Question

using this diagram, please solve for the indicated angle in the next three questions. question 2 1 pts if ( mangle a = 83^{circ} ) and ( mangle b = 45^{circ} ), find ( mangle acd ). question 3 2 pts if ( mangle acd = 139^{circ} ) and ( mangle b = 41^{circ} ), find ( mangle a ). question 4 2 pts if ( mangle acd = 136^{circ} ) and ( mangle a = 84^{circ} ), find ( mangle b ).

Explanation:

Question 2

Step1: Apply the exterior - angle theorem

The exterior - angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. In \(\triangle ABC\), \(\angle ACD\) is an exterior angle, and \(\angle A\) and \(\angle B\) are the non - adjacent interior angles.
So, \(m\angle ACD=m\angle A + m\angle B\).

Step2: Substitute the given values

Given \(m\angle A = 83^{\circ}\) and \(m\angle B=45^{\circ}\).
Substitute into the formula: \(m\angle ACD=83^{\circ}+45^{\circ}\).
\(m\angle ACD = 128^{\circ}\).

Question 3

Step1: Use the exterior - angle theorem

By the exterior - angle theorem \(m\angle ACD=m\angle A + m\angle B\).
We need to solve for \(m\angle A\), so we can rewrite the formula as \(m\angle A=m\angle ACD - m\angle B\).

Step2: Substitute the given values

Given \(m\angle ACD = 139^{\circ}\) and \(m\angle B = 41^{\circ}\).
Substitute into the formula: \(m\angle A=139^{\circ}-41^{\circ}\).
\(m\angle A = 98^{\circ}\).

Question 4

Step1: Apply the exterior - angle theorem

From the exterior - angle theorem \(m\angle ACD=m\angle A + m\angle B\).
We solve for \(m\angle B\), so \(m\angle B=m\angle ACD - m\angle A\).

Step2: Substitute the given values

Given \(m\angle ACD = 136^{\circ}\) and \(m\angle A = 84^{\circ}\).
Substitute into the formula: \(m\angle B=136^{\circ}-84^{\circ}\).
\(m\angle B = 52^{\circ}\).

Answer:

Question 2: \(128^{\circ}\)
Question 3: \(98^{\circ}\)
Question 4: \(52^{\circ}\)