Apply the properties of angles at a point (360°), on a straight line (180°), and vertically opposite angles to find unknown angles in multi-step problems
Angles on a straight line add to 180 degrees when the parts exactly fill the straight angle, with no gaps or overlaps. Angles around a point add to 360 degrees when the parts exactly fill one full turn. A numeric sum alone does not prove a drawn arrangement is such a partition. When two parts are unknown, their combined remainder alone may not determine each; another relation, such as equal size, can do so.
When two straight lines cross, they make four non-reflex openings. Opposite openings share the vertex but no arm. Adjacent openings share one arm and together exactly fill a straight angle, so they add to 180 degrees.
Two straight lines cross. One opening is 70 degrees. Its adjacent opening fills the rest of the straight angle: 180 - 70 = 110 degrees. The opening opposite the 70 degree opening is also 70 degrees, and the opening opposite the 110 degree opening is also 110 degrees. One known angle unlocks all four.
Vertically opposite angles are equal. When two straight lines cross, each pair of opposite openings matches exactly. This is the vertical equality rule. It does not say adjacent openings are equal. Adjacent openings are equal only in the special case where both are 90 degrees.
An angle marked (3x) degrees means 3 times x. When two vertically opposite openings are equal, set their expressions equal: 3x = 75. Solve by dividing both sides by 3: x = 25. This is the one-step equation method.
For an expression like (2x + 30) degrees, first undo the addition, then undo the multiplication. If 2x + 30 = 110, subtract 30 from both sides: 2x = 80. Then divide both sides by 2: x = 40. This is the two-step equation method.
Both sides of an equation may carry x. If 4x = 2x + 60, subtract 2x from both sides: 2x = 60. Then divide by 2: x = 30. A crossing angle of 60 degrees gives a neighbour of 180 - 60 = 120 degrees. Remove a known 30 degree part: 120 - 30 = 90. Split the rest into three equal angles: 90 / 3 = 30 degrees each.
Angles on a straight line add to 180 when the parts exactly fill the straight angle with no gaps or overlaps. Angles around a point add to 360 when the parts exactly fill one full turn. When two straight lines cross, vertically opposite openings are equal, and adjacent openings share one arm and add to 180. Solve one-step, two-step, and both-sides equations to find unknowns.
This stop added ten newly authored tasks and six newly authored extra cards, plus four Khan Academy reading adaptations: Complementary and supplementary angles review (academy:article:7th-engage-ny:x73688785), Vertical angles review (academy:article:7th-engage-ny:x532ac19b), Triangle angles review (academy:article:basic-geo:xe0fea713), and Key ideas: find unknown angle measures within right and straight angles (academy:article:4th-grade-math-eureka-squared-aligned:x5ec47d8527d3c5b0). The corpus records their licenses as null; those null licenses are unverified.
You know the crossing rules: openings straight across match, and adjacent openings share one arm and fill 180° on a line. Two wooden strips cross as straight centre lines in a garden trellis, and you have measured one opening: 70°.
Which angle do you get for free, with no arithmetic at all?
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