IN rhombus ABCD, measure of angle DAB equals two times the measure of angle ADC and CB equals 6. Find measure of angle ACD
Opposite angles of a rhombus are equal are equal. The sum of rhombus angles is 360.
Let angle DAB = angle BCD = a
Then angle ADC = angle ABC = a/2
3a = 360
a = 120
The rhombus internal angles are 120 (twice) and 60 (twice).
ACD is not a rhombus angle. AC is a diagonal, with length 6. ACD is the angle between diagonal AC and side CD. Draw the figure and you will see that ACD is 60 degrees.
It makes no difference what side CD is. That information was not required.
To find the measure of angle ACD, we need to start by knowing more information about the properties of rhombus ABCD. Since CB equals 6, this means that opposite sides of the rhombus are congruent. Therefore, AD is also equal to 6.
Let's denote the measure of angle ADC as x. According to the given information, the measure of angle DAB is two times the measure of angle ADC, so we can say that the measure of angle DAB is 2x.
Since ABCD is a rhombus, the sum of the angles in a rhombus is always 360 degrees. In triangle ABC, we have the angles at A, B, and C. Since the opposite angles of a rhombus are congruent, angle ACD is equal to angle ABC, denoted as y.
Using the sum of the angles in triangle ABC:
y + 2x + y = 180
Combining like terms:
2y + 2x = 180
Dividing both sides by 2:
y + x = 90
Now we have a system of equations. We also know that CB and AD are congruent sides of the rhombus, so they are also congruent triangles.
CB = AD
6 = 6
Since triangles ACD and BDC share side DC, and also have congruent sides AC and CD, they are congruent triangles. This means that angle ACD is equal to angle BCD, denoted as z.
Using the sum of the angles in triangle BCD:
z + 2x + z = 180
Combining like terms:
2z + 2x = 180
Dividing both sides by 2:
z + x = 90
We now have a system of equations:
y + x = 90
z + x = 90
Subtracting x from both sides of both equations:
y = 90 - x
z = 90 - x
Since angle ACD is equal to angle BCD (z), we can substitute z in the first equation:
y = z
Now we have:
y = 90 - x
y = z
Therefore, angle ACD is equal to 90 - x degrees.
To find the measure of angle ACD in rhombus ABCD, we can use the given information that the measure of angle DAB is two times the measure of angle ADC.
Let's assign a variable to the measure of angle ADC. Let's say the measure of angle ADC = x.
Since the measure of angle DAB is two times the measure of angle ADC, we can write it as: measure of angle DAB = 2x.
In a rhombus, opposite angles are equal. So, measure of angle ADC = measure of angle BCD.
Now, we can use the fact that the sum of the angles in a quadrilateral is 360 degrees. In rhombus ABCD, we have four equal angles, so each angle measures 360/4 = 90 degrees.
Since the measure of angle BCD is 90 degrees, we can write it as: measure of angle BCD = 90.
Now we have two equations:
1) measure of angle DAB = 2x
2) measure of angle BCD = 90
Since opposite angles in a rhombus are equal, we know that measure of angle BCD = measure of angle ADC. Therefore, we can substitute 90 for measure of angle BCD in equation 1:
90 = 2x
To solve for x, we divide both sides of the equation by 2:
90/2 = 2x/2
45 = x
Therefore, the measure of angle ADC is 45 degrees.
Finally, to find the measure of angle ACD, we subtract the measure of angle ADC from 90 degrees (since the sum of angles in a triangle is 180 degrees):
measure of angle ACD = 90 - measure of angle ADC
measure of angle ACD = 90 - 45
measure of angle ACD = 45 degrees
So, the measure of angle ACD in rhombus ABCD is 45 degrees.