Rigid Trandformations
What are the three types of rigid transformations that preserve distance and angle measure?
What are translation, rotation, and reflection?
The interior angles of any triangle always sum to this many degrees.
What is 180 degrees?
Two figures are congruent if there is a sequence of these transformations that makes them stack perfectly.
What are rigid transformations?
Name the three triangle congruence criteria that always work.
What are SSS, SAS, and ASA?
A statement that can be proved or disproved is called this.
What is a conjecture?
This transformation moves every point the same distance in the same direction.
What is a translation?
If a triangle has angles measuring 65° and 45°, what is the measure of the third angle?
What is 70°?
If △ABC ≅ △DEF, then ∠A corresponds to which angle?
What is ∠D?
If two triangles have three pairs of corresponding sides congruent, the triangles must be congruent by this criterion.
What is SSS (Side-Side-Side)?
Parts that are in the same position and orientation in congruent figures are called this.
What are corresponding parts?
When two lines intersect, a 180° rotation around the intersection point proves this angle relationship.
What are vertical angles are congruent?
Name two different strategies you could use to prove that the interior angles of a triangle sum to 180°.
What are: (1) construct a line parallel to one side through the opposite vertex and use alternate interior angles, or (2) make copies of the triangle using rigid transformations to arrange the three angles into a straight line?
List all six pairs of corresponding congruent parts for △JKL ≅ △MNP.
What are: ∠J ≅ ∠M, ∠K ≅ ∠N, ∠L ≅ ∠P, JK ≅ MN, KL ≅ NP, JL ≅ MP?
Explain why AAA (three pairs of congruent angles) is not sufficient to prove triangle congruence.
What is triangles can have the same three angle measures but different side lengths - they would be similar but not necessarily congruent?
Explain why precise definitions are important in geometry, using an example from the unit.
What is precise definitions help us communicate clearly and avoid confusion - for example, saying "∠N" is confusing when there are multiple angles at vertex N, but "∠QNP" is specific and clear?
400 - Describe how to use rigid transformations to prove that when parallel lines are cut by a transversal, corresponding angles are congruent.
What is translate one parallel line onto the other, which carries one corresponding angle onto the other, proving they are congruent because rigid transformations preserve angle measures?
Explain how the relationship between parallel lines and transversals helps prove the triangle angle sum theorem.
What is when you construct a line parallel to one side of a triangle through the opposite vertex, alternate interior angles are congruent, allowing you to rearrange the three interior angles of the triangle to form a straight angle (180°)?
Two triangles have all corresponding angles congruent. Explain whether this is sufficient to prove the triangles are congruent and justify your reasoning.
What is no, this is not sufficient - triangles can have all corresponding angles congruent but different side lengths (similar but not congruent triangles); you need corresponding sides to be congruent as well?
Two triangles have two pairs of corresponding sides congruent and one pair of corresponding angles congruent. Under what conditions are the triangles definitely congruent?
What is when the congruent angle is between the two congruent sides (SAS), or when the congruent angle is opposite one of the congruent sides in the correct configuration (AAS)?
Compare and contrast proving something is true versus verifying something is true. Use examples from triangle congruence.
What is verifying checks a specific case (like measuring angles in one triangle), while proving shows it's always true using general reasoning (like using rigid transformations to show any triangle's angles sum to 180°)?
Analyze why the sequence of transformations matters when proving congruence. Create an example where changing the order would affect the outcome.
What is the order determines which vertices correspond - for example, if △ABC ≅ △DEF through a specific sequence, changing the order could incorrectly match vertices, leading to false congruence statements like claiming ∠A ≅ ∠E instead of ∠A ≅ ∠D?
Critique this student's reasoning: "I proved the triangle angle sum by measuring angles in one specific triangle and showing they add to 180°." What's wrong with this approach and how would you improve it?
What is this only proves it for one specific case, not for all triangles - to prove it's always true, you need to use general reasoning (like variables or rigid transformations) that works for any triangle, not just measure one example?
Design a method to determine if two quadrilaterals are congruent using the minimum amount of information. Justify why your method works.
What is verify that all corresponding sides and angles are congruent (4 sides and 4 angles), or find a sequence of rigid transformations that maps one onto the other - this works because congruence is defined by rigid transformations preserving all distances and angle measures?
A student claims that two triangles are congruent because they have two congruent sides and two congruent angles. Analyze when this claim is valid and when it might be false. Provide examples.
What is this is valid when it's ASA or AAS configuration, but could be false in other arrangements - for example, if you have two sides and an angle not between them or not opposite to one of the given sides, you might be able to construct non-congruent triangles with these same measurements?
Evaluate the role of community in mathematical proof. How does the classroom community determine whether an argument is convincing?
What is proofs are created for and validated by a mathematical community - what counts as convincing depends on the shared standards and understanding of that community, and if community members aren't convinced, the argument should be revised to be clearer or stronger?