Unit 1: Reasoning in Geometry
What are the two types of reasoning in geometry that we learned in class? What are their purposes?
What are conditional statements and biconditional statements? What are converse statements?
What is classifying and differentiating?
Inductive reasoning and deductive reasoning. Inductive reasoning's purpose is to make broad generalizations based on specific examples, patterns, or observations, while deductive reasoning's purpose is to use facts, definitions, rules, or laws to draw logically certain conclusions.
Conditional statements are a statement that uses a hypothesis to prove a conclusion "If P, then Q."
Biconditional statements are statements that are true in both its original conditional statement form and its converse statement form (If P, then Q, If Q, then P are both true).
Converse statements are the reverse of conditional statements (If P, then Q (original) ---> If Q, then P (converse)).
Classifying is putting something in a general category and differentiating is finding how that something is different from others of the same category.
What are points, segments, lines, parallel lines, perpendicular lines,transversal lines, vertical angles, measuring segments, measuring angles, and angle pairs?
What is the addition postulate?
Points are marked locations on a line, Segments are lines with dots at the ends to indicate a stop, parallel lines are lines that are across from each other, perpendicular lines are two lines that intersect so that there is a midpoint (imagine a + as perpendicular lines, transversal lines are lines that cross more than one line (often parallel lines are crossed), measuring angles are involves using a measurement tool like a protractor to find and measure angles, and angle pairs are the relationships between two different angles (complementary, supplementary, adjacent, vertical angles, linear pairs, alternate interior angles, alternate exterior angles, corresponding angles, etc).
The addition postulate is a two form postulate that acts as a postulate for angles and segments. The Angle Addition Postulate states that the measure of a whole angle is the sum of the measures of two adjacent angles inside it, while The Segment Addition Postulate states that if point B is between points A and C on a line, then the length of segment AB plus the length of segment BC equals the length of segment AC (AB+BC=AC).
What are transformations, the different types, and what does each one do?
What are preimages?
Transformations are things you can do to get from somewhere on a graph to another desired place. The three different types of transformation are translation (shift to left, right, or up), rotation (turn by degrees), and reflection (flip horizontally or vertically).
A preimage is the original image before a transformation was or is applied.
What are the main conjectures that are used to prove that a triangle is congruent?
How many sets of congruent sides and angles do you need to prove a triangle is congruent with another triangle?
What is the Pythagorean Theorem?
The main conjectures we use are SSS, SAS, AAS, and ASA.
2 sets of congruent sides and 1 congruent pair of angles (SAS), 3 congruent sides (SSS), or two congruent pairs of angles and a side that is included (ASA and sometimes AAS).
The Pythagorean Theorem is a² + b² = c², which can help with finding the area of shapes and more.
What are quadrilaterals and polygons?
What are the different kinds of quadrilaterals?
Quadrilaterals are a specific kind of polygon that has four sides, four angles, and four vertices. Polygons are closed shapes with 3 or more sides.
Squares, rectangles, rhombuses, parallelograms, trapezoids, and kites are all examples of quadrilaterals.
Given the true conditional statement: "If a red berry, helps others cross the berry street, then it is a crossing berry." A red berry named Romeo helped Razberri to cross the road. Decide if the statement is a logical conclusion based only on the information above.
A. Razberri is a crossing berry
Not logical since Razberri is the one receiving the help.
B. Romeo is a crossing berry
Romeo is logically a crossing berry since he fulfills the condition of helping others across the road.
The length of line segment EF is 34 inches and the length of line segment FG is 7 inches. What is the length of line segment DE?
DE = 27, because EF - FG = DE, which is 34 - 7 = 27.
The transformation rules for each transformation type
Translation: (x,y) --> (x+/- __, y +/- __)
Rotation 90: (x,y) --> (-y, x)
Rotation 180: (x,y) --> (-x, -y)
Rotation 270: (x,y) --> (y, -x)
Reflection over x-axis: (x,y) --> (x, -y)
Reflection over y-axis: (x,y) --> (-x, y)
Reflection over line y=x: (x,y) --> (y,x)
If ST is the same as RS, and RS is 10 then what is ST and what conjecture is the triangle if there is one pair of congruent angles included?
ST = 10, as it is equal or congruent to RS.
The conjecture would be SAS, there is two congruent sides and 1 pair of congruent angles.
These shapes have properties that they all share. What are the properties?
What are squares, rectangles, and rhombi?
They all have 4 sides, 2 sets of parallel sides, congruent opposite angles, supplementary consecutive angles, and the diagonals bisect each other.
