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Questions tagged [congruences-geometry]

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

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Find the length x such that the two distances in the triangle are the same

I have been working on the following problem Statement Assume you have a right angle triangle $\Delta ABC$ with cateti $a$, $b$ and hypotenuse $c = \sqrt{a^2 + b^2}$. Find or construct a point $D$ ...
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1answer
34 views

If ratio of sides of two triangles is constant then the triangles have the same angles

If $\triangle ABC$ and $\triangle A'B'C'$ are a pair of triangles such that $$\dfrac{|AB|}{|A'B'|}=\dfrac{|BC|}{|B'C'|}=\dfrac{|AC|}{|A'C'|}$$ then $$\triangle ABC \sim \triangle A'B'C'$$ I have ...
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1answer
42 views

Can anyone help me to find what is the similarity between these two triangles Just check the picture)? [closed]

Can anybody show me how does the red triangle (bigger triangle) is similar to the green triangle (smaller triangle). I need to know how they are similar so that I can do a proportional between them.
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3answers
81 views

Compute $m ( \angle ACD $).

Let $\triangle ABC $ s.t $m (\angle A)=100^{°}, m (\angle B)=20^{°} $. Let $D\in Int (\triangle ABC) $ s.t. $m (\angle BAD)=30^{°} $ and $[BD $ is the bisector of $\angle B $. Compute $m ( \angle ACD ...
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0answers
33 views

Spherical triangles and congruence criteria

I read from different sources that the usual criteria of congruence of triangles work for spherical triangles. However it seems to me that there is a counterexample. Consider two poles on the sphere A ...
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1answer
63 views

In $\triangle ABC$, $D$ is an exterior point such that $AC = CD$ and $CE$ is parallel to $AF$. Find the area of $ABDF$.

In $\triangle ABC$, $CB$ is extended upto $D$ so that $AC$ = $CD$. An angle $\angle DCE$ is drawn at point $C$ so that is equal to $\angle CAB$ and $AB$ meets $CE$ at $I$.$E$ is such an external point ...
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3answers
35 views

Can we find two non-congruent right triangles with whole-number lengths and congruent hypotenuses?

I know some ways to find some Pythagorean triples. And I understand that if $a^2 + b^2 = c^2$ then $(a-b)^2 + (a+b)^2 = 2c^2$. I feel like that suggests a way forward, but I cannot find that way. ...
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2answers
49 views

What is the length of $EF$ in the following diagram?

In triangle $\triangle ABC$, angle $A=50^\circ$ , angle $C=65^\circ$ . Point $F$ is on $AC$ such that, $BF$ is perpendicular to $A$C. $D$ is a point on $BF$ (extended) such that $AD=AB$. E is a ...
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1answer
34 views

Proof that the circumcenters of sub triangles forms a triangle congruent with the original triangle

Let $\triangle ABC$ be a triangle with orthocenter H and let $O_A, O_B, O_C$ be the circumcenters of triangles $\triangle BCH, \triangle CAH, \triangle ABH$, respectively. Prove that the $\triangle ...
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3answers
68 views

Why would two triangles with exact same lengths have the exact same 3 angles? [duplicate]

I know how incredibly stupid this question sounds, and I know any 2 triangles you draw with same line lengths will have the same 3 angles, but I just can't figure out the exact reason why having 3 ...
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1answer
94 views

Congruent Triangles.

In the diagram, CE=CF=EF, EA=BF=2AB, and PA=QB=PC=QC=PD=QD=1, Determine BD. Triangle APD is congruent to Triangle BQD ... So, the areas must be equal too... Please advise.
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2answers
97 views

Learning About Isometry From Serge Lang's Basic Math; Very Confused

I'm trying to self learn Serge Lang's "Basic Mathematics". I'm currently in the "Isometries" section of the book, but I'm extremely confused, by the notations and the concepts Lang uses. Lang first ...
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1answer
123 views

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 ...
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0answers
65 views

How I can solve this conjecture?

