Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [congruences-geometry]

For questions about congruent geometric figures, e.g., determining whether one figure is congruent to another.

Two geometric figures are called congruent if they have the same shape and size.

More formally, two sets of points are called congruent if they are equal up to an isometry.

134 questions
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What is the difference between congruency and equality?

What is the difference between equality and congruency? When should I say that two figures are congruent and when that they are equal?
dexterous
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Proving 2 triangles are congruent

Given $\Delta ABC, \Delta A'B'C'$ s.t $\widehat{BAC}=\widehat{B'A'C'}, BC=B'C', AD=A'D'$ $(AD, A'D'$ are internal bisectors of $\widehat{BAC}, \widehat{B'A'C'}$ respectively). Prove that $\Delta ABC=\Delta A'B'C'$. My attempt: Let $E\in\vec{AB},…
Harry
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The altitude of a triangle bisects a segment joining vertices of squares erected upon two sides of that triangle

We start with $\triangle ABC$ with $AD$ as its altitude. We then construct squares $\square ABEF$ and $\square ACGH$ outwards from $AB$ and $AC$. We then connect $F$ and $H$. $DA$ is extended so it intersects $FH$ at $M$. How do we prove that $FM$…
Eric Zhang
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Can a L-shaped figure be divided into 5 congruent shapes?

Given a square, remove one quarter of it. Can the resulting L-shaped figure be divided into 5 congruent shapes? If not, how can we prove that fact? I tried using circles, triangles and smaller squares, but could not find a solution. Is there a known…
ramana_k
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Prove ABC,A'B'C' are congruent:D is on BC,D' is on B'C', $\angle BAD \angle CAD= \angle B'A'D' \angle C'A'D', AB=A'B', AC=A'C', AD=A'D'$

In $\triangle ABC$ and $\triangle A'B'C'$, $D$ is a point on line segment $BC$ and $D'$ is a point on line segment $B'C'$. $\frac{\angle BAD}{\angle CAD}=\frac{\angle B'A'D'}{\angle C'A'D'}$, $AB=A'B'$, $AC=A'C'$ and $AD=A'D'$. How to prove that…
Dinshey
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How to cut an irregular shape into 2 congruent parts

Is it possible to cut this shape: into 2 congruent parts (equal area and shape). The guy who gave us this teaser said that it's possible. But i can't for the life of me figure out how. In the figure, $ABCD$ is a square, angle $CDE$ is a right…
kuke
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Geometry Problem on Congruent Triangles and Angle Chasing

Okay so the question is: In the diagram below, $\overline{\text{AD}} \cong \overline{\text{BC}}$ and $\alpha + \beta = 180^\circ$ (diagram not to scale). Find the measure of $\theta$. We also get this information. Hint: First extend…
Violet Fields
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Finding all sides and angles of a triangle

So SAS, SSS, ASA, AAS and RHS are reasons for congruent triangles, that means if a triangle, for example, have side lengths of 5, 6 and 8, then the triangle is unique. What I am trying to do is to find an expressions for other sides and angles in…
YesSpoon3
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Prove that $\angle AED = 2 \cdot \angle BEC$

Let $ABCDE$ be a pentagon such that $AE = ED$, $BC = DC + AB$ and $\angle BAE + \angle CDE = 180°$. Prove that $\angle AED = 2 \cdot \angle BEC$. So, by constructing it in Geogebra, I noticed that if I mark a point $F$ in $BC$ such that $CF = CD$…
I'm Ingrid
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$PQ ∥ BC$ for isosceles $\triangle ABC$ and inscribed equilateral $\triangle PQR$ with $R$ being midpoint of $BC$

Triangle $ABC$ is isosceles. An equilateral triangle $PQR$ is inscribed in it with $R$ being the midpoint of $BC$. How can you prove $PQ \parallel BC$?
Afsheen
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In $\Delta ABC,$ side $AC$ and the perpendicular bisector of $BC$ meet at $D$, where $BD$ bisects $\angle ABC$.

In $\Delta ABC,$ side $AC$ and the perpendicular bisector of $BC$ meet at $D$, where $BD$ bisects $\angle ABC$. If $CD = 7$ and $[\Delta ABD] = a\sqrt{5}$ , find $a$ . What I Tried: Here is a picture:- Let the perpendicular bisector of $BC$ pass…
Anonymous
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In $\Delta ABC$, $AB:AC = 4:3$ and $M$ is the midpoint of $BC$ . $E$ is a point on $AB$ and $F$ is a point on $AC$ such that $AE:AF = 2:1$

In $\Delta ABC$, $AB:AC = 4:3$ and $M$ is the midpoint of $BC$ . $E$ is a point on $AB$ and $F$ is a point on $AC$ such that $AE:AF = 2:1$. Also $EF$ and $AM$ intersect at $G$ with $GF = 36$ cm, $GE = x$ cm. Find $x$ . What I Tried: Here is a…
Anonymous
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Proving the ASA, SAS congruence of triangles in absolute geometry.

I am asked to prove the SAS (side-angle-side) and ASA(angle-side-angle) congruence of triangles in absolute geometry ( the geometry based in the axiom system of Euclides with the parallel postulate removed). How do I even prove two triangles are…
John Keeper
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Hexahedron congruent faces

Since I have an interest in polyhedra I've come across https://en.wikipedia.org/wiki/Trigonal_trapezohedron, especially the asymmetric one. So this made me wonder for a classification of convex hexahedron with congruent quadrilateral faces. Let $P$…
Simon Marynissen
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Prove two triangles are congruent

I have found a problem form internet and got stucked trying to proof or disproof it. It says: Given $AD=AE$, $BF=FC$, prove $\triangle ABE\cong\triangle ACD$ Update 1 The @Matrial's solution seems very promising however solving $FC=BF$ is killing…
SuperLucky
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