9.1- demonstrate knowledge of axiomatic systems and of the axioms of non-euclidean geometries

Modus Tollens (MT)

(p > q) :: (~q > ~p), also known as modus tollendo tollens (Latin for "method of removing by taking away") and denying the consequent, is a deductive argument form and a rule of inference.

Double Negation

- If a statement is true, then it is not the case that the statement is not true."

alternate exterior angles: A=-1+14x, B=12x+17..., x=?

1. -1+14x=12x+17 +1 -12x -12x+1 _______________ 2x=18 x=9

The dimensions of the smaller cylinder are two thirds of the larger. The volume of the larger cylinder is 2160 п cm^3. Find the volume of the smaller cylinder.

2^3/3^3 = V/2160п

Solve the proportion, . x/x+3 = 34/40

34x+ 102 = 40x = -6x = -102 = -6x = -102, x = 17

CDEF is a kite. are the diagonals of the kite perpendicular? explain your reasoning, if the slope of the line, line CE in the center is ⅔ and line DF going across is -3/2 then...

the diagonals are perpendicular because the slopes are opposite

If two lines in the same plane are perpendicular to the third line in the plane, then the two lines are parallel. Then you have a parallelogram, and therefore

the opposite sides of a parallelogram are congruent.

If two parallel lines are cut by a transversal

then same-side interior angles are supplementary, corresponding angles are congruent

complimentary angles

two angles that add up to 90 degrees

find the measure of the consecutive interior angles: x+96 and x+96

90° x+96+x+96=180 2x+192=180 -192.-192 2x=-12 x=-6

P. 551 - #61 c. The angle that is neither an interior nor an exterior angle of the triangle is...

<x

Euclid Postulate 3

A circle can be drawn with any center in any radius. Any circle can be drawn from the end or start point of a circle, and the diameter of the circle will be the length of the line segment period

Consequent

A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then".

point

A location in space.

Euclid Postulate 1

A straight line can be drawn from one point to another.

Euclid Postulate 2

A terminated line can be produced indefinitely

acute triangle

A triangle that contains only angles that are less than 90 degrees.

Acute Isosceles Triangle

A triangle with acute angles and exactly 2 equal sides. has 3 vertices and edges

P - 7 Segment Addition Postulate

If B is between A and C, then AB+BC=AC

P-8 Angle Addition Postulate

If Point P lies in the interior of <RST, then m<RSP + m<PST = m<RST

C - 109 - Parallel/Proportionality Conjecture

If a line parallel to one side of a triangle passes through the other two sides, then it divides the other two sides proportionally. Conversely, if a line cuts two sides of a triangle proportionally, then it is parallel to the third side.

P-2 Line Intersection Postulate

The intersection of two distinct lines is exactly one point.

If two angles are not vertical angles, they are not congruent (If ~P then ~Q), is an example of an inverse statement.

The inverse of this statement is false.

C - 68 Intercepted arc

The measure of an inscribed angle in a circle equals one half the measure of its intercepted arc.

C-115 - Altitude of a Right Triangle

If you drop an altitude from the right angle to the hypotenuse of a right angle, it divides the right triangles into two right triangles that are similar to each other and to the original right triangle.

transversal line definition

a line that intersects two or more coplanar line

Conclusion

a statement arrived through reasoning

Conditional Statement (also known as an If Then Statement)

a statement that can be written in the form "If P then Q," where P and Q are sentences. For this conditional statement, P is called the hypothesis and Q is called the conclusion.

If two angles are vertical angles, then they are congruent (If P then Q) is an example of

a statement.

logical argument

an argument based on deductive reasoning, which uses facts, definitions, and accepted properties in a logical order

premise

an assumption; the basis for a conclusion

auxiliary lines definition

are helping lines or an extra line needed to complete a proof in plane geometry.

P - 1 Line Postulate

You can construct exactly one line through any two points.

P - 3 Midpoint Postulate

You can construct exactly one midpoint on any line segment.