I have tried to make this demonstration, but I have not come up with a concrete idea. If someone can give me an idea of how to perform this demonstration, I would appreciate it very much. Study and ...
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1answer
379 views

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 ...
2
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1answer
135 views

GCSE similar/congruent triangles

looking for some advice with my GCSE revision in tackling the following please: Q10 b and Q11: For Question 10, I so far have that EF = 28.8cm and BC = 48cm, using properties for similar triangles. ...
2
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2answers
53 views

Calculate length of one side given all other sides in a triangle.

All the lengths in red had been given in the problem statement. The task is to find $BC$. The green ones were calculated from the information in red. Here's two ways I tried to solve it (and I would ...
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1answer
42 views

Can someone explain how these angles are equivalent?

Linked here is a great tutorial I've been reading on contravariant and covariant representations of vectors. I'm following along well up until page 5 and 6. Here, I'd like to know why in Figure 4.7(a),...
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1answer
63 views

Prove that $2AK=BP+PC$ in isosceles triangle.

Let $ABC$ is isosceles triangle. $AB=AC$. Point $P$ such that $\angle BPC=2\angle BAC$. $PK$ is bisector of $\angle BPD$ and $AK \perp PK$. Prove that $$2AK=BP+PC$$ My attempts: Let point $A'$ and $...
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1answer
36 views

congruence of two triangles

if two corresponding sides of two triangles are equal and their medians drawn on the third sides are also equal, prove two triangles are congruent. I tried to solve it by extending the median by its ...
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1answer
298 views

Find length of chord between two tangents and distance of chord from origin

I am trying to solve 2 questions Is there a way to find an equation for the length of a chord between two tangents in terms of the radius and distance of that chord to the external point? Is there a ...
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0answers
243 views

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 ...
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2answers
140 views

Leg bisectors in a trapezoid and angle equality

Let $PQRS$ be a trapezoid. The bisectors of legs $PS$ and $QR$ intersect the opposite legs at $M$ and $N$ so that triangles $PMS$ and $QNR$ are formed. Prove that $\measuredangle PMS= \measuredangle ...
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2answers
78 views

How does congruence really work?

I was doing the following problem An isoceles triangle is a triangle in which two sides are equal. Prove that the angles opposite to the equal sides are equal. I drew this diagram (sorry for the ...
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2answers
121 views

How to find $ \alpha + \beta $ here

In the below figure, ABGH, BCFG, CDEF are squares with same length. I need to find $ \alpha + \beta $ I spend an hour with this and found that answer is 90° but according to answer key, it's 45°. ...
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1answer
46 views

How to prove $\measuredangle{BCA}=\measuredangle{EAF}$?

$ABC$ and $BCD$ are triangles, $AB\perp AC$, $\measuredangle{ACB}=\measuredangle{ACD}$, $2|BE|=5|DF|$, $|BC|=20$, $|CD|=5$, $|AE|=x$ is given. Find $x$. Here is a diagram for it: First, i assigned ...
2
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1answer
225 views

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 ...
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1answer
148 views

Prove congruence of pentagons with all right angles.

Prove that two pentagons with all right angles are congruent if two opposite sides are congruent. I have done the proof when two adjacent sides are congruent, but with this I have a problem. I ...
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1answer
129 views

||gms OAEB, OBFC, OCGD, ODHA are complated from any point O in ||gm ABCD. Show EFGH is a ||gm

ABCD is a parallelogram and O is any point. The parallelograms OAEB, OBFC, OCGD, ODHA are completed, Prove that EFGH is a parallelogram. How can this be proved using the concept of congruence of ...
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1answer
179 views

Confusion over SSA axiom for congruency

I was browsing through KHAN Academy videos when I met the one which Explained why SSA is not a Congruency postutate. But I had this Diagram in my Mind(Different from the video) Click Here to see ...
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1answer
426 views

Relationship between Perimeter of 2 Similar Triangle

Is there any proof for the relation between perimeter of 2 similar triangles, like there is one for the area ? Pls help and tell if there is any proof ?
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0answers
51 views

Partitioning of square lattice into equivalence classes under congruence relation

Let $m,n$ be positive integers. Consider the set $S^{\small(m)}_n=([1,n]\cap\mathbb N)^m$ having $n^m$ points in the Euclidean space $\mathbb R^m$ arranged in a square (cubic, etc) lattice. Two ...
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2answers
248 views

proving that triangles $ABC$, $A'B'C'$ are congruent

Given $AD$ is a median to $BC$ in triangle $ABC$, and $A'D'$ is a median to $B'C'$ in triangle $A'B'C'$, and $AD=A'D', AC=A'C', AB=A'B'$. How can i prove that triangles $ABC$, $A'B'C'$ are congruent? ...
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1answer
50 views

Solving for length of an unknown side of a triangle.

