�ÚABC ~= �ÚACB
BD perpendicular AC
The sum fo the three angles in any triangle is 180 degree.
Prove: m�ÚCBD = 1/2 m�ÚBAC
angle ABC ~= �angle ACB
BD perpendicular AC
The sum fo the three angles in any triangle is 180 degree.
Prove: m�angle CBD = 1/2 m�angle BAC
let angle CBD = a
in trianagle ABC:
2x + y = 180
in triangle BCD
a + x = 90 or 2a + 2x = 180
then 2a + 2x = 2x + y , both equal to 180
2a = y
a = (1/2y
or
angle CBD = (1/2) angle BAC
To prove that m�ÚCBD = 1/2 m�ÚBAC, we need to use the given information and the properties of triangles.
First, we know that ABC ~= ACB, which means that triangle ABC is similar to triangle ACB. This implies that the corresponding angles in these triangles are equal.
Using this information, we can say that m�ÚABC = m�ÚACB. Let's call this angle x.
Now consider triangle ABC. We are given that BD is perpendicular to AC. This means that angle BDC is a right angle, which measures 90 degrees.
We also know that the sum of the three angles in any triangle is 180 degrees. So, in triangle BDC, we have:
m�ÚBDC + m�ÚCBD + m�ÚCDB = 180 degrees
Since angle BDC is a right angle (90 degrees), we can substitute this in and simplify:
90 + m�ÚCBD + m�ÚCDB = 180 degrees
Subtracting 90 from both sides:
m�ÚCBD + m�ÚCDB = 90 degrees
Now, let's go back to triangle ABC. Since m�ÚABC = m�ÚACB = x, we can replace these angles in triangle BDC:
x + m�ÚCBD + m�ÚCDB = 90 degrees
We want to prove that m�ÚCBD = 1/2 m�ÚBAC, so let's see how angle BAC relates to these angles.
In triangle ABC, we have:
m�ÚABC + m�ÚBAC + m�ÚACB = 180 degrees
Substituting x for m�ÚABC and m�ÚACB:
x + m�ÚBAC + x = 180 degrees
Simplifying:
2x + m�ÚBAC = 180 degrees
Subtracting 2x from both sides:
m�ÚBAC = 180 degrees - 2x
Now back to our equation from triangle BDC:
x + m�ÚCBD + m�ÚCDB = 90 degrees
Since m�ÚCDB is equal to m�ÚBAC (as they are corresponding angles in similar triangles), we can substitute m�ÚBAC into the equation:
x + m�ÚCBD + m�ÚBAC = 90 degrees
Subtracting x from both sides:
m�ÚCBD + m�ÚBAC = 90 degrees - x
Comparing this to our earlier equation for m�ÚBAC:
m�ÚBAC = 180 degrees - 2x
We can substitute this in:
m�ÚCBD + (180 degrees - 2x) = 90 degrees - x
Simplifying:
m�ÚCBD - 2x = -90 degrees - x
Combining like terms:
m�ÚCBD = -90 degrees - x + 2x
Simplifying further:
m�ÚCBD = -90 degrees + x
Now remember that we are trying to prove that m�ÚCBD = 1/2 m�ÚBAC. We can substitute the expression for m�ÚBAC we found earlier:
m�ÚCBD = -90 degrees + (180 degrees - 2x)
Simplifying:
m�ÚCBD = -90 degrees + 180 degrees - 2x
Combining like terms:
m�ÚCBD = 90 degrees - 2x
Finally, we want to show that m�ÚCBD = 1/2 m�ÚBAC. Let's substitute the expression for m�ÚBAC one more time:
m�ÚCBD = 90 degrees - 2x
m�ÚCBD = 90 degrees - 2(180 degrees - 2x)
Expanding:
m�ÚCBD = 90 degrees - 360 degrees + 4x
Combining like terms:
m�ÚCBD = 4x - 270 degrees
Now let's compare this to 1/2 m�ÚBAC:
1/2 m�ÚBAC = 1/2 (180 degrees - 2x)
Expanding:
1/2 m�ÚBAC = 90 degrees - x
As we can see, m�ÚCBD = 4x - 270 degrees and 1/2 m�ÚBAC = 90 degrees - x are equal. Hence, we have proved that m�ÚCBD = 1/2 m�ÚBAC.