In which year was a United States President born? a. 1492 b. 1676 c. 1809 d. 1979

You can eliminate 1492 and 1676 because the person would be a hundred years old or older in 1776, the year the United States became a country. You can eliminate 1979 because anyone born in 1979 would not be old enough to be president -the minimum age is thirty-five. Because you have only four choices and you have eliminated three of them, the remaining choice 1809 must be correct.

unsound argument

a deductive argument that is invalid, has one or more false premises, or both

corollary

a direct or natural consequence or result

Find the midpoint of the segment with the given points as endpoints, (-3,4), (6,7)=

(-3+6/2, 4+7/2)= (3/2, 11/2)

Conjecture: The consecutive angles of a parallelogram are supplementary (C-53) Given: Parallelogram SAND Show: <DSA and <NAS are supplementary

1. SAND is a parallelogram/ 1. Given 2. SN≅AN/ 2. Definition of a Parallelogram 3. <DSA & <NAS/ 3. Definition of Supplementary angles

What are some Linking Proofs About Parallelograms?

AIA Postulate and ASA Postulate both show that a diagonal of a parallelogram divides the parallelogram in two congruent triangles. AIA says that the opposite sides of the parallelogram are congruent. ASA says that the opposite angles of the parallelogram are congruent. Both say the diagonals of a parallelogram bisect each other.

Euclid Postulate 4

All right angles are congruent

Line segment KA || line segment BL What is the length of Line segment CA?

Because triangle CKA is similar to triangle CBL, then 4/4+12 = x/x+18 ¼ = x/x+18 4x = x+18 3x = 18 X=6 Both the ratio (CK/KB = 1/3) and the ratio CA/AL = 1/3

Line segment HT || line segment WH...?

Because triangle HIT similar to triangle WIE, then 48/48+36 = 60/60+y 240+4y = 420 4y = 180 y=45 WH/HI = ¾ and that the ratio of ET/TI=3/4

Prove the conjecture: Conjecture: The base angles of an isosceles trapezoid are congruent.

Construct line PS and line AB perpendicular to TR, intersecting line TR at points S and B. Triangle TSP and Triangle RBA are triangles by the definition of perpendicular, line PS≅line AB by the theorem proved in Exercise 15.8.7, and TP = RA by the definition of isosceles trapezoid. Therefore, TSP≅RBA by the definition of isosceles trapezoid. Therefore, TSP ≅RBA by HL, Congruence Shortcut Therefore, <T≅<R and <TPS≅<RAB by CPCTC. Then the measure of <P=m<TPS+90 degrees = m<A. Therefore, the base angles of an isosceles trapezoid are congruent.

If the four angles of one quadrilateral are congruent to the four corresponding angles of another quadrilateral, then the two quadrilaterals are similar.

False, one possible counterexample is attached. A counterexample Is an example that opposes or contradicts an idea or theory.

If two sides of one triangle are proportional to the sides of another triangle, then the two triangles are similar, true or false

False, one possible counterexample is shown. A counterexample is an example that opposes or contradicts an idea or theory.

Conjecture: If two sides of a quadrilateral are both parallel and congruent, then the quadrilateral is a parallelogram. Task 1 Identify what is given and what you must show.

Given: Two sides of a quadrilateral are both parallel and congruent Show: The quadrilateral is a parallelogram. Task 2: Draw and label the diagram Task 3 Restate what is given and what you must show in terms of the diagram. Given: Quadrilateral FRNK with line FR≅ line KN and line FR ≅ line KN Show: The quadrilateral FRNK is a parallelogram. Often a conjecture is not stated in the form of a conditional. When this occurs, it may be more difficult to recognize what is given what you are trying to show.

Postulate 16 - Coordinate Midpoint Postulate

If (x1, y1) and x2, y2) are the coordinates of the endpoints of a segment, then the coordinate of the midpoint are (x1+x2/2, y1+y2/2)

P - 9 Linear Pair Postulate

If 2 angles form a linear pair, then they are supplementary formerly linear pair conjecture

Euclid Postulate 5

If a straight line falling on 2 other straight lines makes the interior angles on the same side of it taken together less than 2 right angles, then the 2 straight lines if produced indefinitely, meet on the side on which the some of the angles is less than two right angles.

