ABC is an acute angle triangle with points D and E on BC and AC, respectively such that BE and AD are altitudes. AD and BE intersect at H. If ∠BCA=39∘ and ∠EBA=2∠DAB, what is the measure of ∠ABC (in degrees)?
We have a bunch of right triangles here
BCA=39, so CAD = 51
BEA=90, so EHA = 39
vertical angle DHB is thus 39
HDB = 90, so HBD = 51
Now we know all the angles
A = 51+x
B = 51+2x
so, 39+51+x+51+2x = 180
x = 13
ABC = 26
85
To find the measure of angle ABC, we need to use the given information and apply some properties of triangles.
Let's start by considering the relationship between the angles and sides of an acute angle triangle:
1. The sum of angles in a triangle is always 180 degrees.
Now, let's analyze the given information:
2. ∠BCA = 39 degrees (Given)
3. ∠EBA = 2∠DAB (Given)
We know that BE and AD are altitudes of triangle ABC, and they intersect at H. This means triangle ABC is a right triangle, with right angle B.
Let's focus on triangle ABE:
4. Since BE is an altitude, angle EBA is a right angle (∠EBA = 90 degrees).
5. ∠EBA = 2∠DAB (Given)
Therefore, 90 degrees = 2∠DAB.
From this information, we can find the value of ∠DAB:
6. 90 degrees = 2∠DAB
7. Dividing both sides by 2, we get:
45 degrees = ∠DAB
Now, let's find the value of ∠BCA:
8. ∠BCA = 39 degrees (Given)
Since the sum of angles in a triangle is 180 degrees, we can find ∠ABC:
9. ∠ABC = 180 degrees - ∠BCA - ∠BCA
10. ∠ABC = 180 degrees - 39 degrees - 45 degrees
11. ∠ABC = 180 degrees - 84 degrees
12. ∠ABC = 96 degrees
Therefore, the measure of ∠ABC is 96 degrees.