Formulas: (1.) Geometry Formulas
(2.) Mensuration Formulas
For ACT Students
The ACT is a timed exam...60 questions for 60 minutes
This implies that you have to solve each question in one minute.
Some questions will typically take less than a minute a solve.
Some questions will typically take more than a minute to solve.
The goal is to maximize your time. You use the time saved on those questions you solved in less than a minute, to solve the questions that will take more than a minute.
So, you should try to solve each question correctly and timely.
So, it is not just solving a question correctly, but solving it correctly on time.
Please ensure you attempt all ACT questions.
There is no negative penalty for any wrong answer.
For JAMB and CMAT Students
Calculators are not allowed. So, the questions are solved in a way that does not require a calculator.
For NSC Students
For the Questions:
Any space included in a number indicates a comma used to separate digits...separating multiples of three digits from behind.
Any comma included in a number indicates a decimal point.
For the Solutions:
Decimals are used appropriately rather than commas
Commas are used to separate digits appropriately.
Solve all questions
Show all work
(1.) A student made several mathematical statements/sentences or asked questions regarding basic geometry.
As a teacher, how would you respond?
(a.)
The line above is $\overleftrightarrow{ABC}$
(b.) A line segment has a finite number of points because it has two endpoints.
(c.) $\overline{AB}$ and $\overline{CD}$ are parallel because they do not intersect.
(d.) An angle is the intersection of two line segments (or the intersection of the endpoints of two rays).
Can we also define an angle as the union of two half-lines and a common point?
(e.) To make the measure of an angle greater, just extend the rays.
(f.) $\overline{AB}$ ≠ $\overline{BA}$ because:
$\overline{AB}$ starts at A and ends at B, and $\overline{BA}$ starts at B and ends at A.
(g.) There can only be 360 different rays emanating from a point since there are only 360 ° in a circle.
(h.) It is actually impossible to measure an angle, since each angle is the union of two rays that extend infinitely and therefore continue forever.
(i.) If any two planes that do not intersect are parallel, then any two lines that do not intersect should also be parallel.
(j.) If two lines, a and b, are perpendicular to the same line, lines a and b must be parallel.
(k.) A ray has two endpoints.
(l.) $\overrightarrow{MN}$ = $\overrightarrow{NM}$
(m.) For any points M and N, $\overleftrightarrow{MN}$ = $\overleftrightarrow{NM}$
(n.) A line segment contains an infinite number of points.
(o.) Skew lines are coplanar.
(p.) If two distinct planes intersect, their intersection is a line segment.
(q.) These two statements mean the same thing. I can write either way.
$ 1st\;\;Statement:\;\; AB \cong CD \\[3ex] 2nd\;\;Statement:\;\; \overline{AB} \cong \overline{CD} \\[3ex] $ (r.)
(s.)
(t.)
(u.)
(v.)
(w.)
(x.)
(y.)
(z.)
(a.) The line should be named as: $\overleftrightarrow{AC}$
Two points determine a line, so only two points are used to name the line.
(b.) A line segment has two endpoints. However, it also includes all the points between the endpoints.
The number of points in a line segment is comparable to the number of real numbers between 0 and 1.
Just as the number of real numbers between 0 and 1 is infinite, the number of points between the endpoints of a line segment is infinite.
(c.) The diagram shows that the line segment with endpoints A and B does not intersect the line segment with endpoints C and D.
Lines AB and CD extend beyond those points, and lines AB and CD appear to intersect at a point above the figure.
Parallel lines are coplanar lines (lines that lie on the same plane) that do not intersect. Hence, lines AB and CD are not parallel.
(d.) An angle is formed by two sides that share only an endpoint.
Because half-lines are rays without endpoints, they cannot have a common endpoint.
(e.) Extending the rays does not change the angle measure.
(f.) Please note that:
(1st.) Lines and line segments do not have a direction, and any two points uniquely determine both lines and line segments.
(2nd.) Lines do not have endpoints. Line segments do.
(g.) As a degree can be subdivided infinitely, there are an infinite number of points that lie on a circle that can be used for the second point of a ray.
(h.) Angle measurements do not relate to length.
Angles are measured by the amount of "opening" between their sides (the amount of turn between the sides) around their vertex.
(i.) Two distinct lines are parallel if they do not intersect and are in a single plane.
Lines that do not intersect and are not in a single plane are called skew lines.
(j.) Lines a and b could be parallel, but they do not have to be, because they could form different planes with the perpendicular line, and parallel lines must be coplanar.
(k.) A ray has only one endpoint.
(l.) $\overrightarrow{MN}$ ≠ $\overrightarrow{NM}$
This is because $\overrightarrow{MN}$ has an endpoint at point M and extends in the direction of point N while $\overrightarrow{NM}$ has an endpoint at point N and extends in the direction of point M.
(m.) The statement is correct.
Both lines $\overleftrightarrow{MN}$ and $\overleftrightarrow{NM}$ extend in the directions of point M and point N.
(n.) The statement is correct.
(o.) Skew lines are not coplanar.