Law of Syllogism

If p-->q and q-->r are true statements, then p-->r is a true statement. also called reasoning by transitivity, is a valid argument form of deductive reasoning that follows a set pattern. It is similar to the transitive property of equality, which reads: if a = b and b = c then, a = c.

Arc Addition Postulate

If point B is on Arc AC and between points A and C, then marcAB+marcBC=marcAC

P-11 SSS Congruence Postulate

If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent formerly SSS Congruence Conjecture

ASA Congruence Postulate

If two angles and the side between them in one triangle are congruent to two angles and the side between them in another triangle, then the two triangles are congruent

P - 15 AA Similarity Postulate

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

c - 110 - Extended Parallel Proportionality Conjecture

If two or more lines pass through two sides of a triangle parallel to the third side, then they divide the two sides proportionally.

CA Conjecture

If two parallel lines are cut by a transversal, then corresponding angles are congruent

P-10 AIA Postulate

If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Conversely, if two lines are cut by a transversal forming congruent alternate interior angles, then the lines are parallel...formerly AIA Conjecture

SAS Congruence Postulate

If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent formerly SSS Congruence Conjecture

Modus Ponens

If you accept if P then Q as true and you accept P as true, then you must logically accept Q as true.

Determine the measures of angles A and B.

It is the case that tan A = 9/12 = .75 and tan^-1 .75 =37 degrees. Since angle B is complementary to angle A, the measure of angle B is therefore 53 degrees.

Conjecture: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Given: Quadrilateral ABCD with <DAB ≅ <BCD and <ADC ≅ <ABC Show: Quadrilateral ABCD is a parallelogram

Let a = m<A, =m<B, and so on. Because a+b+c+d = 360 degrees, a=c and b=d, we have 2a+2b = 360 degrees or a+b= 180 degrees. Therefore, <A and <B are supplementary. Therefore, line AD|| line BC, similarly b+c = 180 degrees Therefore, line AB||line DC. There ABCD is a parallelogram.

Conjecture: If m<N ≠ m<O in triangle NOT, then NT≠OT Given: Triangle NOT with m<N≠m<O Show: NT≠OT

Paragraph Proof: Assume NT = OT, If NT = OT, then m<N = m<O by the Isosceles Triangle Conjecture. But this contradicts the given fact that m<N ≠ m<O. Therefore, the assumption (NT=OT) is false and thus the opposite (NT ≠ OT) is true.

Conjecture - The diagonals of a trapezoid do not bisect each other. Given: Trapezoid ZOID with parallel bases line AO and line ID and diagonals line DO and IZ intersecting at point Y Show: Diagonals of trapezoid ZOID do not bisect each other. That is, DY≠OY and ZY≠IY

Paragraph Proof: Assume the diagonals of trapezoid ZOID do bisect each other. In Exercise Set 16.1, you proved that if the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram. So ZOID is a parallelogram. This ZOID has two pairs of opposite sides parallel. But because it is a trapezoid, it has exactly one pair of parallel sides. This is contradictory. So the assumption that its diagonals bisect each other is false and the conjecture is true.

Conjecture: The sum of the measures of the three angles of every triangle is 180 degrees (Triangle Sum Conjecture) Given: Triangle ABC Show: m<1+m<2+m<3 = 180 degrees

Plan: A nice way to prove this conjecture is to construct a line through one vertex, parallel to the opposite side. (The Parallel Postulate allows you to construct it.) You can show m<4+m<2+m<5=180 degrees with the help of the Linear Pair Postulate and the Angle Addition Postulate, you can show m<1= m<4 and m<3 = m<5 by the AIA Postulate.

Provide the steps and reasons to prove the logical argument. Premises: (~QàP)à~R TàS ~Rà~S T Conclusion: ~(~PàQ)

Proof: 1. ~PàQ 1. Assume the opposite of the conclusion 2. ~QàP 2. From line 1, LC 3. (~QàP)à~R 3. Premise 4. ~R 4. From lines 2 and 3, using MP 5. ~Rà~S 5. Premise 6. ~S 6. From lines 4 and 5, using MP 7. TàS 7. Premise 8. T 8. Premise 9. S 9. From lines 7 and 8, using MP But lines 6 and 9 contradict each other. Therefore, If ~P then Q, the assumption, is false. ∴ If ~(~ P then Q)

Triangle ABC is similar to Triangle DBA X = ?, y = ?