I have been given the figure below: Figure (click me). I know that $AD=20-x$ and $m\angle ACD=m\angle BCD$. How can I set up a ratio also knowing that $AC=11$ and $BC=14$ in order to find $x$? ...
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3answers
54 views

Find the measure of a side and an angle.

In the figure, $BG=10$, $AG=13$, $DC=12$, and $m\angle DBC=39^\circ$. Given that $AB=BC$, find $AD$ and $m\angle ABC$. Here is the figure: I am inclined to say that since $\overline{AB}\simeq \...
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3answers
212 views

If two of these segments have lengths 8 and 6, what are all possible lengths of the third segment?

Please help me for this question, I can't fully understand the problem and not sure where and how to start. In a plane, two congruent squares share a common vertex but have no other points in common ...
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1answer
1k views

How to prove the similarity of two rectangles?

The task is this: On the ABCD rectangles AB,BC,CD,DA sides we took up P,Q,R,S points, so PR and QS are perpendicular to each other. Let's prove that the middle points of SP,PQ,QR,RS segments form a ...
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1answer
118 views

Congruence of triangles: SSA criteria

It is well known that this criteria does not work in general. I am trying to answer to the following question if two triangles have two sides and the angle NOT between them equal, they are either ...
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0answers
213 views

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 ...
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1answer
166 views

AAS Congruence Included Side?

When proving congruence between two triangles, AAS is a common method. However, like that for SAS the angle must be included, is there any restriction on where the side must be for AAS to work? For ...
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2answers
237 views

Proving area of a square inside of two squares in Euclidean Geometry

Let ABCD and PQRS be squares of the same side length such that P is the center of ABCD (i.e., the intersection of its diagonals). Suppose that BC and PQ intersect each other at X. Suppose also that ...
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1answer
43 views

Every finite group of congruences of the $n$-dimensional Euclidean space has a fixed point

Is there an "absolute" proof of the fact that if $G$ is a finite group of congruences of the $n$-dimensional Euclidean space, then $G$ has a fixed point?
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1answer
56 views

Proving angles in the same corner equal

Suppose we have two line segments, AB and CD, which cross at point X. Now suppose there is an arbitrary point Y somewhere on the segment AX (that is, points A, Y and X are collinear). What is the ...
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3answers
196 views

Disproving $A-S-S(Angle-Side-Side)$ congruence condition…

I know that $A-S-S$$(Angle-Side-Side)$ congruence does not exist.But I cannot disprove it. Every time I draw a figure,I get two congruent triangles. My Attempt- So,we draw two lines such that $AB=...
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3answers
1k views

To find ratio of Length and Breadth of a Rectangle [closed]

Given a rectangular paper sheet. The diagonal vertices of the sheet are brought together and folded so that a line (mark) is formed on the sheet. If this mark length is same as the length of the sheet,...
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1answer
428 views

Without using angle measure, how do I prove that vertical angles are congruent?

Assume that X is a point between A and C, that X is also between B and D, and that these points are not all collinear. Then the angles AXB and CXD are called vertical angles. Prove that vertical ...
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2answers
350 views

Interesting problem in congruence of triangles

While solving the exercises of my book I came across this interesting problem: $\triangle ABC$ is isosceles triangle with $AB=AC$. D is a point on base BC such that $AD$ perpendicular on $BC$. To ...
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2answers
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Diagonal line through rectangle always creates two congruent triangles (?)

1) Is it true that the diagonal line through a rectangle always creates to congruent triangles? 2) If a quadrilateral has two right angles that are opposite (is this the right word to use), as shown,...
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3answers
196 views

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 ...
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4answers
49k views

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?