- It has four right angles OR
- It is a parallelogram AND it has one right angle OR
- it is a parallelogram AND its diagonals are (
7a. Rhombus: All four sides congruent
- A rhombus IS a parallelogram
- Diagonals are (
- Diagonals bisect the vertices
7b. To prove a quadrilateral is a rhombus, prove
- all four sides ( OR
- it is a parallelogram AND two consecutive sides are ( OR
- it is a parallelogram AND it diagonals are (
8a. Square: Four right angles AND all four sides congruent
- A square IS a parallelogram AND a rectangle AND a rhombus
8b. To prove a quadrilateral is a square, prove
- it is BOTH a rectangle AND a rhombus
ANGLE SUMS IN POLYGONS
1. The sum of the EXTERIOR angles of any polygon is 360(
- If the polygon is REGULAR, then each EXTEROIR angle measures [pic]
- If the polygon is REGULAR, then each INTERIOR angle measures [pic]
2. The sum of the INTERIOR angles of a polygon with n sides is 180(n – 2)
- The above formula is the same as drawing diagonals from one vertex and counting triangles
- If the polygon is REGULAR, then each angle measures [pic]
3. An interior angle and an adjacent exterior angle are supplementary.
COORDINATE GEOMETRY
Formulas
1. Slope: [pic]
2. Distance: [pic] (Comes directly from Pythagorean Theorem.)
3: Midpoint: [pic] (Average the x’s; average the y’s.)
CG Proofs
4. To prove to segments CONGRUENT, use DISTANCE FORMULA: SAME LENGTH.
5. To prove to segments PARALLEL, use SLOPE FORMULA: SAME SLOPE.
6. To prove to segments PERPENDICULAR, use DISTANCE FORMULA: OPP. RECIP. SLOPES.
7. To prove to segments BISECT each other, use MIDPOINT FORMULA: SAME MIDPOINT.
8. Parts of a CG proof:
- Graph or diagram (optional but almost always helpful)
- Appropriate computations (slope, distance, midpoint) CLEARLY LABELED
e.g. Midpoint of AB = (2, (3)
- Appropriate CONCLUSIONS and JUSTIFICATIONS based on the computations.
- Final statement should be what was to be proved.
Equations of lines
9. Horizontal line: y = b (b is a NUMBER). Slope is 0
10. Horizontal line: x = a (a is a NUMBER). Slope is UNDEFINED.
11. Slanted line:
Slope-intercept formula: y = mx + b (m and b are NUMBERS)
- m = slope
- b = y-intercept
Point-slope formula: y ( y1 = m(x ( x1)
- m = slope
- (x1, y1) = coordinates of a point on the line
Solving system of equations graphically
12. To solve a system of two equations in two unknowns:
- Graph both equations on the same set of axes. LABEL the graphs.
- Find where the two graphs intersect. WRITE DOWN the coordinates of the intersection point(s).
SIMILARITY
Similar polygons
1. Two polygons are similar if
- All pairs corresponding angles congruent AND
- all pairs of corresponding sides in proportion (same ratio)
2. In similar polygons,
- all pairs of corresponding LENGTHS will have the same ratio
- corresponding AREAS will have (ratio)2
- corresponding volumes will have (ratio)3
Similar triangles
3. To prove two triangles similar:
- AA
- SAS~
- SSS~
4. If [pic] then (PQC ~ (ABC
[pic]
[pic]
5. If M and N are MIDPOINTS of the sides, then
- [pic]
- [pic]
- (MNC ~ (ABC with ratio 1:2
6. If [pic] is the altitude to the hypotenuse of
RIGHT (ABC, then
- ALL THREE triangles are similar
- a2 = Pp
- l2 = ph and L2 = Ph
TRANSFORMATIONS
Basic Transformations
1. Line reflection: Straight to the line ((), same distance beyond.
Image congruent to preimage.
Distance, angle measure preserved.
Orientation not preserved.