Redraw the two triangles so that the corresponding angles are in the same position. 6/5 = 5/x and 9/6 = y/5

The Jones family paid $150 to a painting contractor to stain their 12 ft by 15ft back deck. The Smiths, their neighbors have a similar deck that measures 16ft by 20ft. If the Smiths wish to "keep up with the Joneses." What is a proportional price the Smiths family should expect to pay to have their deck stained by the contractor?

The fact that the two decks are similar rectangles is irrelevant. The painting contractor charges by time or square footage, and thus $150/ 12x15 = 5x/16 x20

If two angles are not congruent, then they are not vertical angles, (If ~Q then ~P).

This is an example of a contrapositive statement.

P - 5 Parallel Postulate

Through a point not on a line, there is one and only one line parallel to the given line.

P - 6 Perpendicular Postulate

Through a point not on a line, there is one and only one line perpendicular to the given line.

The Ring a Ding Sisters Circus has come to town. P.T. Barnone is the star of the show. She does a juggling act atop a stool that sits atop of a rotating ball that spins at the top of a 20-meter pole. The diameter of the ball is 4 meters, and P.T.'s eye is 2 meters above the ball. The circus manager needs to know the radius of the circular region beneath the ball in which spectators would be unable to see eye to eye with P.T. Find the radius to the nearest tenth meter for the manager so that he can put in the seats for the show. (Use 1.7 for square root 3)

Triangle ABE is similar to triangle ADC AB/BE = AD/CD

If a line parallel to one side of a triangle passes through the other two sides, then it divides them proportionally, true or false.

True, if a line divides two sides of a triangle proportionally, then it is parallel to the third side.

P - 17 Parallel Slope Postulate

Two non-vertical lines have the same slope if and only if they are parallel

Vertical Angles Theorem

Vertical angles are congruent

Three approaches to proving logical arguments:

direct proofs, conditional proofs, and indirect proofs.

Law of Contrapositive

if a conditional statement is true then the contrapositive is true

congruent angles theorem

if a transversal intersects two parallel lines, the corresponding <'s are congruent

valid

if the conclusion follows from the assumptions by applying legal mathematical operations to arrive at the conclusion.

Perpendicular Slope Postulate

if two lines are perpendicular to each other, then the product of their slope is -1. In a coordinate plane, two lines are perpendicular if and only if their slopes are negative reciprocals of each other .

Contrapositive

is exchanging the hypothesis and conclusion of a conditional statement and negating both hypothesis and conclusion. For example, the contrapositive of "if A then B" is "if not-B then not-A". The contrapositive of a conditional statement is a combination of the converse and inverse.

negation of a statement

is the opposite of the given mathematical statement. If "P" is a statement, then the negation of statement P is represented by ~P. The symbols used to represent the negation of a statement are "~" or "¬".

In the figure, line MT|| line segment LU. Does triangle LUV appear to be similar to triangle MTV? Given: Triangle LUV with line MT || line segment LU Show: Triangle LUV is similar to Triangle MTV

line MT || line segment LU, then <1 ≅<2 and <3≅<4 by the Corresponding Angle Conjecture If <1 ≅<2 and <3≅<4, then Triangle LUV is similar to Triangle MTV By the AA Similarity Conjecture If the two triangles are similar, then the corresponding sides are proportional. In the figure, LV/MV = VU/VT = LU/MT

inverse

of a conditional statement is when both the hypothesis and conclusion are negated; the "If" part or p is negated and the "then" part or q is negated. In Geometry the conditional statement is referred to as p → q. The Inverse is referred to as ~p → ~q where ~ stands for NOT or negating the statement.

If two angles are congruent, then they are vertical angles. (If Q then P), the converse of this statement is false is an example

of a converse statement.

converse

of a statement is formed by switching the hypothesis and the conclusion. The converse of "If two lines don't intersect, then they are parallel" is "If two lines are parallel, then they don't intersect." The converse of "if p, then q" is "if q, then p."

