congruence, theorems, and proofs

Congruence, Theorems, and Proofs

By Claire Latsha

​Congruence is keeping shape and angle measurement in transformations or two shapes being congruent. Transformations such as reflections, rotations, and translations preserve congruence. There are series of transformations you can take to map figures onto each other. This image shows two shapes that are the same size and if you were to map B onto A it would prove congruence.

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​Congruence

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•​There are also theorems you can use to prove congruence(These also play into theorems and proofs). The ones I am going over are Side-Angle-Side, Side-Side-Side, Angle-Angle-Side, and Angle-Side-Angle.

•​If two sides and an angle of a triangle have the same as another triangle they are congruent by SAS.

•​If three sides of a triangle are the same as another triangle they are congruent by SSS.

•​​If two angles and a non-included side of one triangle are the same as another triangle, the triangles are congruent by Angle-Angle-Side.

•​If two angles and the included side of a triangle are the same as another triangle they are congruent by ASA. For example, this picture here is showing two triangles congruent by ASA.

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​Congruence Continued

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Multiple Choice

What are these two triangles congruent by?

• ASA

• SSS

• SAS

• SSA

Multiple Choice

Which of these transformations does NOT preserve congruence?

• Translations

• Reflections

• Rotations

• Dilations

​Theorems deal with lines, angles, triangles, quadrilaterals, e.t.c. and they go into solving proofs. Certain definitions and postulates can also go along with some theorems. There are rules and things associated with lines, angles, triangles, and quadrilaterals. Listed below will be 3 rules for lines, angles, triangles, and quadrilaterals.

Lines:

•​​The intersection of two lines makes a point.

•​​Lines that intersect at a right angle are perpendicular.

•​​Two lines in a plane that never intersect are parallel.​

​Angles:​

•​Complementary angles are two angles that add up to 90 degrees.

​•​Supplementary angles are two angles that add up to 180 degrees.

•​​The alternate interior angles theorem says that if two lines cut by a transversal are parallel, then alternate interior angles are congruent.

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​Theorems

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​Triangles:​

•​The Pythagorean theorem says that for right triangles, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

​•​The triangle sum theorem says that the interior angles of a triangle add up to 180 degrees.

•​​The alternate interior angles theorem says that if two lines cut by a transversal are parallel, then alternate interior angles are congruent.

Quadrilaterals:

​•​​If both pairs of opposite sides of a quadrilateral are congruent, that quadrilateral is a parallelogram.

​•The diagonals of a parallelogram bisect each other.​

​•If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.​

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​Theorems Continued

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Multiple Choice

What are two angles that add up to 90 degrees called?

• Complementary

• Supplementary

• Congruent

• Perpendicular

Multiple Choice

Which one of these is NOT a characteristic of a parallelogram?

• The diagonals bisect

• Opposite angles are congruent

• Opposite sides are congruent

• The sum of the interior angles is 180 degrees

Multiple Select

WHO IS THE MOST SLAYFUL QUEEN???💅💅💅💅💅💅

When solving problems, justifying statements, or completing proofs, remember the definitions, postulates, and theorems that may apply. Look at this proof on the right, line 5 of the proof is missing. Here are your answer choices:

A. m∠1 = m∠7; Transitive property of equality

B​. m∠7 = m ∠3; Definition of complementary angles

C​. m∠1 = m ∠7; Definition of complementary angles

D​. m∠7 = m∠3; Transitive property of equality

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It is already given that angles 1 and 7 are supplementary and angles 1 and 3 are a linear pair. Line 5 of the proof goes into proving lines m and n are parallel. Line 4 uses the reason subtraction property of equality, and line 6 uses the reason definition of congruent angles. You need an equality relationship for line 5, so answers B and C are ruled out. As you see line 4 says m∠7 = 180° - m∠1 and m∠3 = 180° - m∠1. This shows they are both equal to 180° - m∠1. Therefore the answer must be D as there is an equality relationship between m∠7 and m∠3​.

Proofs​

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​Complete the missing part of this proof. Here are your answer choices:

A. ​SSS congruence

B. AAS congruence

C. ​SAS congruence

​D. SSA congruence

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​You need to prove ABD and CBD are congruent to help prove ABC is isosceles. You can do this using one of the congruence theorems listed above. It is shown angles A and C are congruent. It has also been proven the right angles are congruent. △ABD and △CBD both have two congruent angles. Line BD subtends the congruent angles. If you have two angles and a side it proves △ABD and △CBD are congruent by AAS(Angle-Angle-Side). So the answer that goes in the missing blank is B. AAS congruence.

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​Proofs Continued

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Multiple Choice

What goes in the missing blank?

• Alternate exterior angles are congruent

• Vertical angles are congruent

• Reflexive property of e

• Alternate interior angles are congruent

Multiple Choice

What goes in the missing blank?

• Given

• Congruent angles

• Transitive property

• Reflexive property

Open Ended

What do you think best fits in the missing blank?

Poll

How did you like my quiz?(Be serious in answering).

• It was slaying in every category

• It was good, but I have advice to improve it

• It wasn't the best, but I can give you tips to make it better

• I didn't really like it and I can point out things to help you improve