In triangle ABC, AB = BC. Calculate the angle a for ADC.
It is two triangles (A triangle within a triangle) and I'm trying to figure out the angles for them all.
Image link (I hope this is ok, I don't know how else to relay this): i.imgur.com/wKOva19.png
One angle is 31 degrees. Looking at the picture, if AB=BC then A and C must be 74.5 degrees. That totals 180 and should be right.
Now the issue is getting what getting the small a angle is. I assume I need to solve the left side first before the right side, but how does this work? I understand how to get the outer degree/etc but getting D or the split up A is confusing me.
Your link worked fine, but I think we need one more piece of information
perhaps AD = BC ??
In that case angle BAD = 31° and you can find the rest
Hello Reiny!
They do not specify any more then that. That's what confuses me the most.
To find the angle 'a' for triangle ADC, let's break down the problem step by step using the given information:
1. Start by observing that triangle ABC is an isosceles triangle because AB = BC. This means that the base angles (angles A and C) are congruent.
2. Since angle A is 74.5 degrees (as you correctly deduced from the given information), angle C is also 74.5 degrees.
3. Now, let's focus on triangle ADC. We already know that angle C is 74.5 degrees.
4. To find the angle 'a', we need to consider the angles within triangle ADC. Let's represent the angle at vertex D as angle DAD' (where D' is on line BC).
5. Notice that angle ADC is the exterior angle formed by the triangle ABC at vertex C. According to the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.
6. Therefore, we can conclude that angle ADC = angle DAD' + angle DAC.
7. We already know that angle DAC is 74.5 degrees (from step 2). Let's call angle DAD' as 'x' for now.
8. Since the total measure of angle ADC is given as 31 degrees, we can write the equation: 31 = x + 74.5.
9. To solve for 'x', subtract 74.5 from both sides of the equation: 31 - 74.5 = x, resulting in x = -43.5 degrees.
10. However, angles cannot have negative measures in this context. So, we need to consider the positive measure of 'x'. Since angle DAD' lies on line BC, it can be considered as an external angle of triangle ABC at vertex B.
11. The measure of an external angle of a triangle is equal to the sum of the measures of the two opposite interior angles. In this case, angle DAD' = angle DAB + angle ABD.
12. Knowing that angle A is 74.5 degrees (from step 2), we can rewrite the equation: x = 74.5 + angle ABD.
13. Since angle ABD is positive (as it's an interior angle), we can solve for 'x' by subtracting 74.5 from both sides of the equation: x - 74.5 = angle ABD.
14. Therefore, x - 74.5 is the measure of angle ABD.
So, the final conclusion is that the measure of angle 'a' (the small angle in triangle ADC) is equal to the measure of angle ABD, which is x - 74.5 degrees.