Indirect Proofs

when all of the possibilities that can be true are recognized. Next, all but one possibility are eliminated when they are shown to contradict some given fact or accepted idea.

Distance formula on a coordinate plane

which is an application of the Pythagorean theorem. We can rewrite the Pythagorean theorem as d = √[( x₂ - x₁)² + (y₂ - y₁)²] to find the distance between any two points.

Solve the proportion, 4/x = x/9

x^2 = 36, x = ±6

P - 4 Angle Bisector Postulate

you can construct exactly one angle bisector in any angle

The line through points (1,4) and (4, -2) is parallel to the line y=-2x-5, true or false...

True, -2-4/4-1 = -2

If the segment with endpoints (2, -3) and (-1,5) is reflected about the y-axis, the slope of the segment in the new position is 8/3, true or false...

True, The slope of the original segment AB is given by: m = (y2-y1) / (x2-x1). Substituting the coordinate of the points A and B: m = (5-(-3)) / ((-1)-2)= 8/(-3) =-8/3 The reflection rule for the y-axis is (-x,y); Let's denote the original endpoints as A(2, -3) and and B(-1, 5) , and the reflected endpoints as A'(-2, -3) and B'(-1, 5). m' = (5-(-3)) / ((1)-(-2)) = 8/3 =8/3

The sum of the lengths of any two sides of a triangle is always greater than the length of the third side, true or false...

True, the sum of two angles in a triangle may not be greater than the third angle.

P.552 - #71- True or false? if one interior angle of a triangle is a right angle, then the other 2 interior angles of the triangle are complimentary angles.

True. the sum of all 3 interior angles is 180ﾟ. If one of the 3 angles is a right angle, then the other 2 will also have their measure some of 90ﾟ. Therefore, the other 2 interior angles of the triangle are complimentary.

The sum of the measures of the three angles of an obtuse triangle is greater than the sum of the measures of the three angles of an acute triangle, true or false...

False, In an acute triangle, All the three interior angles of an acute triangle measures less than 90°. The angles of an acute triangle add up to 180°.In and obtuse triangle, if one angle measures more than 90°, then the sum of the remaining two angles is less than 90°.

In a triangle, the angle with greatest measure is opposite the shortest side, true or false...

False, The angle with the greatest measure is opposite the longest side.

angle bisector

a ray that divides an angle into 2 congruent angles

consecutive interior angles

interior angles that lie on the same side of the transversal

Find the midpoint of the segment with the given points as endpoints, (15,0), (7,5)=

(15+7/2, 0+5/2) = (11, 5/2)

Find the midpoint of the segment with the given points as endpoints, (5, -8), (-3,-8) =

(5-3/2, -8-8/2) = (1, -8)

Obtuse Isosceles Triangle

One obtuse angle two congruent sides

vertical angles

A pair of opposite congruent angles formed by intersecting lines

definition of a parallelogram

A quadrilateral with two pairs of parallel sides

corresponding angles

Angles in the same place on different lines

Transversal

a line that intersects two or more lines

transversal

a line that intersects two or more lines

consecutive interior angles: A=100°, B=?...?

80°

skew lines

Lines that do not intersect and are not coplanar

subset

a set that is part of a larger set

Theorem

a statement that can be proven

postulate/axiom

a statement that is accepted as true without proof

isosceles triangle

a triangle with at least two congruent sides

in a triangle, <A, <B, & <C= 60° a piece, the sides: a, b, & c= 8.6 a piece, this triangle is:

equilateral

in a triangle, <A=45°, <B=45°, the sides: a=4.8, b=4.8, & c= 6.8, this triangle is:

right isosceles triangle

Midpoint

(x1+x2/2, y1+y2/2)

alternate interior angles: A=111°, B=?...?

