Question

question 1 5 pts use the diagram to prove the exterior angle theorem. given: abc is a triangle and b, c, and d are collinear prove: ( mangle a + mangle b = mangle acd ) abc is a triangle and b, c, and d collinear. select select triangle sum the ( mangle acd + mangle acb = 180^{circ} ) select ( mangle b + mangle a + mangle acb = mangle acd + mangle acb ) select ( mangle b + mangle a = mangle acd ) select

Explanation:

Step1: State the given information

ABC is a triangle and B, C, and D are collinear. This is given in the problem statement.

Step2: Use the Triangle Sum Theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). So, \(m\angle A + m\angle B + m\angle ACB=180^{\circ}\).

Step3: Use the definition of a linear pair

Since B, C, and D are collinear, \(\angle ACD\) and \(\angle ACB\) form a linear pair. By the definition of a linear pair, \(m\angle ACD + m\angle ACB = 180^{\circ}\).

Step4: Set the two equations equal

Because \(m\angle A + m\angle B + m\angle ACB=180^{\circ}\) (from the Triangle Sum Theorem) and \(m\angle ACD + m\angle ACB = 180^{\circ}\) (from the linear - pair definition), we can substitute. So, \(m\angle B + m\angle A + m\angle ACB=m\angle ACD + m\angle ACB\).

Step5: Subtract \(m\angle ACB\) from both sides

Using the subtraction property of equality, if \(a + c=b + c\), then \(a = b\). Here, \(a=m\angle A + m\angle B\), \(b = m\angle ACD\), and \(c=m\angle ACB\). So, \(m\angle B + m\angle A=m\angle ACD\).

Answer:

The first blank (reason for the first statement) is "Given". The second statement is "\(m\angle A + m\angle B + m\angle ACB = 180^{\circ}\)". The reason for \(m\angle ACD + m\angle ACB = 180^{\circ}\) is "Definition of a linear pair". The reason for \(m\angle B + m\angle A + m\angle ACB=m\angle ACD + m\angle ACB\) is "Substitution Property (since both equal \(180^{\circ}\))". The reason for \(m\angle B + m\angle A=m\angle ACD\) is "Subtraction Property of Equality".