THE GEOMETRY OF POLYGONS

$$x=180-115=65°$$

$$x=73° \text{ (vertically opposite angles are equal)}$$

$$90+x+x+110=360 \Rightarrow 2x=160 \Rightarrow x=80°$$

$$180-58=122°$$

$$58°$$

$$58°$$

Co-interior: $180-70=110°$. Alternate: $70°$ (equal).

Since the alternate angle equals the original angle (70°), and the co-interior angle is $180°$ minus the original angle (110°), the co-interior and alternate angles must sum to $70+110=180°$ — they are supplementary, following logically from the two facts combined.

Vertically opposite angles occur at a SINGLE intersection of two lines (opposite each other across the crossing point) — they don't require any parallel lines at all. Corresponding angles occur specifically when a transversal crosses TWO PARALLEL lines, comparing an angle at one intersection to the angle in the 'matching' position at the other intersection.

Any two straight lines crossing at a single point create two pairs of vertically opposite angles, regardless of whether any other lines are parallel — e.g. simply drawing an X shape creates vertically opposite angles with no parallel lines needed at all.

$$63° \text{ (corresponding angles are equal)}$$

If the corresponding angle isn't exactly equal, this suggests the two tracks are NOT perfectly parallel — a genuine surveying result like this could indicate the tracks have a very slight, real-world deviation from true parallel alignment.

$(3x+15)+(2x+25)=180 \Rightarrow 5x+40=180 \Rightarrow 5x=140 \Rightarrow x=28$.

First angle: $3(28)+15=99°$. Second angle: $2(28)+25=81°$. (Check: $99+81=180$ ?.)

$x+2x+75+(x+15)=360 \Rightarrow 4x+90=360 \Rightarrow 4x=270 \Rightarrow x=67.5$.

$67.5°$, $135°$, $75°$, $82.5°$. Sum: $67.5+135+75+82.5=360°$ ?.

Alternate angle: $62°$ (equal). Co-interior angle: $180-62=118°$ (supplementary).

The two transversals are entirely INDEPENDENT — each transversal creates its OWN set of related angles (alternate, corresponding, co-interior) based on ITS OWN angle of crossing, using the SAME parallel-line angle rules. Knowing the 62° transversal's angles gives no direct information about the 40° transversal's angles, since they are separate, unrelated intersecting lines, even though both interact with the same pair of parallel lines and follow the same general angle rules.

Vertically opposite: $50°$ (equal). Sum: $50+50=100°$.

Vertically opposite: $73°$. Sum: $73+73=146°$ — DIFFERENT from the earlier 100°. Since vertically opposite angles are always EQUAL to each other (not a fixed value like 100°), their sum is always DOUBLE the original angle — which changes depending on what the original angle actually is; there's no single fixed 'sum' value for all vertically-opposite-angle pairs.

Calling them 'opposites' is misleading because they're not fundamentally contrasting concepts — alternate angles and co-interior angles are actually closely CONNECTED: for any pair of parallel lines cut by a transversal, the alternate angle and co-interior angle on the SAME side are always SUPPLEMENTARY to each other (summing to 180°), which follows directly from the fact that alternate angles are equal to the original angle, while co-interior angles are supplementary to it — they're two related consequences of the same parallel-line geometry, not opposing ideas.

'The road acts as a transversal crossing two parallel railway tracks. Since co-interior (allied) angles between parallel lines cut by a transversal always sum to 180°, the co-interior angle on the other side is $180°-58°=122°$. This follows from the property that interior angles on the same side of a transversal are supplementary when the two lines being crossed are parallel.'

$$\frac{180-100}{2}=40° \text{ each}$$

If base angle $=35°$ (the minimum allowed), apex angle $=180-2(35)=110°$. So the MAXIMUM apex angle while still meeting the 35° base-angle minimum is $110°$ — note this means a LARGER apex angle actually corresponds to the smallest allowed base angle, since apex and base angles trade off inversely: increasing the apex angle decreases each base angle (and vice versa), so the architect must balance a taller-looking roof (larger apex) against maintaining sufficient base-angle support.

If the two ladders are parallel, and BOTH meet the (also parallel, since both ladders meet the same flat ground) horizontal ground, then by corresponding angle rules, the second ladder MUST also make exactly a 72° angle with the ground — parallel lines crossing another line (or a common direction like 'the ground') always create equal corresponding angles.