# Geometry Semester 2 Finals

##### by Flo Spiekerman 06/06/2012

# Geometry Semester 2 Finals

by Flo Spiekerman# 1. Vocabulary

## 1.1. Circumcenter

### 1.1.1. point of intersection of the perpendicular bisectors

## 1.2. Concurrent

### 1.2.1. lines or line segments that intersect in a single point

## 1.3. Incenter

### 1.3.1. point of intersection of bisectors

## 1.4. Circumscribe

### 1.4.1. to draw a geometric figure AROUND another figure so that the two are in contact but do not intersect

## 1.5. Inscribe

### 1.5.1. to draw a figure INSIDE another so that their boundaries touch but do not intersect

## 1.6. Median

### 1.6.1. a line segment that connects the midpoint and the vertex of the opposite side

## 1.7. Centroid

### 1.7.1. the point of intersections of the medians of a triangle

## 1.8. Altitude

### 1.8.1. a perpendicular line segment to a side of a triangle that passes through the opposite vertex

## 1.9. Orthocenter

### 1.9.1. points of intersection of the altitudes of a triangle

## 1.10. Midsegment

### 1.10.1. a line segment whose endpoints are midpoints of an included and an opposite line

## 1.11. Conjecture

### 1.11.1. an educated guess through observation

## 1.12. Similarity

### 1.12.1. same shape but NOT the same size

### 1.12.2. Similar polgyons

1.12.2.1. all corresponding angles are congruent, all corresponding sides are in proportion

## 1.13. Proportion

### 1.13.1. an equation in which 2 fractions are set equal to each other

## 1.14. Pre-image

### 1.14.1. the first set of coordinates of a shape

## 1.15. Image

### 1.15.1. the set of coordinates after a transformation has been applied

### 1.15.2. the set of coordinates after a transformation has been applied

# 2. Postulates and Theorems

## 2.1. Angle Side Angle

### 2.1.1. If two angles and the included side of one triangle are congruent to two angles and the included side of another, then the two triangles are congruent.

## 2.2. CPCTC

### 2.2.1. Corresponding Parts of Congruent Triangles are Congruent

2.2.1.1. Since ____ ≅ ____, _____≅______, and _______≅_______, triangle ____ is congruent to triangle ____ by _______. Since corresponding part of congruent triangles are congruent, _________.

2.2.1.2. Since ____ ≅ ____, _____≅______, and _______≅_______, triangle ____ is congruent to triangle ____ by _______. Since corresponding part of congruent triangles are congruent, _________.

## 2.3. Hypotenuse Leg Theorem

### 2.3.1. If the hypotenuses are congruent and a pair of legs is congruent then the triangles are congruent.

## 2.4. Side Side Side

### 2.4.1. If three sides of one triangle are congruent to three sides of another triangle then the triangles are congruent.

## 2.5. Side Angle Side

### 2.5.1. If two sides and the included angle of a triangle are congruent to two side and an included angle of another triangle, then the triangles are congruent.

## 2.6. Triangle Angle and Opposite Side Comparison Theorem

### 2.6.1. If two sides of a triangle are not congruent, then the larger angle lies opposite the longest side.

## 2.7. Corollary to the Triangle Exterior Angle Theorem

### 2.7.1. The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles.

## 2.8. Triangle Inequality Theorem

### 2.8.1. The sum of the lengths of any two sides if a triangle is greater than the length of the third side.

## 2.9. Angle Angle Similarity

### 2.9.1. If two angles from two different triangles are congruent, then the two triangles are similar.

2.9.1.1. by the Triangle Angle Sum Theorem

## 2.10. Triangle Angle Sum Theorem

### 2.10.1. The sum of all angles in a triangle is 180.

## 2.11. Corresponding Angles Theorem

### 2.11.1. When there are 2 parallel lines crossed by a transversal, then corresponding angles are congruent.

## 2.12. Pythagorean Theorem

### 2.12.1. leg₁² + leg₂² = hypotenuse²

## 2.13. Side Splitter Theorem

### 2.13.1. a+b/c = (a/c) + (b/c)

## 2.14. Base Angle Theorem

### 2.14.1. in an isosceles triangle, base angles must have the same measure

## 2.15. Vertical Angles Theorem

### 2.15.1. vertical angles are always congruent.

# 3. Proofs

## 3.1. Indirect proofs

### 3.1.1. A proof in which the statement to be proved is assumed to be false. You then prove that the statement is not possible, confirming it is a false statement.

