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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Quadrilateral congruence through SSSD?

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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Let $O$ be the centre of the circumcircle of $\Delta ABC$, $P$ and $Q$ be the midpoint of $AO$ and $BC$, respectively.

Let $O$ be the centre of the circumcircle of $\Delta ABC$, $P$ and $Q$ be the midpoint of $AO$ and $BC$, respectively. Suppose $\angle CBA = 4\angle OPQ$ and $\angle ACB = 6\angle OPQ$ . FiNd $\angle ...
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Planar convex sets whose self-intersections are similar to themselves

Let $S\subset \mathbb{R}^2$ be a bounded convex set. For which $S$ can we take $S'$ congruent to $S$ (i.e., the image of $S$ under an isometry of the Euclidean plane) such that $S\cap S'$ is similar ...
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In $\Delta ABC$, $AB = 14, BC = 16, AC = 26$. $M$ is the midpoint of $BC$ and $D$ is the point on $BC$ such that $AD$ bisects $\angle BAC$.

In $\Delta ABC$, $AB = 14, BC = 16, AC = 26$. $M$ is the midpoint of $BC$ and $D$ is the point on $BC$ such that $AD$ bisects $\angle BAC$. Let $P$ be the foot of the perpendicular from $B$ to $AD$ . ...
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In $\triangle ABC, AB = 28, BC = 21$ and $CA = 14$. Points $D$ and $E$ are on $AB$ with $AD = 7$ and $\angle ACD = \angle BCE$

In $\triangle ABC, AB = 28, BC = 21$ and $CA = 14$. Points $D$ and $E$ are on $AB$ with $AD = 7$ and $\angle ACD = \angle BCE$. Find $BE$. What I Tried: Here is a picture :- I know the side-lengths ...
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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 = ...
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How to prove that a triangle is uniquely determined by an angle, its opposite side and its perpendicular height.

I am trying to solve a homework problem and as part of this problem, I need to show that a certain situation is impossible. I have the following situation. Given some line, I have $2$ points $H$ and $...
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When extending the sides $AB,BC,CA$ of $\Delta ABC$ to $B',C',A'$ respectively, such that $AB' = 2AB$ , $CC' = 2BC$ , $AA' = 3CA$ .

When extending the sides $AB,BC,CA$ of $\Delta ABC$ to $B',C',A'$ respectively, such that $AB' = 2AB$ , $CC' = 2BC$ , $AA' = 3CA$ . If $[\Delta ABC] = 1$, find $[\Delta A'B'C']$ . What I Tried: Here ...
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In $\Delta ABC$, angle bisector of $\angle ABC$ and median on side $BC$ intersect perpendicularly

In $\Delta ABC$, $BE$ is the angle bisector of $\angle ABC$, $AD$ is the median on side $BC$. $AD$ intersects $BE$ at $O$ perpendicularly. If $AD = BE = 4$, find the lengths of each side of $\Delta ...
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2 votes
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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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1 vote
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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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-1 votes
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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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2 votes
4 answers
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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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1 vote
1 answer
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$ABCD$ is a square. $E$ and $F$ are points respectively on $BC$ and $CD$ such that $\angle EAF = 45^\circ$.

$ABCD$ is a square. $E$ and $F$ are points respectively on $BC$ and $CD$ such that $\angle EAF = 45^\circ$. $AE$ and $AF$ cut the diagonal $BD$ at $P,Q$ respectively. Find $\frac{[\Delta AEF]}{[\Delta ...
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1 vote
1 answer
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Let $ABCD$ be a rectangle where $\Delta PAB$ is isosceles. The radius of the circles are $3$,$4$,$3$ cm respectively.

Let $ABCD$ be a rectangle where $\Delta PAB$ is isosceles. The radius of each of the smaller circles is $3$ cm and the radius of the bigger circle is $4$ cm. Find the length and breadth of the ...
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2 votes
1 answer
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$ABCD$ is a square, with $K$ and $L$ are points on $BC$ and $DC$ respectively . If $AM \perp LK$ AND $\angle AKM = \angle AKB$ , Find $\angle LAK$ .

Here is a diagram if needed :- What I Tried :- I did angle-chasing , considered $\angle MKA = \angle AKB = x$ and then I did not get any information for some other angles , so I considered $\angle ...
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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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1 vote
2 answers
51 views

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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  • 79
1 vote
2 answers
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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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1 vote
1 answer
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Proof that a spherical triangle is congruent with its dual

I'm currently studying spherical geometry and ran into an exercise problem that I'm having trouble understanding. The book first defines a dual spherical triangle $\triangle^* ABC$ of original $\...
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1 vote
1 answer
267 views

Triangle congruence when the longest sides, the largest angles, and one of the other sides are congruent?

Are two scalene triangles (a scalene triangle is one with no two equal sides) congruent if their longest sides, largest angles, and shortest sides are congruent? I believe the case described above is ...
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3 votes
2 answers
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Avoiding Circular Reasoning: How to Define Congruent Shapes

I apologize for being overly verbose here, but the question I want to know is at the very bottom. I am going to be honest and say I have no idea how to axiomatically handle congruence in geometry and ...
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3 votes
3 answers
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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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2 votes
2 answers
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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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2 votes
3 answers
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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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3 votes
1 answer
299 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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1 answer
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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 ...
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