111°

P. 551 - #61 b. The angles that are exterior angles.are.(attached).?

y, z

Find the midpoint of the segment with the given points as endpoints, (6,4), (12, -2) =

(6+12/2, 4-2/2) = (9,1)

line (geometry)

(geometry) a set of points that extends infinitely in opposite directions

equilateral triangle area

(s^2√3)/4

P. 552, #75 - If AB and CD intersect at point O, and <AOC=<BOC, explain why AB⟂CD.

1. <AOC and <BOD are vertical angles. 2. Therefore, <BOC and < AOD are vertical angles. 3. Therefore, <BOC = <AOD 4. From the statement, < AOC=<BOC, now <BOD = <AOD 5. That is only possible when AB⟂CD then all the 4 angles will be equal. ​

p. 602 - #18 -The measure of <a...?

1. <a=180°- (98°+(180°-159°))-- the sum of the interior & exterior angles is 180° 2. <a=180°-119°--the sum of the measures of the interior angles of a triangle is 180°.

Given: ABCD is a parallelogram, line segment AW≅line segment XC Prove: line segment WY≅line segment YX

1. ABCD is a parallelogram, given 2. Line segment AW≅line segment XC, given 3. Line segment AB ∥line segment DC, definition of parallelogram 4. <WAC≅<XCA and <AWX≅<CXW, if parallel lines are cut by a transversal, alternate interior <'s are congruent 5. △AWY≅△CXY, angle side angle 6. Line segment WY≅ line segment YX, Corresponding parts of congruent triangles are congruent

Steps to use a protractor

1. Place the center mark of the protractor on the vertex of the angle. 2. Rotate the zero edge of the protractor to line up with the side of the angle. 3. Read the measure of the angle where th other side of the angle crosses the protractor's scale.

P. 552, #73 - Find the sum of the measures of angles X,Y, and Z in the diagram attached.

1. The angles a and Y are supplementary angles. Therefore, the sum of the angles is: a+y= 180°, y=180°-a. 2. <c and <X are supplementary angles. the sum of the angles: c+x=180°, x=180°-c. 3. the angles, B and Z are supplementary angles. therefore the sum of the angles is: b+z=180°, z=180°-b 4. Add the 3 equations: X + Y + Z, 180°-c+(180°-a)+(180°-b). 5. the sum of 3 angles is a+ B+ C=180ﾟ x+y+z=540°-(a+b+c), x+y+z=360°

P. 552, #66 - Given that <y=130°, find the measures of <a and <b.

1. The objective is to find the measures of < a and <b. 2. For convenience, the other interior angles of the triangle is name as <C and <d. 3. from the diagram, <D and <Y are supplementary angles. 4. <d+<y=180° <d+130°=180°, <d=50° 5. The sum of angles in a triangle is 180°. Therefore, <a+<c+<d=180°, <a+90°+50°=180°, <a=40° 6. From the diagram, <b and <a are supplementary angles. <b+40°=180°, <b=140°

P. 552, #74- Explain why < a + <B =< X. Write a rule that describes the relationship between an exterior angle of a triangle and the opposite interior angles.

1. The sum of 3 <'s is <a+<b+<c=180° 2. From the equation (1)<c+<x=180°, <c+<x= <a+<b+<c, <x=<a+<b 3. Therefore, It can be said that an exterior angle of a triangle is equal to the sum of the opposite interior angle. 4. The angles B & z are supplementary angles, therefore, b+z=180°. 5. The sum of 3<'s in a triangle is <a+<b+<c=180° 6. From the equation (2) <b+<x=180°, <b+<x=<a+<b+<c, hence <x=<a+<c ​

P. 550, #51, Given that l₁∥ l₂, find the measures of angles of a and b

1. To find the values of a and B, use the equality of corresponding angles and supplementary angles. Corresponding angles are the angles on the same side of the transversal angle 2. angle equals 38ﾟ 3. < a + <b are supplementary angles, so < a + <b= 180ﾟ 4. substituting the value of <a, 38°+ <b=180°, <b=180°-38°, <b=142°

alternate interior angles: A=?, B=110°...?

110°

alternate interior angles: A=?°, B=113°...?

113°

alternate interior angles: A=125°, B=?...?