If an angle is obtuse, then the angle has a measure more than 90 degrees. A triangle has angle BCA and it is obtuse. What is a statement that is a logical conclusion and what is a statement that is NOT a logical statement, based on the information above?
A logical statement could be that angle BCA is more than 90 degrees because it is logically an angle that fulfills the condition of being an obtuse angle
An illogical statement could be that angle BCA is the biggest angle in the triangle, which is wrong because it is not the biggest angle in the triangle if it is the only angle in the triangle.
Slope formula: m= y2 - y1 over x2 - x1
E: (-4,1)
F: (1,3)
G: (-3,0)
A: (2,2)
What are the slopes of EF and GA with formulas showing the way to get the answer for them? What are these lines?
What best describes the lines?
EF = 2/5 (3-1 over 1-(-4)
GA = 2/5 (2-0 over 2 - (-3).
Parallel best describes the lines the lines are facing each other
A's coordinates are (5,12). If it is translated according to the rule (x,y) --> (x+ 3, y + 2), what would your new coordinates be?
The new coordinates would be (8, 14).
A car starts at the origin of a coordinate plane, then drives 4 units to the right and 7 units down.
What are the new coordinates? How far is the car from the origin in units?
The new coordinates of the car are (4, -7), and in units, it is 8.06 units away from the origin.
Complete these statements with "always", "sometimes", or "never".
1. A rectangle is _____ a square.
2. A rhombus is _____ a parallelogram.
1. A rectangle is sometimes a square
2. A rhombus is always a parallelogram.
Dictionary.com defines car as "a road vehicle with an engine, typically four wheels, designed to carry people, often called a motorcar or automobile". What is the classification in this definition and what is the differentiation?
Classification: Vehicle
Differentiation: has an engine, four wheels, is designed to carry people, and is usually called an automobile.
Find the line whose equation is (y+5) = 4(x+2) on the xy plane of a graph.
How to solve:
y+5 = 4x + 8 (use distributive property)
y =4x+8 - 5 (move 5 to right)
y = 4x+3
Give J'K'L'M's coordinates after applying the translation rule of (x,y) --> (x - 1, y + 9). The original coordinates are
J: (-6, 3)
K: (-5, 6)
L: (-2, 7)
M: (-5, 0)
The new coordinates applying the rule are:
J': (-7, 12)
K': (-6, 15)
L': (-3, 16)
M': (-6, 9)
Find the distance between (-9, -6) and (11, -13).
d = (11-(-9))2 + (-13 - (-6))2 = 449 (square root this to get the final answer, 21.2)
Is a concave polygon FIRE that is drawn so that m<S is exactly 90 degrees possible? Explain.
No, it is not possible because <R being equal to exactly 90 degrees means a 360 degree sum of interior angles. FIRE is also a quadrilateral.
33 students were asked what music platforms they use.
22 used Spotify, 13 used Pandora, 5 used Soundcloud.
No student used all three platforms, all Soundcloud users also used Spotify, 15 students who used Spotify don't use the others, and every student used at least one of these platforms.
How many students use only Pandora?
Only 13 students use only Pandora, because 33= 15+5 + x + 0 = 33=20 + x = x= 33-20 = 13.
On a figure, point B isn't shown but has coordinates (9,a). CD and EF are perpendicular,but then what is the value of a?
How to solve: mEF = -1-4 over 4-2 = -5/2, mCD - (-5/2), mCD x (-5/2) = -1, mCD = -1/(-5/2) = 2/5, mCD = a-1/a-(-1), 2/5 = a-1/10 (multiply both sides by 10), 4 = a- 1, a = 4 + 1, a = 5.
The value of a is 5.
Line segment PR has end points P (5, 10) and R (5,8). Find the coordinates of P' and P'' after being translated 4 units left and then being reflected over the y-axis.
P' (1, 10) --> P'' (-1,10)
Let A = (7, -1) and B = (2,y). AB= 133, find two possible y values.
The two possible y values are 11 and -13.
It is solved like this:
132 = (2-7)2 + (y-(-1)2
1692 = (5)2 + (y+1)2
169 = 25 + (y+1)2
169 - 25 = (y+1)2
144= (y+1)2 (square root this whole section)
+12 = y + 1 = y = 12 + 1 = 11 and y = -12 - 1 = -13
Parallelogram LIFE is congruent by a certain conjecture. What is it and what are the congruent proofs in LIFE?
First, the congruent proofs we can find in LIFE are:
1. <LFE ≅ <LIE
2. <IFL ≅ < FEI
3. △LFE ≅ △IEL
Now that we know the proofs, we know we have the ASA or SSS conjecture as our statement.