origami is the japanese art of paper folding. the diagram below represents an unfolded paper kabuto, a samurai warriors helmet. from the kabuto below, which of the following are pairs of congruent angles?

check all that apply.

a. $\angle qrt$ and $\angle qru$
b. $\angle irf$ and $\angle mrn$
c. $\angle ont$ and $\angle mtn$
d. $\angle cru$ and $\angle irm$

Question

origami is the japanese art of paper folding. the diagram below represents an unfolded paper kabuto, a samurai warriors helmet. from the kabuto below, which of the following are pairs of congruent angles?

check all that apply.

a. $\angle qrt$ and $\angle qru$
b. $\angle irf$ and $\angle mrn$
c. $\angle ont$ and $\angle mtn$
d. $\angle cru$ and $\angle irm$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "direct_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Angle Types", "Origami Geometry" ], "new_concepts": [ "Symmetry in Origami Crease Patterns" ], "current_concepts": [ "Angle Types", "Origami Geometry", "Symmetry in Origami Crease Patterns" ] } ## Analyze the symmetry of the crease pattern Using the *Origami Geometry* knowledge point, we observe that the crease pattern of the unfolded paper kabuto has vertical symmetry along the line $FN$ and horizontal symmetry along the line $AK$. The central intersection point is $R$. The vertical line of symmetry passes through points $F, W, V, U, R, T, N$. The horizontal line of symmetry passes through points $A, Q, R, S, K$. ## Evaluate Option A: $\angle QRT$ and $\angle QRU$ Using the *Angle Types* knowledge point $$ \begin{aligned} &\text{Line } AK \text{ is perpendicular to line } FN \text{ at point } R.\ &\angle QRT \text{ is formed by ray } RQ \text{ and ray } RT \text{ (pointing straight down). Thus, } \angle QRT = 90^\circ.\ &\angle QRU \text{ is formed by ray } RQ \text{ and ray } RU \text{ (pointing straight up). Thus, } \angle QRU = 90^\circ.\ &\angle QRT \cong \angle QRU \text{ because both are right angles.} \end{aligned} $$ ## Evaluate Option B: $\angle IRF$ and $\angle MRN$ Using the *Angle Types* knowledge point $$ \begin{aligned} &\text{Line } IM \text{ and line } FN \text{ intersect at point } R.\ &\angle IRF \text{ and } \angle MRN \text{ are vertical angles formed by the intersection of lines } IM \text{ and } FN.\ &\angle IRF \cong \angle MRN \text{ because vertical angles are always congruent.} \end{aligned} $$ ## Evaluate Option C: $\angle ONT$ and $\angle MTN$ Using the *Angle Types* knowledge point $$ \begin{aligned} &\angle ONT \text{ is an angle in } \triangle ONT \text{ at vertex } N.\ &\angle MTN \text{ is a right angle } (90^\circ) \text{ formed by the perpendicular lines } OM \text{ and } FN \text{ at point } T.\ &\angle ONT \text{ is an acute angle in the right triangle } \triangle OTN \text{, so } \angle ONT < 90^\circ.\ &\angle ONT ot\cong \angle MTN. \end{aligned} $$ ## Evaluate Option D: $\angle CRU$ and $\angle IRM$ Using the *Angle Types* knowledge point $$ \begin{aligned} &\angle CRU \text{ is formed by ray } RC \text{ and ray } RU.\ &\angle IRM \text{ is formed by ray } RI \text{ and ray } RM.\ &\text{By vertical symmetry across line } FN\text{, ray } RC \text{ reflects to ray } RI\text{, and ray } RU \text{ reflects to itself. Thus, } \angle CRU \cong \angle IRU.\ &\text{Since ray } RM \text{ is the opposite ray of } RI\text{, } \angle IRM = 180^\circ \text{ (a straight angle).}\ &\angle CRU ot\cong \angle IRM. \end{aligned} $$

A. $\angle QRT$ and $\angle QRU$

B. $\angle IRF$ and $\angle MRN$

C. $\angle ONT$ and $\angle MTN$

D. $\angle CRU$ and $\angle IRM$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Angle Types" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the symmetry of the crease pattern

Evaluate Option A: \(\angle QRT\) and \(\angle QRU\)

Evaluate Option B: \(\angle IRF\) and \(\angle MRN\)

Evaluate Option C: \(\angle ONT\) and \(\angle MTN\)

Evaluate Option D: \(\angle CRU\) and \(\angle IRM\)

Answer: