Questions tagged [congruences-geometry]
For questions about congruent geometric figures, e.g., determining whether one figure is congruent to another.
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Simple request for help with geometric transformations and a problem I have.
The question
I have a problem where I have been given four options, but as far as I can tell, none of them are correct. Can someone help me understand the process of proving similarity?
So to my ...
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reflexive property of congruence
While exploring basic geometry with my artist friend, she raised a question that neither of us could answer satisfactorily. Until today I thought that Hilbert's axiomatization of Euclidean geometry is ...
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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=\...
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How to prove the SSS triangle congruence without isosceles triangles or circles?
The SSS triangle congruence is the following theorem from elementary geometry:
If three sides of a triangle are equal to the three sides of another triangle, then the two triangles are congruent.
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How to define congruent angles without geometric transformations or measures?
I'm trying to build a proof of triangle SAS congruence (see the existing proof I know of) that does not base on geometric transformations, but only on basic axioms and definitions.
Along my journey to ...
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The sides and 3 diagonals of two convex hexagons is a rigid graph [closed]
Convex hexagon $ABCDEF$, is the graph of the sides and 3 diagonals a rigid graph?
I attempted to apply Menger's theorem: $V(ABCD)=0\iff$
$$\det {\begin{bmatrix}0&d(AB)^{2}&d(AC)^{2}&d(AD)^...
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(Confusing) Sine/Cosine rule for Question Regarding Obtuse Triangle / GCSE
Background: This question was taken from Pearson's Edexcel GCSE (9-1) Mathematics Algebra and Shape Workbook
Question focus: 3(a)
Triangle WXZ - Diagram (link to image, provided by stack exchange)
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How to prove $EN=\dfrac{AI}{2}?$ and $KN=\dfrac{CJ}{2}$? and $\dfrac{Area(ABD)}{2}+\dfrac{Area(BCD)}{2}=\dfrac{Area(ABCD)}{2}$?
Given
\begin{aligned}
\operatorname{Area}(E F G H) & =E K \cdot F G \\
& =(E N+K N) \cdot \frac{1}{2} B D \\
& =\frac{1}{2} B D \cdot E N+\frac{1}{2} B D \cdot K N \\
& =\frac{1}{4} ...
user1231517
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Geometry question with congruent triangles and isosceles triangles
In the diagram below $AD\equiv BC$ and $\alpha + \beta=180^{\circ}$. Find the measure of $\theta$.
I'm given a hint:
First extend $DC$ past $C$ to the point $E$ where $CE \equiv AB$, then draw $AE$ ...
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Proving two triangles congruent given two congruent sides and a congruent median
The title was a bit too short for me to fit the full details, so here's the scenario I have.
Prove that two triangles are congruent if in two triangles, the median from the common vertex and two sides ...
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Geometry question on angle chasing concerning 3 squares
I am kind of stuck on this problem.
We know that the 9 points present in this sketch form three squares, HBAI, CFGB and DEFC.
We also know, that the lines DI and AF intersect in S. The question for ...
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Let $ABC$ be a triangle. Let $D$ and $E$ such that $AB \perp BD$, $AC \perp CE$Prove that $\bigtriangleup FBC$ is an isosceles right angled.
PROBLEM:
Let $ABC$ be a triangle in which the measures $\angle ABC, \angle ACB$ are smaller than $45$. We consider that the points $D$ and $E$ such that $AB \perp BD$, $AB=BD$, $AC \perp CE$, $AC=CE$, ...
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ABCD is a parallelogram, and AXY is a straight line through A meeting BC at X and DC at Y. Prove that BX.DY is constant.
I'm not able to figure out what "constant" here means. I proved it till AB.AD = BX.DY but how is this value constant I don't get it. If we put different values of sides AB and AD we would ...
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Two triangles have an equal angle, inradius, circumradius. Are these two traingles necessarily congruent?
Two triangles have an equal angle, inradius, circumradius. Are these two triangles necessarily congruent?
Let $A_1,A_2$ be the areas if the triangles respectively, $s_1,s_2$ be the semi perimeter ...
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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 ...
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Why is superposition not rigorous in Geometry?
In the Elements things that are equal can coincide perfectly with one another. However we cannot show things to be equal by "applying" one to another.
If two lines are equal, why can't we ...
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High School Mathematics: A problem on parallelograms [duplicate]
$ABCD$ is a parallelogram and $O$ is any point. The parallelograms $OAEB$,
$OBFC$, $OCGD$, $ODHA$ are completed. Show that $EFGH$ is a parallelogram
I did see this question but it doesn't have the ...
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Using sine rule to prove triangle congruence
The following problem looks like it should be easy, but I don't know how to prove it rigorously. All I know is the sine rule should be applied somewhere.
Let $ABC$ be triangle with angles $\alpha$, $\...
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Show the congruence $YZ+JK=JF$ in this square with trapezoids, triangles constructed with square's midpoints
Show that $YZ+JK=JF$
I tried Pythagorean theorem.
OG: Area of a square inside a square created by connecting point-opposite midpoint
user685056
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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 ...
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Show the equality of two congruence transformations (geometry)
I have some problems in the following task of my geometry studies:
Let $A,B,C$ three non-collinear points in $\mathbb{R}^2$ and let $T_1,T_2:\mathbb{R}^2\rightarrow\mathbb{R}^2$ two congruence ...
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Definition of congruence of triangles.
In school level geometry,we studied congruence of triangles.Where it is defined that two triangles are congruent if they have same shape and size.But this is not a precise definition,what does it mean ...
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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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$PQ ∥ BC$ 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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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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