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Quiz about Prove it Mathematically of course

Prove it! Mathematically, of course. Quiz

A quick quiz on how the professionals know something is true. Know your proofs!

A multiple-choice quiz by Mercenary_Elk. Estimated time: 6 mins.

Author
Mercenary_Elk

Time
6 mins

Type
Multiple Choice

Quiz #
306,817

Updated
Dec 03 21

# Qns
10

Difficulty
Difficult

Avg Score
5 / 10

Plays
540

One at a Time - Single Page - Timed Game

Question 1 of 10

1. When you make a conclusion based upon a pattern found in a series of examples, what is that called? Hint

Logical reasoning

Deductive reasoning

Inductive reasoning

Converse reasoning

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Question 2 of 10

2. When you prove a theorem by proving that the opposite of the theorem contradicts known facts, what is that called? Hint

Direct proof

Converse proof

Deductive proof

Indirect proof

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Question 3 of 10

3. Given a theorem in "if A, then B" format, what would the converse be?
(A and B are conditions in the theorem.) Hint

if B, then not A

if B, then A

if A, then not B

if A, then B

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Question 4 of 10

4. What is the format used for a proved theorem with a proved converse?
(A and B are conditions of the theorem.) Hint

if B, then only A

if A, then B and only B

A, if and only if B

if A, then only B

5. What form of proof meets these requirements?
-A form of proof for problems involving only natural numbers
-To prove, you must prove it true for n=1
-To prove, you must also prove it true for n=k+1, if n=k is true Hint

Scientific Induction

Scientific Deduction

Mathematical Deduction

Mathematical Induction

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Question 6 of 10

6. The rest of the questions are all about certain proofs! YAY!
What theorem is this?
The tangent segments from an external point to a circle are equal in length. Hint

Equal Tangents Theorem

External Point Theorem

External Tangents Theorem

Equal Point-Tangents Theorem

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Question 7 of 10

7. What theorem is this?
"Suppose one side of a triangle is extended. Then the exterior angle formed is equal to the sum of the two interior and opposite angles." Hint

Exterior Angle Theorem

Interior Triangle Theorem

Double Interior Angle Theorem

Exterior Triangle Theorem

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Question 8 of 10

8. What theorem is this?
Given a semicircle with diameter AB, the angle APB will be 90 degrees for any point on the semicircle P. Hint

Semicircle Theorem

Angle Semicircle Theorem

Right Semicircle Theorem

Right Angle Semicircle Theorem

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Question 9 of 10

9. What theorem is this?
The line segment joining the midpoints of two sides of a triangle is parallel to the third side and one-half as long as the third side. Hint

Semitriangle Theorem

Triangle Bisection Theorem

Midtriangle Theorem

Side-Splitting Theorem

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Question 10 of 10

10. Hard one!
Let a and d be any two natural numbers with no common factor. Then the infinite arithmetic sequence a, a+d, a+2d, a+3d...contains infinitely many prime numbers. Hint

Euclid's Proof

Dirichlet's Theorem

Lejeune's Theorem

Chab's Last Proof

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Quiz Answer Key and Fun Facts

1. When you make a conclusion based upon a pattern found in a series of examples, what is that called?

Answer: Inductive reasoning

Inductive reasoning is generally not accepted as a form of proof; it is usually used to find an original conclusion to further prove. Many ideas that have come out of inductive reasoning have been proved incorrect by counter-examples.

2. When you prove a theorem by proving that the opposite of the theorem contradicts known facts, what is that called?

Answer: Indirect proof

Most indirect proofs follow a set template, because it works. Assume 'true' is what the theorem is attempting to prove and 'false' is the opposite.

Either 'true' or 'false'
Assume 'false'
-->Proof that 'false' contradicts known axioms
Therefore, the assumption of 'false' was incorrect.
Hence, 'true' is the only correct answer.

3. Given a theorem in "if A, then B" format, what would the converse be? (A and B are conditions in the theorem.)

Answer: if B, then A

The converse is just the theorem written the other way around. It is possible for theorems to be true despite false converses and for theorems to be false, despite true converses.

One example is:
All rectangles are parallelograms. --> True
All parallelograms are rectangles. --> False

4. What is the format used for a proved theorem with a proved converse? (A and B are conditions of the theorem.)

Answer: A, if and only if B

This is called a combined form. Given the format, it could also have been written 'B, if and only if A'. One example is the corresponding-angles theorem:

A transversal intersects two lines 'a' and 'b'.
The lines are parallel if and only if the corresponding angles are equal.

5. What form of proof meets these requirements? -A form of proof for problems involving only natural numbers -To prove, you must prove it true for n=1 -To prove, you must also prove it true for n=k+1, if n=k is true

Answer: Mathematical Induction

No relation to inductive reasoning. Do not make that mistake.
This is often used for sequential proofs like:
1+3+5+7+...+(2n-1)=n^2 for all natural numbers n
Or it can be used for division proofs like:
(n^3+2n) MOD 3 = 0 for all natural numbers n

This is my personal favourite method of proof.

6. The rest of the questions are all about certain proofs! YAY! What theorem is this? The tangent segments from an external point to a circle are equal in length.

Answer: Equal Tangents Theorem

This seems kind of like common sense, but a lot of mathematical theorems do.
The problem is that it is hard to prove common sense.

7. What theorem is this? "Suppose one side of a triangle is extended. Then the exterior angle formed is equal to the sum of the two interior and opposite angles."

Answer: Exterior Angle Theorem

This can be proven by or used to prove the Angle Sum Theorem. Since the triangle's angles will always add up to 180 and the exterior and interior angle will always add up to 180, we can determine that the exterior angle is equal to the other two interior angles.

8. What theorem is this? Given a semicircle with diameter AB, the angle APB will be 90 degrees for any point on the semicircle P.

Answer: Semicircle Theorem

Using the formula x^2+y^2=r^2, it is possible to make an easy coordinate proof by proving that the slopes of AP and BP multiply to -1.

9. What theorem is this? The line segment joining the midpoints of two sides of a triangle is parallel to the third side and one-half as long as the third side.

Answer: Side-Splitting Theorem

You don't know hard it was for me not to write a comment like "laughably easy" or something.

10. Hard one! Let a and d be any two natural numbers with no common factor. Then the infinite arithmetic sequence a, a+d, a+2d, a+3d...contains infinitely many prime numbers.

Answer: Dirichlet's Theorem

Lejeune is Dirichlet's first name.
Euclid's proof is quite different from this. Euclid's proof simply shows that there are an infinite number of primes. That's all.
I have no idea who Chab is; I'm fairly certain I just made it up.

Good job finishing the quiz!

Source: Author Mercenary_Elk

This quiz was reviewed by FunTrivia editor crisw before going online.
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