# Questions tagged [congruences-geometry]

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

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### polyhedrons congruent if faces are all congruent + same connection status?

In 2 dimension, a unit square and a unit rhombus(of certain angle) has the same list of edges and connection status. But the unit square and the unit rhombus are not generally congruent. Can we find ...
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### How to prove that angle bisector of right angle triangle ABC right angled at B is perpendicular bisector of third side AC. [closed]

I have tried using sine theoram, angle bisector theoram, congruency of type RHS,AA,ASA but haven't been able to do this.
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### Congruent triangles that do not use reflections

This is a simple question on terminology in geometry: Suppose $ABC$ and $DEF$ are two triangles that are congruent, but the transformations from one to the other are based solely only on translation ...
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### Sources for IMO Geometry Theorems

The International Mathematical Olympiad contains highly non-standard problems involving the geometry of circles, triangles and lines which usually draws upon quite a few theorems and facts not ...
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it's the first time I'm posting so please tell me if I get anything wrong in formatting, information etc. below. We are learning about congruence rules in class (Y10 Edexcel GCSE Maths course) and we ...
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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 $PQ \parallel BC$?
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### How to proof the mid-point theorem without triangle congruence theorems?

Is there a way for me to proof the mid-point theorem without using the triangle congruence theorems? I can't seem to find one that's not using the triangle congruence theorems. The reason I'm trying ...
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### Is triangle congruence considered as theorems or postulates?

Is triangle congruence like $SSS$, $SAS$, $ASA$ and $AAS$ considered as theorems or postulates? I've seen some people calling it as theorems and some other people calling it as postulates. What I know ...
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### Does there exist an efficient way to determine if two $n$-simplices in $\mathbb{R}^n$ are congruent?

Given two $n$-simplices in $\mathbb{R}^n$, does there exist an algorithm to determine if they are congruent which has a time complexity of the form $O(n^c)$ where $c$ is a constant? I'm assuming that ...
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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 $B'C'$. $\frac{\angle BAD}{\angle CAD}=\frac{\angle B'A'D'}{\angle C'A'D'}$, $AB=A'B'$...
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### Triangle and Median related question

Let BD be a median in triangle ABC. The points E and F divide the median BD in three equal parts, such that BE = EF = FD. If AB = 1 and AF = AD, find the length of the line segment CE. I have tried ...
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### Find the measure of $\angle G$ in a triangle

In the following figure, $\Delta$ADB, $\Delta$PCB and $\Delta$EFG are right triangles. $PB=AE$,$AC=CB$. Question: What is the value of $\angle G$? I figured out $\angle A = \angle CPB$ , $EF=AC$. ...
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### Geometry find the angle

Good day, excuse me, I am looking at congruence themes in Geometry, and I am trying to solve this problem, but I am not sure how to proceed. The coloured numbers is what I have so far (although now ...
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### Similarity by SSA (side-side-angle) in obtuse triangles

Source: Challenge and Thrills of Pre-College Mathematics, Page 74, Problem 54: "In two obtuse triangles, an acute angle of the one is equal to an angle of the other, and sides about the other ...
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### Finding side lengths of a trapezoid given the distance between its diagonal intersection and midpoint of a diagonal

The Question My friend recently gave me a problem that I was interested in, but could not completely solve. I've already constructed a diagram that roughly represents the problem, and it's below. ...
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### Let $ABCD$ be a square. Points $A,B,G$ are collinear. $AC$ and $DG$ meet at $E$ , and $DG$ and $BC$ meet at $F$.

Let $ABCD$ be a square. Points $A,B,G$ are collinear. $AC$ and $DG$ meet at $E$ , and $DG$ and $BC$ meet at $F$. If $DE = 15$ cm, $EF = 9$ cm, find $FG$. What I Tried: Here is a picture :- I have ...
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### In the diagram, $AB$ $||$ $EF$ $||$ $DC$ . Given that $AC + BD = 250$ , $BC = 100$ and $EC + ED = 150$, find $CF$.

In the diagram, $AB$ $||$ $EF$ $||$ $DC$ . Given that $AC + BD = 250$, $BC = 100$ and $EC + ED = 150$, find $CF$. What I Tried: Here is the diagram :- I have assigned variables for the different ...
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### $\Delta ABC$, $AC = 2BC$ and $\angle C = 90^\circ$ and $D$ is the foot of the altitude from $C$ onto $AB$. Find $AE : EC$ .