2. Point reflection: Straight to the same distance, same distance beyond.
Image congruent to preimage.
Distance, angle measure and orientation all preserved.
3. Rotation: All points rotate the same angle around the same vertex point.
Positive rotations are counterclockwise.
Image congruent to preimage.
Distance, angle measure and orientation all preserved.
4. Translation: All point move the same distance, the same direction.
Image congruent to preimage.
Distance, angle measure and orientation all preserved.
5. Dilation: All points move toward or away from a given point by the same factor (multiplication).
Image similar to preimage.
Angle measure and orientation preserved.
Distance not preserved.
Special Transformations, Coordinate Formulas
6. Reflection in the x-axis: NEGATE y. rx(x, y) = (x, (y)
7. Reflection in the y-axis: NEGATE x. ry(x, y) = ((x, y)
8. Reflection over the line y = x: SWITCH the coordinates. Ry = x(x, y) = (y, x)
9. Reflection in the origin: NEGATE BOTH. RO (x, y) = ((x, (y)
10. 90( rotation about the origin: GRAPH IT. R90((x, y) = ((y, x)
11. Translation: ADD. Ta, b(x, y) = (x + a, y + b)
12. Dilation in the origin: MULTIPLY. Dk(x, y) = (kx, ky)
Compositions
13. Work right to left. T1, -2 ( Dk means dilate FIRST, then translate.
Not always commutative.
14. Glide reflection: A composition of a line reflection and a translation parallel to the line.
Vocabulary
15. Orientation: Clockwise or counterclockwise.
16. Isometry: Preserves distance. NOT dilation.
17. Direct isometry: An isometry that preserves orientation. NOT line reflection or glide reflection.
18. Opposite isometry: An isometry that switches orientation. Line reflection or glide reflection.
LOCUS
Five basic loci
1. All points a given distance r from ONE POINT C:
Circle, center at C, radius r.
2. All points equidistant from TWO POINTS A and B:
One line, the perpendicular bisector of AB
3. All points a given distance d from ONE LINE l:
TWO LINES parallel to l, one on each side, d units away
4. All point equidistance from TWO PARALLEL LINES l and k:
ONE LINE, parallel to l and k and exactly halfway between them.
5. All point equidistance from TWO INTERSECTING LINES l and k:
TWO LINES, the angle bisectors of the angles formed by lines l and k
Compound loci (Intersection of loci)
6. Draw both loci on the SAME diagram. Find the points where the two loci INTERSECT.
Equation of a circle
7. (x – h)2 + (y ( k)2 = r2
(h, k) = coordinates of the center (Note: signs are opposite from signs in the equation.
r = radius
CIRCLES
Arcs
1. A circle has 360(.
2. Arcs between parallel chords are (
3. Congruent chords cut off ( arcs.
Angles
4. A tangent and a radius intersect at right angles.
5. Measure of a central angle equals measure of the arc is intercepts.
( = arc
6. Measure of an inscribed angle is half the measure of the arc it intercepts.
( = [pic]
7. Measure of an interior angle equals half the sum of the intercepted arcs.
( = [pic]
8. Measure of an exterior angle equals half the difference of the intercepted arcs.
( = [pic]
Segments
9. Two tangent segments from the same exterior point are congruent.
10. If a radius (diameter) is perpendicular to a chord, it bisects the chord
(and conversely).
11. If two chords are equidistant from the center, they are congruent.
12. When two chords intersect inside a circle,
PART ( PART = PART ( PART ab = cd
13. When two secant segments, or a secant segment and a tangent segment,
intersect outside a circle,
OUTSIDE ( WHOLE THING = OUTSIDE ( WHOLE THING
a(a + b) = c(c + d)
a(a + b) = c2
BASIC CONSTRUCTIONS
Notes:
1. All segments are drawn with a straight-edge.
2. All segments are measured with a compass (not a ruler!).
3. The various arcs and segments created are called “construction marks.” Do not erase them when you’re done. If they do not all appear on your paper, you get no credit for the problem!