125°

find the measure of the corresponding angles: 5x+10 and 6x

60° 5x+10=6x -5x. -5x 10=x

definition geometry

A branch of mathematics that deals with the measurement, properties, and relationships of points, lines, angles, solids, and surfaces.

angle

A figure formed by two rays with a common endpoint

Ray (geometry)

A line with a start point but no end point (it goes to infinity)

obtuse triangle

A triangle with one angle that is greater than 90 degrees.

equilateral triangle

A triangle with three congruent sides with 3 vertices and edges

exterior angle

An angle formed by one side of a polygon and the extension of an adjacent side

corresponding angles

Angles in the same place (matching corners) on different lines

corresponding angles...?

Angles in the same place on different lines same angle measurement in a different position <1 and <5 <2 and <6 <3 and <7 <4 and <8

adjacent angles

Angles that have a common side and a common vertex (corner point).

alternate exterior angles

Angles that lie outside a pair of lines and on opposite sides of a transversal.

The base angles of an isosceles triangle are supplementary, true or false.

False, the base angles of an isosceles triangle are just congruent and equal not always supplementary.

The centroid of a triangle is the point of intersection of the three altitudes.

False, the centroid of a triangle is the intersection of three medians. The orthocenter is the area where three altitudes meet.

P. 551, # 60...True or False (see attached) <c and <d are supplementary if l₁ and l₂ are parallel...

False. From the diagram (attached) angle C and angle D or supplementary angles. It has nothing to do with parallelism. ​

P. 551, # 57...True or False (see attached) <b and <c have the same measure even if l₁ and l₂ are not parallel...

False. alternate interior angles are 2 non adjacent angles that are on opposite sides of the transversal. They have the same measure when 2 lines are parallel lines. If 2 lines are not parallel, then the 2 angles being b &c do not have the same measure.

corresponding angles theorem

If a transversal intersects two parallel lines, then corresponding angles are congruent.

Alternate Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.

Consecutive Interior Angles Theorem

If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.

Proportional Parts Conjecture

If two triangles are similar, then the corresponding altitudes, corresponding medians, and corresponding angle bisectors are proportional to the corresponding sides.

The measures of angles for A & B on the attached are: A= 37° and B= 53°, true or false...?

It is the case that tan A = 9/12= .75 and tan^-1 .75= 37°. Since angle B is complementary to angle A, the measure of angle B is therefore 53ﾟ. 9.1- Demonstrate knowledge of axiomatic systems and of the axioms of non Euclidean geometries.

P. 552, #65 - Given that <y=45°, find the measures of <a and <b.

Step 1: <a and <y are vertically opposite angles; therefore, <a=45°. Step2: Now consider ⧍ABC, the sum of the <'s in the ⧍ are 180°. Step 3: m<a+<BCA+m<ABC =180° 45°+90°+<mABC=180°, m<ABC=180°-135° =45° Step 4: The measure of a straight < is 180° m<ABC+m<b=180° 45°+m<b=180° m<b=180°-45°=> 135°

Given that <a=95° and <b=70°, find the measures of <x and <y...

Step1: <a=95° and <70° Step2: To find the measures of angle X and angle Y, the other 2 interior angles are: <C and <D • < Y and < B are vertical angles <y=<b=70°, <y=70° step 3: <a and <C or supplementary angles, <a+<c=180°, 95°+<c=180°, <c=85° step 4: observed in the diagram <b, <c, and <d=180°, 70°+85°+<d=180°, 155°+<d=180°, <d=25° Step 5 angle D and angle X are supplementary angles angle D plus angle X equals a 180ﾟ <x=155°

P.546- Given that <a=45° and <x=100°, find the measures of angles b,c, and y...see attached...

Strategy 1. To find the measure of angle B in angle C use the definition of supplementary angles. To find the measure of angle Y use the fact that angle C and angle Y are vertical angles. Solution 2. <b+<x=180°, <b+100°=180°, <b=80° <a+<b+<c= 180° 45°+80°+<c=180° 125°+<c=180° <y=<c=55°

Line AB & cD are parallel and measure of angle AXF equals a 140°. then what is the measure of <CYE.

