Question

what is the value of x in each figure? see example 4 20. 21. 22. 23. for exercises 24 - 27, find the measure of each

Explanation:

Step1: Use the exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.

For problem 20:

Step1: Apply the exterior angle theorem

The exterior angle \(x\) is equal to the sum of the two non - adjacent interior angles.
\(x = 41^{\circ}+98^{\circ}\)

Step2: Calculate the value of \(x\)

\(x=139^{\circ}\)

For problem 21:

Step1: Apply the exterior angle theorem

The exterior angle \(x\) is equal to the sum of the two non - adjacent interior angles.
\(x = 42^{\circ}+79^{\circ}\)

Step2: Calculate the value of \(x\)

\(x = 121^{\circ}\)

For problem 22:

Step1: Find the non - adjacent interior angle related to the given exterior angle

First, find the adjacent interior angle to \(119^{\circ}\). Since an interior angle and its adjacent exterior angle are supplementary (\(a + b=180^{\circ}\)), the adjacent interior angle \(y=180^{\circ}- 119^{\circ}=61^{\circ}\)

Step2: Use the triangle angle - sum property (the sum of interior angles of a triangle is \(180^{\circ}\))

Let the third angle be \(x\). We know that \(46^{\circ}+61^{\circ}+x = 180^{\circ}\). Another way is to use the exterior angle theorem. The exterior angle \(119^{\circ}\) and the non - adjacent interior angles: \(x=119^{\circ}-46^{\circ}\)

Step3: Calculate the value of \(x\)

\(x = 73^{\circ}\)

For problem 23:

Step1: Use the exterior angle theorem

The exterior angle (let's assume the non - labeled exterior angle related to the \(23^{\circ}\) and the other non - adjacent angle) concept. First, find the non - adjacent interior angles. The sum of the two non - adjacent interior angles to the exterior angle (formed by extending a side) is equal to the exterior angle.
We know that \(x+23^{\circ}=74^{\circ}\)

Step2: Calculate the value of \(x\)

\(x=74^{\circ}-23^{\circ}\)
\(x = 51^{\circ}\)

Answer:

1. \(139^{\circ}\)
2. \(121^{\circ}\)
3. \(73^{\circ}\)
4. \(51^{\circ}\)