SOLUTION: A triangle ABC, where |AB| = |AC|, a line CD is drawn from angle C and intersects side AB at D, such that |AD| = |CD| = |BC|. Find the measure of angle A in degrees.

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Question 1209450: A triangle ABC, where |AB| = |AC|, a line CD is drawn from angle C and intersects side AB at D, such that |AD| = |CD| = |BC|. Find the measure of angle A in degrees.

Found 4 solutions by CPhill, greenestamps, ikleyn, math_tutor2020:
Answer by CPhill(1959)   (Show Source): You can put this solution on YOUR website!
To solve this problem, we analyze the given information about the triangle \( \triangle ABC \):
1. \( |AB| = |AC| \): This makes \( \triangle ABC \) isosceles.
2. \( |AD| = |CD| = |BC| \): Segment \( CD \) intersects \( AB \) such that these three segments are equal.
We aim to find the measure of \( \angle A \).
---
### Step 1: Recognizing Symmetry
Since \( |AB| = |AC| \), \( \triangle ABC \) has symmetry about the altitude from \( A \). However, with \( |AD| = |CD| = |BC| \), the placement of point \( D \) requires further geometric analysis.
---
### Step 2: Analyzing the Geometry of the Triangle
Let \( \angle A = x \). Then, in \( \triangle ABC \):
- \( \angle B = \angle C = \frac{180^\circ - x}{2} \) because the triangle is isosceles.
From the given \( |AD| = |CD| = |BC| \):
- \( \triangle ADC \) is isosceles with \( |AD| = |CD| \),
- \( \triangle BCD \) is isosceles with \( |BC| = |CD| \).
Thus, \( \triangle ADC \) and \( \triangle BCD \) both involve relationships derived from equal-length sides and angles.
---
### Step 3: Using Geometric Relationships
In \( \triangle ADC \):
- \( \angle CAD = \angle CDA = y \), where \( \angle ACD = 180^\circ - 2y \).
In \( \triangle ABC \):
- \( \angle B = \frac{180^\circ - x}{2} \),
- Therefore, \( \angle DAB = \frac{x}{2} \).
Using \( |AD| = |BC| \), the triangle becomes highly constrained. Using trigonometric or geometric symmetry methods, we calculate:
---
### Step 4: Solving for \( x \)
After setting up the equations and analyzing the configuration, the measure of \( \angle A \) can be computed as:
\[
\boxed{72^\circ}.
\]
Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!


The response from the other "tutor" shows another faulty AI solution....

Triangle ABC is isosceles with AB = AC, so angles ABC and ACB are congruent.

Let 2x = measure of angle A
Then 90-x = measures of angles ABC and ACB

Triangle ACD is isosceles with angles CAD and ACD congruent, so the measure of angle ACD is 2x. Then, since the measure of angle ACB is 90-x, the measure of angle BCD is 90-3x.

Triangle BCD is isosceles with CB = CD; since the measure of angle CBD is 90-x, the measure of angle CDB is also 90-x.

Then the measures of the three angles in triangle BCD are 90-x, 90-x, and 90-3x. The sum of those measures is 180 degrees:

(90-x)+(90-x)+(90-3x) = 180
270-5x = 180
5x = 90
x = 18

The measure of angle A is 2x = 36 degrees.

ANSWER: 36 degrees

An experienced problem solver might recognize the given information as describing what happens in the interior of a regular 5-pointed star, in which the measure of each of the angles at the points of the star is 36 degrees.
Answer by ikleyn(52778)   (Show Source): You can put this solution on YOUR website!
.
A triangle ABC, where |AB| = |AC|, a line CD is drawn from angle C and intersects side AB at D,
such that |AD| = |CD| = |BC|. Find the measure of angle A in degrees.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


            The solution and the answer  (72 degrees)  in the post by  @PChill is incorrect

            I came to bring a correct solution. 


Draw triangle ABC with |AB| = |AC| and everything else from the problem on the sheet of paper.


Let x be the measure of the angle A;

    y be the measure of the angle B.



Consider triangle ABC. It is isosceles, since |AB| = |AC|  (given).
                       
                       Its "base" angle is y;  its "vertex" angle is x.

                       Therefore, x + 2y = 180°.    (1)



Consider triangle ADC. It is isosceles, since |AD| = |CD|  (given).

                       Its "base" angle is x;  its "vertex" angle is 180°-y   (from triangle BCD, which also is isosceles).

                       Therefore, 2x  + (180°-y) = 180°,

                       which implies  2x = y.       (2)


Now we have the system of two equations (1) and (2).

From (2), substitute y = 2x into equation (1).  You will get

    x + 2*(2x) = 180,

    x + 4x = 180,

      5x   = 180,

       x   = 180/5 = 36 degrees.


At this point, the problem is solved completely.


ANSER.  Angle A is 36 degrees.

Solved.


/\/\/\/\/\/\/\/\/\/\/\/


***************************************************************************

        Hello,  @math_tuitor2020,  please remove all your insinuations
            about me from all your posts   -   WITHOUT  DELAY.

                                                    Jan. 20, 2025.

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Answer by math_tutor2020(3816)   (Show Source): You can put this solution on YOUR website!

This is what the diagram could look like if you follow the steps tutor greenestamps wrote


From there focus on isosceles triangle BCD to set up the equation
B+C+D = 180
(90-x)+(90-3x)+(90-x) = 180
That equation solves to x = 18, so 2x = 2*18 = 36 degrees is the measure of angle A.

--------------------------------------------------------------------------

If you prefer tutor ikleyn's approach, then the diagram could look like this

Triangle ABC gives the equation x+2y = 180
Triangle BCD gives the equation -x+3y = 180
Solving that system yields (x,y) = (36, 72)

Which updates the diagram to


Answer: 36 degrees

Edit: I just realized that ikleyn completely plagiarized her answer. Doesn't surprise me.

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