Recognise angles where they meet at a point, are on a straight line, or are vertically opposite; find missing angles using these properties
Angles can meet at a single point and fill all the space around it. When they fill that whole space with no gaps or overlaps, they add up to 360 degrees, a full turn. Angles can also sit side by side along a straight line. When two angles exactly fill a straight angle with no gaps or overlaps, they add up to 180 degrees.
Two angles exactly filling a straight angle with no gaps or overlaps add up to 180 degrees. This works for any exact partition of a straight angle, not only two angles. Two straight lines cross and make four angles. The angles across from each other share the same vertex but no ray. They are called vertically opposite angles. They are always equal. Adjacent angles share a ray and exactly fill a straight angle, so they add to 180.
Two straight lines cross. One angle is 70 degrees. The angle vertically opposite it is also 70 degrees. The angle next to it on the straight angle is 180 - 70 = 110 degrees. The angle opposite that is also 110 degrees. One known angle unlocks all four.
Three angles fill one full turn with no gaps or overlaps. One is 90 degrees and the other two are equal. Subtract: 360 - 90 = 270. Divide: 270 / 2 = 135 degrees each. Two straight lines cross. One opening is 50 degrees, so its neighbour on the straight line is 180 - 50 = 130 degrees. A ray splits that neighbour into two equal parts. Divide: 130 / 2 = 65 degrees each.
To find a missing angle, first identify the relationship. If two angles are vertically opposite, copy the equal measure. If angles exactly fill a straight angle, subtract the known parts from 180. If they fill one full turn, subtract from 360. If remaining parts are equal, divide the remainder equally. Adding all parts back checks a partition. For vertically opposite angles, equality checks the opposite measures.
Angles filling one full turn add to 360. Angles exactly filling a straight angle add to 180. Vertically opposite angles are equal. Use one known angle to find all four at a crossing.
Ten fresh original tasks (x1-x10) and two fresh original readings (reading3, reading4) authored for this topic. Two Khan Academy articles retained and adapted: Angle types review (academy:article:4th-engage-ny:xdd5cdcb6) and Key ideas: find unknown angle measures around a point (academy:article:4th-grade-math-eureka-squared-aligned:xb18687f13f63ec41). Three Khan Academy videos retained: Complementary & supplementary angles (academy:video:7th-engage-ny:263439064), Vertical angles (academy:video:7th-engage-ny:263439063), and Measuring angles using a protractor (academy:video:4th-engage-ny:263439062). All corpus licenses are null and unverified.
A pair of scissors opens, and its two blades cross like an X through the pivot. Model each blade as a straight line through the pivot. Two straight lines crossing make four non-reflex openings. The angles opposite each other share the same vertex but no arm. They are called vertically opposite angles, and they are always equal. One opening between the blades is 70°.
What is the opening on the opposite side of the crossing?
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