Question

solve for b.
68.5° 49.5°
b

b = ?°

Explanation:

Step1: Find the straight - line angle sum

A straight line forms an angle of \(180^{\circ}\). So the angle at the top vertex of the triangle (the one between the two given angles) is \(180-(68.5 + 49.5)\) degrees.
First, calculate \(68.5+49.5 = 118\) degrees. Then \(180 - 118=62\) degrees.

Step2: Use the triangle angle - sum property

The sum of the interior angles of a triangle is \(180^{\circ}\). In an isosceles triangle (since two sides are parallel as indicated by the arrows, so the triangle is isosceles with \(b\) being equal to the angle we just found? Wait, no, actually, the two parallel lines and the transversal form an isosceles triangle? Wait, no, the sum of angles in a triangle is \(180^{\circ}\), and we know that in a triangle, if two sides are parallel (the two arrows), then the base angles are equal? Wait, no, let's re - think.
Wait, the triangle has two angles: one is \(b\), one is \(b\) (because the lines are parallel, so alternate interior angles are equal), and the third angle is \(62^{\circ}\) (from step 1). So using the triangle angle sum formula \(b + b+62 = 180\).
Combine like terms: \(2b+62 = 180\).
Subtract 62 from both sides: \(2b=180 - 62=118\).
Divide both sides by 2: \(b=\frac{118}{2}=59\) degrees. Wait, no, wait, maybe I made a mistake. Wait, actually, the triangle is isosceles because the two lines with arrows are parallel, so the base angles are equal. The angle at the top is \(180-(68.5 + 49.5)=62^{\circ}\). Then the sum of the other two angles (both \(b\)) is \(180 - 62 = 118^{\circ}\), so each \(b=\frac{118}{2}=59^{\circ}\)? Wait, no, wait, let's check again.
Wait, the correct way: The angle adjacent to the two given angles is \(180-(68.5 + 49.5)=62^{\circ}\). Then, in the triangle, since the two sides are parallel (the arrows), the triangle is isosceles with \(b\) equal to the angle we just found? No, wait, the sum of angles in a triangle is \(180^{\circ}\). If the triangle is isosceles (because of the parallel lines), then the two base angles are equal. So let the two base angles be \(b\) each. Then \(b + b+62=180\), so \(2b = 118\), so \(b = 59\). Wait, but let's do it again.
First, find the angle at the top of the triangle:
\(\theta=180-(68.5 + 49.5)=180 - 118 = 62^{\circ}\)
Then, since the triangle has two equal angles (because the two lines are parallel, so the alternate interior angles are equal), let the two equal angles be \(b\). Then:
\(b + b+\theta=180\)
\(2b+62 = 180\)
\(2b=180 - 62=118\)
\(b = 59\)

Answer:

\(59\)