Geometry Semester 2 Finals

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Geometry Semester 2 Finals by Mind Map: Geometry Semester 2 Finals

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°

8. DIY's

8.1. construct a perpendicular bisector

8.1.1. on the segment, spread compass to a little more than halfway, and draw an arch crossing over the line. Repeat on the other end, and draw a line through where the arches meet.

8.2. circumscribe

8.2.1. Put compass at circumcenter, letting it touch one vertice. It should pass through all vertices when you make a circle.

9. Sum of the measures of a quadrilateral is 360.