There are seven basic constructions:
1. Construct a segment congruent to a given segment [pic].
1. Make a point A' to be one endpoint of the new segment.
2. Measure segment [pic]. Keeping compass the same size, move compass point to A' and make arc in the area where you want the new segment to end.
3. Draw a segment from A' to any point on the arc, B'.
2. Bisect a given segment [pic].
1. Open compass to width greater than [pic].
2. With compass point on A, make large arc crossing [pic] twice.
(Or two small arcs, one above, one below [pic].)
Keeping compass the same size, repeat with compass point on B.
3. With straight-edge, connect intersections of arcs.
3. Construct an angle congruent to a given angle ABC.
1. Draw ray [pic] (it is not necessary that it be congruent to [pic]).
2. With compass point on B, draw arc over both sides of (ABC. (The points where the arc intersects the sides of the angle have been labeled P and Q for convenience.)
3. Without changing compass size, move point to B' and draw similar arc.
4. Measure [pic]. Keeping compass the same size, place compass point on P' and draw arc intersecting original arc. This intersection is Q'.
5. Draw ray [pic].
4. Bisect a given angle ABC.
1. With point on vertex B, draw an arc through both sides of (ABC.
(The points where the arc intersects the sides of the angle have been
labeled P and Q only for convenience.)
2. With point on P, draw arc shown in diagram. Without changing compass size, repeat with point at Q.
3. Draw bisector from vertex out through intersection of arcs from 2.
5. Construct a perpendicular to a given line l through a given point P not on l.
1. With point on P and compass opened wider than distance P to l, draw two arcs on l (or one big arc intersecting l twice).
2. With point on intersection of first arc and l, draw new arc below l. Keeping compass the same size, repeat with point on intersection of second arc and l.
3. Draw segment from P through intersection of the newest two arcs.
6. Construct a perpendicular to a given line l through a given point P on l.
1. With point on P, make two arcs on l, one each side of P.
2. Open compass wider. With point on intersection of one arc and l, make an arc above P. Keeping compass the same size, repeat with point on intersection of second arc and l.
3. Draw segment from intersection of newest two arcs through P.
7. Construct a line parallel to a given line l through a given point P not on l.
1. Draw any convenient line through P and intersecting l.
2. Follow directions for constructing congruent angles to construct (b congruent to (a.
3D GEOMETRY
Lines and planes in space
1. If a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by them.
2. Through a given point there passes one and only one plane perpendicular to a given line.
3. Through a given point there passes one and only one line perpendicular to a given plane.
.
4. Two lines perpendicular to the same plane are coplanar.
5. Two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane.
.
6. If a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the given plane.
7. If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane.
8. If a plane intersects two parallel planes, then the intersection is two parallel lines.
9. If two planes are perpendicular to the same line, they are parallel.
Solid geometry
10. The lateral edges of a prism are congruent and parallel.
11. The volume of a prism is the product of the area of the base and the altitude.
12. Two prisms have equal volumes if their bases have equal areas and their altitudes are equal.
13. a. The lateral edges of a regular pyramid are congruent.
b. The lateral faces of a regular pyramid are congruent isosceles triangles.
c. The volume of a regular pyramid is one-third the product of the area of the base and the altitude.
14. a. The bases of a cylinder are congruent.
b. The volume of a cylinder is the product of the area of the base and the altitude.
c. The lateral area of a right circular cylinder equals the product of an altitude and the circumference of the
base.
15. a. The lateral area of a right circular cone is one-half the product of the slant height and the circumference
of its base.
b. The lateral area of a right circular cone is one-third the product of the area of its base and its altitude.
16. a. The intersection of a plane and a sphere is a circle.
b. A great circle is the largest circle that can be drawn on a sphere.
c. Two planes equidistant from the center of the sphere and intersecting the sphere do so in congruent circles.
d. The surface area of a sphere is [pic]. (This formula is on the reference sheet.)
e. The volume area of a sphere is [pic]. (This formula is on the reference sheet.)
