Explore Angle Theorems with Expert Help

In a triangle ABC, $∠ B A E = ∠ E A D = ∠ D A F = ∠ F A C$ , and $B E : E D : D F : F C = 2 : 1 : 1 : 2.$ .
Why is $A E : A F = E D : D F$ ? And why $B E : E D = B A : A D$ ?
Are there any results in ratio or any theorems that I can use to show the equality of the ratio?

skilpadw3 2022-07-18

What is the angle of b?

So first off, I know how to find the missing length of the leg of the triangle using the pythagorean theorem. $6^{2} + b^{2} = c^{2}$ .
$36 + b^{2} = 100$
$100 - 36 = 64$
$\sqrt{64} = 8$
So angle angle A is going to be $\frac{adjacent}{hypotenuse}$ .
$\cos (angle) = \frac{adjacent}{hypotenuse}$
$\cos (angle) = \frac{8}{10}$
$\cos^{- 1} (0.8) = {36.86989765}^{\circ}$
Here is where I run into my problem. So assuming that B is going to be $\frac{opposite}{hypotenuse}$ .
$\sin (angle) = \frac{6}{10}$
$\sin^{- 1} (0.6) = {36.86989765}^{\circ}$
However, when I use the law of cosines I get the same answer for angle A, but a different angle for B, which comes out as 53.13. It would be greatly appreciated if someone could help me figure out where my mistakes are.

Israel Hale 2022-07-18

Points A, B, C lie on a circle with center O. Show that $∡ O A C = 90 - ∡ C B A$

pliwraih 2022-07-17

Angle between chord AB and tangent at A is the same as subtended by segment AB at any point on the circumference.
How to prove this theorem?

Leila Jennings 2022-07-16

Question of Triangle of Geometry
In triangle ABC,we have $A B > A C$ . If A' is the mid point of BC, AD is the altitude through A and if the internal and external bisectors of angle A meet at BC at X and X' respectively ,prove that $A^{'} D = (c^{2} - b^{2}) / 2 a$
I have tried to use obtuse angle theorem to find height then solve the question but stuck on double variable.

Alex Baird 2022-07-16

Find the length of one side of a triangle given two other sides and the length of the angle bisector
For an arbitrary triangle ABC.
If given the length of side $A B = 15$ and $A C = 12$ and the length of the angle bisector that connects angle A to side $B C = 10$ . What is the length of BC.

kominis3q 2022-07-16

In this image, $M N ∥ A T$ , $C D = D A$ and $∠ P D T = 90 º$ then arc $N P$ =?

Ishaan Booker 2022-07-16

In $△ A B C$ , side BC is divided by D in a ratio of 5 to 2 and BA is divided by E in a ratio of 3 to 4 as shown in Figure. Find the ration in which F divides the cevians AD and CE i.e. find $E F : F C$ and $D F : F A$ .

Lilliana Livingston 2022-07-15

Justify the small angle approximation for tangent
Assume that angle x is small, using small angle approximation,
$\sin (x) = x$
$\cos (x) = 1 - \frac{x^{2}}{2}$ ;
and $\tan (x) = x$ .
I am able to justify the first two using Maclaurin's Theorem but not the last one. How do we justify the last one?

Ximena Skinner 2022-07-15

Let quadrilateral ABCD satisfy $∠ B A C = ∠ C A D = 2 ∠ A C D = 40^{\circ}$ and $∠ A C B = 70^{\circ}$ . Find $∠ A D B$ .

scherezade29pc 2022-07-14

We have $A B = B C$ , $A C = C D$ , $∠ A C D = 90^{\circ}$ . If the radius of the circle is 'r' , find BC in terms of r .

Angles are an important part of mathematics, and angle theorems help us understand the properties of angles and how they relate to each other. Angle theorems allow us to calculate the size of angles and to determine the relationships between them. They are used in geometry and trigonometry to help us solve various problems. Some of the most common angle theorems include the angle sum theorem, the exterior angle theorem, and the triangle angle bisector theorem. These theorems can help us calculate angles in triangles and quadrilaterals, as well as other shapes. Angle theorems are very useful in mathematics and can help us understand the properties of angles and how they relate to each other.

Angle theorems Uncovered: Discover the Secrets with Plainmath's Expert Help