They cannot be contained in a single plane.
(p.) If two distinct planes intersect, their intersection is a line.
Their intersection is not a line segment.
(q.) That is incorrect.
A line is different from a line segment in that a line has no endpoints (extends infinitely in both direcitions) while a line segment has two endpoints.
$ 1st\;\;Statement:\;\; AB \cong CD...Lines \\[3ex] 2nd\;\;Statement:\;\; \overline{AB} \cong \overline{CD}...Line\;\;Segments \\[3ex] $
(r.)
(s.)
(t.)
(u.)
(v.)
(w.)
(x.)
(y.)
(z.)
As a teacher, how would you respond?
(a.)
The line above is $\overleftrightarrow{ABC}$
(b.) A line segment has a finite number of points because it has two endpoints.
(c.) $\overline{AB}$ and $\overline{CD}$ are parallel because they do not intersect.
(d.) An angle is the intersection of two line segments (or the intersection of the endpoints of two rays).
Can we also define an angle as the union of two half-lines and a common point?
(e.) To make the measure of an angle greater, just extend the rays.
(f.) $\overline{AB}$ ≠ $\overline{BA}$ because:
$\overline{AB}$ starts at A and ends at B, and $\overline{BA}$ starts at B and ends at A.
(g.) There can only be 360 different rays emanating from a point since there are only 360 ° in a circle.
(h.) It is actually impossible to measure an angle, since each angle is the union of two rays that extend infinitely and therefore continue forever.
(i.) If any two planes that do not intersect are parallel, then any two lines that do not intersect should also be parallel.
(j.) If two lines, a and b, are perpendicular to the same line, lines a and b must be parallel.
(k.) A ray has two endpoints.
(l.) $\overrightarrow{MN}$ = $\overrightarrow{NM}$
(m.) For any points M and N, $\overleftrightarrow{MN}$ = $\overleftrightarrow{NM}$
(n.) A line segment contains an infinite number of points.
(o.) Skew lines are coplanar.
(p.) If two distinct planes intersect, their intersection is a line segment.
(q.) These two statements mean the same thing. I can write either way.
$ 1st\;\;Statement:\;\; AB \cong CD \\[3ex] 2nd\;\;Statement:\;\; \overline{AB} \cong \overline{CD} \\[3ex] $ (r.)
(s.)
(t.)
(u.)
(v.)
(w.)
(x.)
(y.)
(z.)
(a.) The line should be named as: $\overleftrightarrow{AC}$
Two points determine a line, so only two points are used to name the line.
(b.) A line segment has two endpoints. However, it also includes all the points between the endpoints.
The number of points in a line segment is comparable to the number of real numbers between 0 and 1.
Just as the number of real numbers between 0 and 1 is infinite, the number of points between the endpoints of a line segment is infinite.
(c.) The diagram shows that the line segment with endpoints A and B does not intersect the line segment with endpoints C and D.
Lines AB and CD extend beyond those points, and lines AB and CD appear to intersect at a point above the figure.
Parallel lines are coplanar lines (lines that lie on the same plane) that do not intersect. Hence, lines AB and CD are not parallel.
(d.) An angle is formed by two sides that share only an endpoint.
Because half-lines are rays without endpoints, they cannot have a common endpoint.
(e.) Extending the rays does not change the angle measure.
(f.) Please note that:
(1st.) Lines and line segments do not have a direction, and any two points uniquely determine both lines and line segments.
(2nd.) Lines do not have endpoints. Line segments do.
(g.) As a degree can be subdivided infinitely, there are an infinite number of points that lie on a circle that can be used for the second point of a ray.
(h.) Angle measurements do not relate to length.
Angles are measured by the amount of "opening" between their sides (the amount of turn between the sides) around their vertex.
(i.) Two distinct lines are parallel if they do not intersect and are in a single plane.
Lines that do not intersect and are not in a single plane are called skew lines.
(j.) Lines a and b could be parallel, but they do not have to be, because they could form different planes with the perpendicular line, and parallel lines must be coplanar.
(k.) A ray has only one endpoint.
(l.) $\overrightarrow{MN}$ ≠ $\overrightarrow{NM}$
This is because $\overrightarrow{MN}$ has an endpoint at point M and extends in the direction of point N while $\overrightarrow{NM}$ has an endpoint at point N and extends in the direction of point M.
(m.) The statement is correct.
Both lines $\overleftrightarrow{MN}$ and $\overleftrightarrow{NM}$ extend in the directions of point M and point N.
(n.) The statement is correct.
(o.) Skew lines are not coplanar.
They cannot be contained in a single plane.
(p.) If two distinct planes intersect, their intersection is a line.
Their intersection is not a line segment.
(q.) That is incorrect.
A line is different from a line segment in that a line has no endpoints (extends infinitely in both direcitions) while a line segment has two endpoints.
$ 1st\;\;Statement:\;\; AB \cong CD...Lines \\[3ex] 2nd\;\;Statement:\;\; \overline{AB} \cong \overline{CD}...Line\;\;Segments \\[3ex] $
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(w.)
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