$\Delta ABC$, $AC = 2BC$ and $\angle C = 90^\circ$ and $D$ is the foot of the altitude from $C$ onto $AB$. A circle with diameter $AD$ intersects the segment $AC$ at $E$. Find $AE : EC$. What I Tried:...
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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:...
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### In $\Delta ABC$, $AC = BC$ and $\angle C = 120^\circ$. $M$ is on side $AC$ and $N$ is on side $BC$ .

In $\Delta ABC$, $AC = BC$ and $\angle C = 120^\circ$. $M$ is on side $AC$ and $N$ is on side $BC$ such that $\angle BAN = 50^\circ$ and $\angle ABM = 60^\circ$. Find $\angle NMB$ . What I Tried: ...
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### In right $\Delta ABC$, $\angle C = 90^\circ$. $E$ is on $BC$ such that $AC = BE$. $D$ is on $AB$ such that $DE \perp BC$ .

In right $\Delta ABC$, $\angle C = 90^\circ$. $E$ is on $BC$ such that $AC = BE$. $D$ is on $AB$ such that $DE \perp BC$ . Given that $DE + BC = 1$ and $BD = \frac{1}{2}$, find $\angle B$. What I ...
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### In an isosceles right $\Delta ABC$, $\angle B = 90^\circ$. AD is the median on BC. Let $AB = BC = a$.

In an isosceles right $\Delta ABC$, $\angle B = 90^\circ$. AD is the median on BC. Let $AB = BC = a$. If $BE \perp AD$, intersecting $AC$ at $E$, and $EF \perp BC$ at $F$, find $EF$ in terms of $a$. ...
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### $ABCD$ is a rectangle of area $210$ cm$^2$. $L$ is a mid-point of $CD$ . $P,Q$ trisect $AB$ . $AC$ cuts $LP,LQ$ at $M,N$ respectively.

$ABCD$ is a rectangle of area $210$ cm$^2$. $L$ is a mid-point of $CD$ . $P,Q$ trisect $AB$ . $AC$ cuts $LP,LQ$ at $M,N$ respectively. Find $[\Delta LMN]$ What I Tried: For a geometry Problem, it's ...
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### $ABCD$ is a square. $E$ is the midpoint of $CB$, $AF$ is drawn perpendicular to $DE$. If the side of the square is $2016$ cm , find $BF$.

$ABCD$ is a square. $E$ is the midpoint of $CB$, $AF$ is drawn perpendicular to $DE$. If the side of the square is $2016$ cm , find $BF$. What I Tried: Here is a picture, I used a really peculiar ...
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### is still there a contradiction in the proof by contradiction of the proposition 6 of Elements using this triangle?

The proposition 6 says: If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another. And the proof of Euclid in his Elements basically is (for ...
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### Is the criterion SAA of congruent triangles valid?

I think the fundamental criterions for triangles congruence are: SAS (Side-Angle-Side) ASA (Angle-Side-Angle) SSS (Side-Side-Side) But some proofs like this one: https://www.youtube.com/watch?v=...
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### Does congruent triangles apply to this question?

Two identical rods $BA$ and $CA$ are hinged at $A$. When $BC = 8\ \textrm{cm}$, $\angle BAC = 30^\circ$ and when $BC = 4\ \textrm{cm}$, $\angle BAC = \alpha$. Show that \cos\alpha = \frac{6+\sqrt 3}{...
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### Area of a rectangle using congruency

I don't understand the lecturer, solving the question, says that the rectangles are congruent each other, so the result can be obtained by proportioning them to each other. However, AFAIK two ...
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### Problem on an isosceles right triangle, involving similarity and congruence

Given that $ABC$ is an isosceles right angled triangle with angle $\widehat{ACB}=90$ degrees. $D$ is the midpoint of $BC$, $CE$ is perpendicular to $AD$, intersecting $AB$ and $AD$ at $E$ and $F$ ...
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### Congruency and parallelism in a triangle

For the following example, one of the solutions says $\frac{6-x}{6} = \frac{x}{18}$ if one of the side of the square is called as $x$. I don't understand how the proportion and the equality are ...
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### minimum number of congruent rectangles

consider a 6*6 square which is dissected into 9 rectangles by lines parallel to its sides such that all the rectangles have integral sides.the question is - what is the minimum number of congruent ...
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### 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 ...
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### 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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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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### 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 a ...