(i just need help filling in the ? with the reason name)
Given: Ray AC is the angle bisector of <BAD, m<BAC = 13x and m<CAD = 6x + 21.
Prove: x = 3.
statement: reasons:
Ray AC bisects <BAD (Given)
m<BAC = 13x (Given)
m<CAD = 6x + 21 (?)
m<BAC = m<CAD (Definition of angle bisector)
13x = 6x + 21 (?)
13x - 6x = 6x - 6x + 21 (Addition/Subtraction Property)
7x = 21 (Substitution Property)
7x/7 = 21/7 (?)
x = 3 Substitution Property
statement: reasons:
Ray AC bisects <BAD (Given)
m<BAC = 13x (Given)
m<CAD = 6x + 21 ****GIVEN ****
m<BAC = m<CAD (Definition of angle bisector)
13x = 6x + 21 SUBSTITUION OF VALUES
13x - 6x = 6x - 6x + 21 (Addition/Subtraction Property)
7x = 21 (Substitution Property)
7x/7 = 21/7 MULTIPLY BOTH SIDES BY 1/7
x = 3 Substitution Property
Here is a step-by-step breakdown of the proof and the reasons for each statement:
1. Ray AC bisects <BAD (Given)
- This statement is given in the problem.
2. m<BAC = 13x (Given)
- This statement is given in the problem.
3. m<CAD = 6x + 21 (?)
- The reason for this statement is not explicitly given. We can infer it from the fact that Ray AC bisects angle <BAD. Since Ray AC is the angle bisector, m<BAC is equal to m<CAD. Therefore, we can write m<CAD = m<BAC = 13x.
4. m<BAC = m<CAD (Definition of angle bisector)
- This statement follows from the fact that Ray AC bisects angle <BAD.
5. 13x = 6x + 21 (?)
- This statement is the result of substituting the expressions for m<BAC and m<CAD from statements 2 and 3.
6. 13x - 6x = 6x - 6x + 21 (Addition/Subtraction Property)
- This step involves simplification. We subtract 6x from both sides of the equation to isolate the term with the variable.
7. 7x = 21 (Substitution Property)
- This step comes from simplifying the left side of the equation.
8. 7x/7 = 21/7 (?)
- This step involves dividing both sides of the equation by 7 to isolate the variable.
9. x = 3 (Substitution Property)
- This step follows from simplification, where we divide both sides of the equation to solve for x.
So, the reason you can use for statement 3 is the understanding that Ray AC bisects angle <BAD, thus making m<CAD equal to m<BAC.