Questions tagged [congruences-geometry]

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

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Divide in two congruence piece [closed]

Divide the shape in two congruence piece (cut it into two identical shape). I saw this problem in a geometry book and spend about 4 hours on it but I couldn't find a solution, so any advise would be ...
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3answers
73 views

Congruence of angles axiom

In my geometry book, the following statement is provided as axiom of congruence. Let $ABC$ and $A'B'C'$ be two triangles. If $AB \equiv A'B'$, $AC \equiv A'C'$ and $ \angle BAC \equiv B'A'C'$ then $ \...
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2answers
51 views

When we have two line segments $AB$ and $CD$, what does $AB=CD$ mean?

Suppose that we have two line segments, AB and CD. We know that they have the same length. I know that $\overline{AB}=\overline{CD}$ means $AB$ is identical to CD (aka. They are the same lines), and ...
2
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3answers
93 views

Proving two triangles are congruent

The two circles shown are identical and pass through each other's centre. The line $AC$ passes through the centres of both circles. I would like to prove that triangles $ABD$ and $BCE$ are congruent ...
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1answer
83 views

Olympiad Geometry | Homothety

Let $C$ be a point on line segment $AB$, and construct the circles with diameters $AB$ and $AC$. Let $M$ be the midpoint of $BC,$ and let $MD$ be a tangent to the smaller circle (with $D$ on the ...
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1answer
55 views

Proving that DEF is also an equilateral triangle

The question gives that ABC is an equilateral triangle, AD = BE = CF and ∠DAB = ∠EBC. I tried using only cases of triangle congruences to prove it, but I'm not sure if I did it correctly. Thus, take ...
2
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2answers
46 views

How to solve this congruence of triangles geometrically?

Find x. Solution with trigonometry: First there are the missing angles ... Superior = $180 - 13 - 32 = 135$ Bottom = $180 - 13 - 24 = 143$ By the breast theorem we have to $$...
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2answers
60 views

Prove that ED=EF [closed]

In the diagram below $AD=DC$ and $AE=EB$ and both triangles $AEB$ and $ADC$ are right angel triangles and $M$ is the midpoint of $BC$ also $MD=MF$. Prove that $ED=EF$
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4answers
184 views

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
40 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
44 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
91 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
111 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
68 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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4answers
89 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
68 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
38 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
75 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
106 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
122 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
135 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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72 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
666 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 ...
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1answer
184 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. ...
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2answers
55 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
43 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
70 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
38 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
411 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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275 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
146 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
87 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
180 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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6answers
15k views

Prove the perpendicular bisector of chord passes through the centre of the circle

Hello, can someone please give me a simple proof to the following theorem: "The perpendicular bisector a chord passes through the centre of the circle." I have attached a diagram of what I mean and ...
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1answer
50 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 ...
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1answer
381 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
171 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
289 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
198 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
442 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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53 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
396 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
51 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
62 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
220 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
169 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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237 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
183 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
271 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 ...