Area and perimeter formulas
17. Perimeter of a polygon: Add up all the sides
17. Area of a triangle: [pic]
18. Area of a parallelogram: A = bh
19. Area of a trapezoid: [pic]
20. Area of a circle: A = (r2
21. Circumference of a circle: C = (d
Geometry Review Notes: Proofs
Major Definitions
Perpendicular
Bisect/bisector
Midpoint
Vertical angles
Isosceles triangle
Parallelogram
Rectangle
Rhombus
Square
Trapezoid
Major Postulates
Reflexive
Transitive
Substitution
Addition
Subtraction
Division
Major Theorems
All right angles are congruent.
Vertical angles are congruent.
If two sides of a triangle are congruent, the angles opposite those sides are also congruent.
If two angles of a triangle are congruent, the sides opposite those angles are also congruent.
If two parallel lines are cut by a transversal, corresponding angles are congruent.
If two parallel lines are cut by a transversal, alternate interior angles are congruent.
If two lines are cut by a transversal and corresponding angles are congruent, the lines are parallel.
If two lines are cut by a transversal and alternate interior angles are congruent, the lines are parallel.
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
The diagonals of a parallelogram bisect each other.
The diagonals of a rectangle are congruent.
The diagonals of a rhombus are perpendicular.
The diagonals of a rhombus bisect the vertices.
The diagonals of an isosceles trapezoid are congruent.
Proving Two Triangles Congruent
SSS
SAS
ASA
AAS
RHL
-----------------------
AAA and ASS (SSA) do NOT prove two triangles congruent.
A means corresponding angles (
S means corresponding sides in proportion
A
B
C
P
Q
a
b
f
d
c
e
A
B
C
M
N
A
D
B
C
a
L
l
pp
P
H
O
O
O
O
k
l
m
P
l
A
A
l
P
A
P
A
l
P
m
P
P
Q
l
k
m
l
P
A
l
P
P
Q
k
l
R
l
P
Q
All parallel and congruent.
h
B
h
B
Lateral edges all congruent
Lateral faces all congruent isosceles triangles.
[pic] (This formula is on the reference sheet.)
V = Bh*
= (r2h if the bases are circles.
L = 2(rh* for right circular cylinder.
(*These formulas are on the reference sheet.)
*
Bases congruent.
L = (rl
[pic]
(Both formulas are on the
reference sheet.)
l
h
B
A
B
A'
B'
3
2
1
A
B
2
3
A
C
B
Q
P
2
P'
B'
C'
Q'
1
3
4
5
A
C
B
Q
P
1
3
2
•
P
l
1
2
3
P
l
1
2
3
P
b
a
l
1
2
1
4
5
6
7
2
8
3
l1
t
l2
l2
t
l1
3
8
2
7
6
5
4
1
If l1||l2, then
a. (1 ( (5, (2 ( (6, (3 ( (7, (4 ( (8
b. (3 ( (5, (4 ( (6
c. m(3 + m(6 = 180, m(4 + m(5 = 180
1
4
5
6
7
2
8
3
l1
t
l2
a. If (1 ( (5 (or (2 ( (6 or (3 ( (7, or (4 ( (8), then l1||l2.
b. If (3 ( (5 (or (4 ( (6), then l1||l2.
c. If m(3 + m(6 = 180 (or m(4 + m(5 = 180), then l1||l2.
1
4
2
3
(1 ( (3 and (2 ( (4
A
A'
C'
B'
B
C
C
P
C
B
B'
A'
C'
A
P
C
B
B'
A'
C'
A
(
l
C
B
B'
C'
A'
A
B
B'
C'
A'
A
(
arc
(
(
(
arc
arc
ARC
arc
ARC
a
b
c
d
d
c
b
a
c
b
a
................
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