## 3.2. Coordinate proofs

### 3.2.1. prove distances are equal

### 3.2.2. prove slopes are equal, which means they are parallel

### 3.2.3. slopes multiply to -1, which means they are perpendicular

# 4. Properties

## 4.1. Comparison Property of Inequality

### 4.1.1. 'if a=b+c and c>0, then a>b'

## 4.2. Equality

### 4.2.1. Reflexive

4.2.1.1. a = a

### 4.2.2. Symmetric

4.2.2.1. if a = b, b = a

### 4.2.3. Transitive

4.2.3.1. if a = b and b = c, then a = c

### 4.2.4. Addition

4.2.4.1. if a = b, then a+c=b+c

### 4.2.5. Subtraction

4.2.5.1. if a = b, then a-c=b-c

### 4.2.6. Subsitution

4.2.6.1. if a = b, then a may be replaced with b

## 4.3. Congruence

### 4.3.1. Reflexive

4.3.1.1. AB≅AB

### 4.3.2. Symmetric

4.3.2.1. If AB≅CD, then CD≅AB

### 4.3.3. Transitive

4.3.3.1. If AB≅CD and CD≅EF, then AB≅EF

# 5. Shapes

## 5.1. Rhombus

### 5.1.1. 4 congruent sides

## 5.2. Parallelogram

### 5.2.1. 2 pairs of parallel sides

### 5.2.2. a parallelogram has two pairs of congruent sides. If the length of one side is changed, the length of the opposite side will change to the same measure

### 5.2.3. opposite angles of a parallelogram have the same measure, and if one is changed, the measure of the opposite angle equally changes.

## 5.3. Rectangle

### 5.3.1. 4 right angles, 2 pairs of congruent sides, all adjacent sides perpendicular

### 5.3.2. in a rectangle, diagonals bisect. Diagonals in a rectangle are congruent.

## 5.4. Trapezoid

### 5.4.1. exactly one pair of parallel sides, exactly one pair of congruent sides

## 5.5. Square

### 5.5.1. 4 congruent sides, all adjacent sides perpendicular, 4 right angles

## 5.6. Kite

### 5.6.1. has no parallel sides, 2 congruent sides

### 5.6.2. opposite sides are NOT congruent

# 6. Formulas

## 6.1. distance

### 6.1.1. √(x₂-x₁)² + (y₂-y₁)²

## 6.2. midpoint

### 6.2.1. (average x, average y)

6.2.1.1. ((y₂-y₁) / 2), ((x₂-x₁) / 2)

## 6.3. slope

### 6.3.1. (change in y/change in x)

6.3.1.1. (y₂-y₁) / (x₂-x₁)

## 6.4. Surface Area

### 6.4.1. Cube

6.4.1.1. 6a²

### 6.4.2. Rectangular Prism

6.4.2.1. 2ab + 2bc + 2ac

### 6.4.3. Sphere

6.4.3.1. 4Πr²

### 6.4.4. Cylinder

6.4.4.1. 2Πr² + 2Πrh

### 6.4.5. Square Pyramid

6.4.5.1. s²+2sl

### 6.4.6. Cone

6.4.6.1. Πr² + Πrl

## 6.5. Trigonometry

### 6.5.1. SOHCAHTOA

6.5.1.1. Finding measure of a side

6.5.1.1.1. Finding hypotenuse

6.5.1.1.2. Finding adjacent

6.5.1.1.3. Finding opposite

6.5.1.2. Finding measure of an angle

6.5.1.2.1. Inverse of trigonometry

### 6.5.2. Trigonometric ratio

6.5.2.1. ratio you can write for a particular angle in a right angle

## 6.6. Volume

### 6.6.1. Cylinder

6.6.1.1. Πr² x h

### 6.6.2. Right-angled Triangular Prism

6.6.2.1. ½ (b x h) x l

### 6.6.3. Triangular Prism

6.6.3.1. ½ (b x h)

### 6.6.4. Cube

6.6.4.1. l x h x w

## 6.7. Area

### 6.7.1. triangle

6.7.1.1. ½bh

### 6.7.2. trapezoid

6.7.2.1. ½h (b₁+b₂)

### 6.7.3. rectangle/parallelogram

6.7.3.1. bh

### 6.7.4. rhombus/kite

6.7.4.1. ½d₁d₂

### 6.7.5. regular polygon

6.7.5.1. ½ap

### 6.7.6. circle

6.7.6.1. Πr²

### 6.7.7. F+V = E +2

6.7.7.1. Faces+Vertices = Edges +2

# 7. Transformations

## 7.1. Translation

### 7.1.1. slide

7.1.1.1. T (6, -7)

7.1.1.1.1. Add 6 to the x-coordinate, subtract seven from the y-coordinate

### 7.1.2. reflections

7.1.2.1. x-axis

7.1.2.1.1. the x-coordinates remain of the same value, but the y coordinate becomes it's opposite (1,2)→(1,-2). The image is mirrored over the x-axis.

7.1.2.2. y-axis

7.1.2.2.1. the y-coordinates don't change, but the x-coordinates become their opposite value (1,2)→(-1,2). The image is mirrored over the y-axis.

7.1.2.3. y=x

7.1.2.3.1. the value of y becomes the value of x, and vice versa (1,3)→(3,1).

7.1.2.4. x = 1

7.1.2.4.1. reflect over x=1, setting 1x as the line of reflection. Do the same for y or a different value.

### 7.1.3. rotation

7.1.3.1. you take the image and rotate it a certain amount of degrees. Then you pretend that the turned axes are the axes straight up, and take what the coordinates would be.

7.1.3.1.1. Clockwise.

7.1.3.1.2. Counter-clockwise.

### 7.1.4. dilation

7.1.4.1. multiply the coordinates of the pre-image by the scale factor,

### 7.1.5. composition

7.1.5.1. one transformation followed by another

7.1.5.1.1. T (5,-1) ºR 90°