Triangle ABC has AB=AC and angle BAC =X, with X being less than 60 degrees. Point D lies on AB such that CB = CD Point E lies on AC such that CE= DE Determine angle DEC in terms of X



Answer to a math question Triangle ABC has AB=AC and angle BAC =X, with X being less than 60 degrees. Point D lies on AB such that CB = CD Point E lies on AC such that CE= DE Determine angle DEC in terms of X

Expert avatar
61 Answers
To determine angle DEC in terms of X, we need to first analyze the given information and then apply some geometric relationships.

Given: Triangle ABC with AB = AC and angle BAC = X.

Let's label the angles of triangle ABC as follows:

Angle BAC = X
Angle ABC = Y
Angle BCA = Z

We know that AB = AC, which means that angles ABC and ACB are also equal. Hence, ABC = ACB = Y.

Since angle BAC + angle ABC + angle BCA = 180 degrees (angle sum property of a triangle), we can write:

X + Y + Z = 180

Now, we need to determine the relationship between angle DEC and angle A. Since AB = AC, we have AD = AE. Furthermore, we know that CB = CD and CE = DE.

Using these equalities, we can conclude that triangles ACD and ADE are congruent by the Side-Angle-Side (SAS) congruence criterion.

Now, let's analyze triangle ACD. We know that the sum of angles in a triangle is 180 degrees. Therefore:

Angle ADC + angle CDA + angle CAD = 180

Since angle ADC = angle CDA (as triangles ACD and ADE are congruent), we can substitute them with x, resulting in:

x + x + (180 - 2x) = 180

Simplifying this equation, we get:

2x - 2x + 180 = 180

0 + 180 = 180

So, the equation is true, which means that our assumption is correct.

Therefore, angle DEC is equal to angle ACD, which is x.

Answer: Angle DEC = x (in terms of X)

Frequently asked questions (FAQs)
What is the solution to the inequality -4x + 3 > 7?
Question: What is the variance of a set of numbers: 6, 9, 12, 15, 18?
What is the volume of a right circular cylinder with height 8 units and radius 5 units?
New questions in Mathematics
calculate the derivative by the limit definition: f(x) = 6x^3 + 2
10! - 8! =
1. Suppose we have a good whose quantity supplied changed from 100 to 120 units when the price increased from $10 to $12 per unit. Compute the price elasticity of supply using the midpoint method
An integer is taken at random from the first 40 positive integers. What is the probability that the integer is divisible by 5 or 6?
Solve this mathematical problem if 3/5 of a roll of tape measures 2m. How long is the complete roll?
Emma is on a 50 m high bridge and sees two boats anchored below. From her position, boat A has a bearing of 230° and boat B has a bearing of 120°. Emma estimates the angles of depression to be about 38° for boat A and 35° for boat B. How far apart are the boats to the nearest meter?
4x + 8y = 5 2x + 4y = 10
In the telephone exchange of a certain university, calls come in at a rate of 5 every 2 minutes. Assuming a Poisson distribution, the average number of calls per second is: a) 1/8 b) 1/12 c) 1/10 d) 2/5 e) 1/24
A storage maker price is $2.50 per square feet. Find the price of a custom shed 4 yards long, and 5yards wide and 8 feet tall
A function is considered exponential when it has a base with positive values greater than zero and different from one, where the exponent is an unknown. An important characteristic of exponential functions is that they show rapid growth or decay as an independent variable increases or decreases. Given the function 25^(x+3)=125, it is calculated that x has the value of
9 x² + 2x + 1 = 0
Show work on 4108 divided by 4
For what values of m is point P (m, 1 - 2m) in the 2⁰ quadrant?
How many cards do you expect to pull from a poker deck until you get an ACE?
Gender and communication : Answer the question ( 1 paragraph is ok) . Please can you write about women? Compared to your other identities, how much of a role does gender play in your life? And has your own sex/gender offered you privileges or disadvantages? How so?
Let I be an interval and let f : I → R be a continuous function such that f(I) ⊂ Q. Show (in symbols) that f is constant.