The angles AXF &cYE lie on one side of the transversal. Line EF and inside the 2 lines AB & CD -- consecutive interior angles. Since lines AB is parallel to CD, by consecutive inter angles theorem, angle AXF and angle CYE or supplementary. , That is the measure of angle AXF plus the measure of angle CYE equal a 180ﾟ but measure angle AXF equals a 140ﾟ substitute and solve 140ﾟ plus the measure of angle CYE equal a 180ﾟ m<Cye=140ﾟ

P. 551, # 58...True or False (see attached) <a and <b have the same measure even if l₁ and l₂ are not parallel...

True. 2 angles that are on the opposite sides of the intersection of 2 lines are called vertical angles and vertical angles have the same same measure. angle a an angle B are vertical angles if the 2 lines are not parallel.

P. 551, # 59...True or False (see attached) <a and <d are supplementary if l₁ and l₂ are parallel...

True. Since l₁ and l₂ are parallel,<a=<c as they are corresponding angles. therefore, it can be said that <a and <D are supplementary provided l₁ and l₂ are parallel. ​

adjacent angles

Two angles that share a common side and have the same vertex, these angles are supplementary

supplementary angles

Two angles whose sum is 180 degrees

scalene triangle

a triangle with no congruent sides

P. 551 - #61 a. The angles that are interior angles.are.(attached).?

a,b,c

P. 551, # 62...a. <a+<b+<c =?....b. <a+y and <a+z =? (See attached)...?

a. 180° b. 180°

in a triangle, <A=57°, <B=79°, & <C= 44°, the sides: a=6.1, b=8.7, & c= 7.4, this triangle is:

acute scalene triangle

alternate interior angles

angles between 2 lines and on opposite sides of a transversal

alternate exterior angles

angles between 2 lines and on opposite sides of a transversal <2 and <8 <1 and <7

consecutive interior angles...?

angles that are on the same side of the transversal and inside the two lines ​

corresponding parts of congruent triangles

are congruent

plane geometry

branch of mathematics that shows how points, lines, angles, and surfaces relate to one another

9.1- #60 - line AB lies on plane M and line CD lies on plane N. Planes M & N are parallel. which of the following statements must be true about lines AB &c D,

competency 9 this question requires the examinee to demonstrate knowledge of axiomatic systems. Lines AB &c D are contained in parallel plains and parallel planes do not intersect. Thus lines AB &c D will have no point in common.

The leaning tower of pizzaThe leaning tower of pisa is the bell tower of the cathedral in Pisa, Italy. Its construction began on August 9, 1173, and continued for for about 200 years. The tower was designed to be vertical, but...,

it started to lean during its construction. By 1350, it was 2.5ﾟ off from the vertical; by 1817, it was 5.1ﾟ off; and by 1990, it was 5.5ﾟoff. In 2001, work on the constructor returned its list to 5ﾟ was completed. (Source: Time Magazine, June 25th 2001, page 34-35)

Line AB lies on plane m and line CD lies on plane n. Planes m and n are parallel. which of the following statements must be true about the lines AB and CD?

lines AB and CD are contained in parallel planes, and parallel planes do not intersect. Thus, Lines AB and CD will have no point in Common.

triangle

lines cut by a transversal that are not parallel but intersect at 3 points. each of the 3 points of intersection is the vertex of four angles.- again - 3 sided polygon...

Right Scalene Triangle

one right angle and no congruent sides

a trapezoid is a quadrilateral that has exactly one pair of parallel opposite sides. Is ABCD a trapezoid? if line AB (long side) and line BC (short side) have the same slope 1/3 (up one unit, right 3 units) then they are

parallel

line segment

part of a line with two endpoints

Alternate Exterior Angles - Congruent

two parallel lines are cut by a transversal, then alternate exterior angles are congruent.

C-106 - The angle bisector in a triangle divides the opposite side into two segments

whose lengths are in the same ratio as the length of the two sides forming the angle.

P.549 - #41 - Given that side side AO⟂ OB, express in terms of x the number of degrees in <BOC...see attached...

x+90°<BOC= 180° x+<BOC= 180°-90° x+<BOC= 90° <BOC= 90°-x