# Questions tagged [angle]

An object formed by two rays joining at a common point, or a measure of rotation. In the latter form, it is commonly in degrees or radians. Please do not use this tag just because an angle is involved in the question/attempt; use it for questions where the main concern is about angles. This tag can also be used alongside (geometry).

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### How to express an angle between two angle bisectors in interior angles of a convex quadrilateral?

Given a convex quadrilateral $ABCD$, I would like to express the angle between angle bisector of internal and external angles of the opposite vertices in interior angles of $ABCD$. Here is the drawing ...
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### 2D angle from known 3D angle. [closed]

I have $2$ vectors in $3$D space: $\hat{u}$ and $\hat{v}$. The angle between them is $\varphi$, so that $$\hat{u}\cdot\hat{v} = u_{x}v_{x} + u_{y}v_{y} + u_{z}v_{z} = \cos\left(\varphi\right)$$ ...
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### How do you solve for the side length of this square?

I came across this question which had 3 parts. The first 2 were about showing what sin(a) equals which I managed to get, but the third part was show that $x^4-56x^2+640=0$ and solve for $x$, but how ...
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### Geometric Inequality with Angle Bisectors in a Triangle

Given triangle ABC with angle ABC = 60°. AP is a bisector of angle BAC. AQ is a bisector of angle CAP. Prove that BC > 4PQ. So far, I've managed to express the equality of the products of sides ...
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### Invariant angle between focus and intersections of tangents to an ellipse

Let $A,B$ be two fixed points on the ellipse $\mathcal{E}$ and $X$ a movable point moving along the ellipse $\mathcal{E}$. Let $M,N$ be the intersections of the tangent to $\mathcal{E}$ passing ...
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### How to calculate angle between vectors in a way that is invariant under translation?

Normally you just take the two vectors $\vec A$ and $\vec B$ and simply dot them, so $\vec A \cdot \vec B$ then divide by the product of there lengths and finally take the cosine inverse. But this ...
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### What is an angle? [duplicate]

I am a high school student and recently I started trigonometry and one question that comes to my mind every time is that "What is an angle?" I mean when we say that angle between two sides ...
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### Let $ABCD$ be a convex quadrilateral. If the measure of the angles $A=90, C=96, D=78$ and $BC=2*AB$, then the measure of the angle $ABD$ is?

The problem Let $ABCD$ be a convex quadrilateral. If the measure of the angles $A=90°, C=96°, D=78°$ and $BC=2*AB$, then the measure of the angle $ABD$ is...? The idea As you can see I calculated ...
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### A result concerning heights in an arbitrary triangle

I am trying to understand if the following claim is true or false (probably true?). The claim says that each of the three heights $\{AA_1,BB_1,CC_1\}$ bisects the relevant angle of the blue triangle. ...
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### solution-verification | compare to angles in a rectangular parallelipiped

the problem Let $ABCDA'B'C'D'$ be a rectangular parallelepiped with $AB=3a, BC=2a,AA'=6a, a>0$. We denote by $M$ and $N$ the means of $AA'$ and $CC'$ and by $P$ the intersection of the lines $BA'$ ...
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### solution-verification| calculate a trigonometric function of the angle of the plane (ABC) with the plane $\alpha$

the problem The triangle ABC has vertex A in a plane $\alpha$ and is projected onto this plane according to the isosceles right triangle AMN, with $AM=MN=a\sqrt{2}$ M being the projection of B and N ...
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### How to find the tangent when converting from 2D angles to Spherical to Cartesian coordinates?

I am making a spherical/ball-in-socket joint, and I want to limit the movement of the bodies relative to each other. The limit is defined as 2 angles $\alpha$ and $\beta$ which make a 2D rectangle. I ...
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### Find the smallest integer, $k>3$, such that the angle is on the y-axis

You have an angle $(5π)/6$ multiplied by $k$. So $(5π*k)/6.$ Find the smallest integer for $k$ that is greater than $3$ such that the angle is either $π/2$ or $(3π)/2$. Find the integer value of $k$. ...
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### Does every spherical sector of a circle in $\mathrm{M}_n(\mathbb{C})$ contain an invertible matrix?

Some context : (Notations : For the rest of the post, I will identify $\mathrm{M}_n(\mathbb{C})$ with $\mathbb{C}^{n^2}$, and I will note $\mathcal{S}^{n^2-1}$ the unit sphere of $\mathbb{C}^{n^2}$) I ...
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### How to find the angle of an ellipse's semi-major axis given a rectangle with known width and height, and tangent to the ellipse on 4 sides?

