Find the values of x, y, and z. The diagram is not to scale.
Triangle A B C is drawn with point D on side B C. A segment is drawn from A to D. Angle A C D measures 63 degrees. Angle C A D measures 36 degrees. Angle D A B measures 13 degrees. Angle C D A measures x degrees. Angle C B A measures y degrees. Angle A D B measures z degrees.
First, let's label the diagram:
Triangle ABC:
- Angle ACD = 63 degrees
- Angle CAD = 36 degrees
- Angle DAB = 13 degrees
- Angle CDA = x degrees
- Angle CBA = y degrees
- Angle ADB = z degrees
Now, let's use the property that the sum of the angles in a triangle is always 180 degrees.
In triangle ACD, we have:
- Angle CAD + Angle ACD + Angle CDA = 180 degrees
- 36 + 63 + x = 180
- x + 99 = 180
- x = 180 - 99
- x = 81 degrees
In triangle ACB, we have:
- Angle CBA + Angle CAB + Angle BAC = 180 degrees
- y + 36 + 63 = 180
- y + 99 = 180
- y = 180 - 99
- y = 81 degrees
Now, to find z, we can use the fact that the sum of the angles in a triangle is always 180 degrees. In triangle ABD, we have:
- Angle ADB + Angle DAB + Angle ABD = 180 degrees
- z + 13 + 180 = 180
- z + 13 = 180 - 180
- z + 13 = 0
- z = 0 - 13
- z = -13 degrees
The values of x, y, and z are:
- x = 81 degrees
- y = 81 degrees
- z = -13 degrees
To find the values of x, y, and z, we will start by focusing on the angles given in the problem:
1. Angle ACD measures 63 degrees.
2. Angle CAD measures 36 degrees.
3. Angle DAB measures 13 degrees.
We can use the fact that the sum of the angles in a triangle is 180 degrees to find the measurement of angle CDA:
Angle CDA = 180 - Angle ACD - Angle CAD
Angle CDA = 180 - 63 - 36
Angle CDA = 81 degrees
Now, let's move on to finding the value of angle CBA (y degrees). We know that the sum of the angles in a triangle is 180 degrees. Therefore, we can find angle CBA using the following formula:
Angle CBA = 180 - Angle CAD - Angle CDA
Angle CBA = 180 - 36 - 81
Angle CBA = 63 degrees
Finally, we can find the value of angle ADB (z degrees) using the same approach:
Angle ADB = 180 - Angle DAB
Angle ADB = 180 - 13
Angle ADB = 167 degrees
Therefore, the values of x, y, and z are:
x = Angle CDA = 81 degrees
y = Angle CBA = 63 degrees
z = Angle ADB = 167 degrees