Angles, parallel lines and polygon angle sums

Example 1: same-side interior angles

Problem Lines p and q are parallel, and a transversal crosses both. Between the parallel lines, on the same side of the transversal, one angle measures $(2x + 14)^\circ$ and the other measures $(4x - 20)^\circ$. Find x and both angles.

1. These are same-side interior angles, so they add to 180°. They are not equal.
   $$(2x + 14) + (4x - 20) = 180$$
2. Combine like terms.
   $$6x - 6 = 180$$
3. Add 6 to both sides.
   $$6x = 186$$
4. Divide by 6.
   $$x = 31$$
5. Plug 31 into the first angle.
   $$2(31) + 14 = 76$$
6. Plug 31 into the second angle.
   $$4(31) - 20 = 104$$
7. Check: $76 + 104 = 180$. One angle is acute and one is obtuse, which matches the picture.

Answer $x = 31$. The angles are $76^\circ$ and $104^\circ$.

Example 2: the exterior angle of a triangle

Problem In triangle ABC, side BC is extended past C to a point D. The exterior angle ACD measures $(5x - 8)^\circ$. Angle A measures $(2x + 10)^\circ$ and angle B measures $(x + 22)^\circ$. Find the measure of angle ACB.

1. The exterior angle equals the sum of the two remote interior angles, A and B.
   $$5x - 8 = (2x + 10) + (x + 22)$$
2. Simplify the right side.
   $$5x - 8 = 3x + 32$$
3. Subtract $3x$ from both sides, then add 8.
   $$2x = 40$$
4. Divide by 2.
   $$x = 20$$
5. The exterior angle is $5(20) - 8 = 92^\circ$. Angle ACB sits next to it on the straight line BD, so the two add to 180°.
   $$180 - 92 = 88$$
6. Check: angle A is 50°, angle B is 42°, and $50 + 42 + 88 = 180$.

Answer $x = 20$, so angle ACB measures $88^\circ$.

Example 3: sides of a regular polygon

Problem Each interior angle of a regular polygon measures 156°. How many sides does it have?

1. Set the one-angle formula equal to 156.
   $$\frac{(n - 2) \cdot 180}{n} = 156$$
2. Multiply both sides by n.
   $$(n - 2) \cdot 180 = 156n$$
3. Distribute the 180.
   $$180n - 360 = 156n$$
4. Subtract $156n$ from both sides and add 360.
   $$24n = 360$$
5. Divide by 24.
   $$n = 15$$
6. Faster way: each exterior angle is $180 - 156 = 24$ degrees, and the exterior angles add to 360°, so there are $360 \div 24 = 15$ of them.

Answer 15 sides.

Example 4 (test-hard): a bent path between parallel lines

Problem Lines p and q are parallel, with p above q. Point A is on line p and point B is on line q. Point P lies between the lines, to the right of both A and B. Segment AP makes a 38° angle with the part of line p to the right of A. Segment BP makes a 47° angle with the part of line q to the right of B. Find angle APB.

1. No rule connects the two given angles yet. Draw a helper line through P, parallel to p and q.
2. The helper line splits angle APB into an upper part (touching AP) and a lower part (touching BP).
3. Upper part: AP is a transversal of line p and the helper line. The upper part and the 38° angle are alternate interior angles, so the upper part is 38°.
4. Lower part: BP is a transversal of the helper line and line q. By the same rule, the lower part is 47°.
5. Add the two parts.
   $$38 + 47 = 85$$

Answer Angle APB measures $85^\circ$.