Given a rectangle that contains an ellipse, and the ellipse is tangent on all 4 sides, how can I find the angle and lengths of the semi-major and semi-minor axes? I'm mainly interested in getting the ...
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### Deriving the Unit Quaternion to Tait-Bryan Angles conversion.

Let me start by saying I have a working solution. But I just don't understand how to get there. I've followed the well-written paper Technical Concepts Orientation, Rotation, Velocity and Acceleration,...
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### How do you define a triangle from its orthographic projections on the 3 axis

I've spent hours on this problem and i'm running out of idea. Say you have an angle in a 3D space made of the points BAC, where A = (0,0,0), B and C can be moved and lengths AB and AC are always equal ...
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### How can I calculate the angles of a given orthographic camera perspective?

In Blender, I can set up an orthographic / isometric camera like so: With the X, Y, and Z rotation set to those values, a cube has the following properties: all sides are rendered at equal length, ...
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### Reflecting point across bisector of acute angle leads to perpendicular elsewhere

Exercise. Let $\angle BAC$ be an acute angle. In the interior of $\angle BAC$ there is a point $P$ whose projections onto the sides $AB$ and $AC$ are precicely $B$ and $C$. Draw the bisector of the ...
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### Can angles be defined by norms that are not induced by inner products?

Can angles be (well-)defined in a normed vector space where the parallelogram law does not hold? In other words, if the norm is not induced by an inner product in a normed space (say $L^1$ space), can ...
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### Show that the points $M, N, P, Q$ are coplanar and concyclic.

the question Consider the tetrahedron $ABCD$, with the perpendicular edge $AD$ on the $(BCD)$ plane. Let $X$ be some point of the edge $AD$ and the point $Y$ on $(AD)$ such that $m(∠Y CD) = m(∠DAB)$. ...
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### Looking for alternative proofs of this statement about angles

This is the theorem to prove. Below is my proof that I consider rather long and complex. The given data is on this drawing: Construct $\angle DCE = \angle DCB$. The point $E$ on ray $CE$ is chosen in ...
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### Show that triangle $ABC$ is isosceles.

the question Consider the triangle $ABC$ and a point $M$ inside the triangle such that $\angle MAB = 10 ,\angle MAC = 40 ,\angle MCA = 30$ and $\angle MBA = 20$ . Show that triangle $ABC$ is ...
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### Lengths of divided hypotenuse by bisector of right triangle of a triangle, that has circle inscribed in it

A circle is inscribed in a triangle ABC. Knowing that |AB| = 20cm, |AC| = 16cm and |BC| = 12cm, calculate the lengths of the segments into which the bisector of angle ACB divides side AB. So from the ...
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### Looking for a simple proof of this statement about angle bisectors

Theorem: Two angle bisectors make angles with the sides opposite to the internal angles they bisect. If we construct angle bisectors from two formed angles which are opposite to the same side of the ...
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### Angle in a triangle

The angle of triangle ABC also belongs to triangle AHC. The problem states that $\tan (\angle A) = 5$ $\frac{CH}{AH} = 5$ or $\frac{BC}{AC} = 5$ or both.
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### Would Dividing circles in 2520 degrees instead of 360 degrees give extra advantage? [duplicate]

I have two questions I am aware of the fact that historically babylonians used sexagecimal systems and it had mathematical advantage that the number $360^\circ$ has all divisors from $1$ to $10$ ...
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### Knowing that the area of a lateral face is equal to the area of the base, find the measure of the angle formed by the planes $(MBN)$ and $(ABC)$.

the question Let $VABCD$ be a regular quadrilateral pyramid, $M$ the midpoint of the edge $VC$, $N$ the midpoint of the edge $AD$. Knowing that the area of a lateral face is equal to the area of the ...
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### The volume of a tetrahedron $ABCD$ is $\frac{1}{6}$. Determine $CD$ knowing that $\angle ACB=45$ and that $AD+BC+\frac{AC \sqrt{2}}{2}=3$.

the question The volume of a tetrahedron $ABCD$ is $\frac{1}{6}$. Determine $CD$ knowing that $\angle ACB=45°$ and that $AD+BC+\frac{AC \sqrt{2}}{2}=3$. my drawing my idea From the equality we can ...
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### How to prove triangle is equilateral without triangle congruence or similarity theorems?

I'm examining the following basic theorem: An isosceles triangle with vertex angle equal to $60^\circ$ has other two angles equal. (I know this is true for base angle as well, but I'm interested ...
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### Circle Angle chasing problem: Compute APC+BQD [closed]

I was wondering if someone could give me a hint as to how to solve this problem: Let circle C1 intersect circle C2 at P and Q. A line intersects C1 at A and B, and C2 at C and D, such that the four ...