Combine adjacent angles and find missing measures

Learning goal: Explain angle measure as additive and use equations to find unknown angles in diagrams and situations. Adjacent angle parts share a vertex and a boundary ray. When their interiors do not overlap, their measures add to the measure of the combined opening. Counting one-degree turns in each part counts the turns in the whole exactly once.

The illustration shows a starting ray pointing right, a middle ray and an ending ray. The first opening is 35°. The second opening is 55°. They share the middle ray but no interior region. The complete opening from the starting ray to the ending ray measures 35 + 55 = 90°. It is a right angle. The middle ray separates the whole into two parts without changing the whole opening.

A missing part can be found by subtraction. Suppose the whole right angle measures 90° and the first part measures 35°. Let a represent the unknown second part. Write 35 + a = 90. Subtract thirty-five from ninety to obtain a = 55°. Check by adding 35 + 55 = 90. The unknown is a degree measure, so include the degree symbol in the final answer.

A whole does not have to be a right angle. If an opening measures 125° and is split into a 40° part and an unknown part, write 40 + b = 125. The missing measure is 85°. The whole is obtuse, even though both smaller parts are acute. The type of each part does not determine the type of the total without considering their sum.

More than two parts can be combined. Three adjacent nonoverlapping openings of 20°, 45° and 30° total 95°. Add in a convenient order: 20 + 30 = 50, then 50 + 45 = 95. If the whole is known instead, subtract the sum of the known parts. For a 150° opening with known parts of 35° and 45°, the remaining part is 150 − (35 + 45) = 70°.

Straight angles and full turns provide useful whole measures. Two adjacent parts that exactly fill a straight opening total 180°. If one is 65°, the other is 115°. If three nonoverlapping turns exactly complete a circle and two measure 90° and 140°, the remaining turn measures 360 − 230 = 130°. The diagram must show that the parts actually fill the stated whole; a familiar-looking picture is not enough.

Overlapping angles cannot simply be added to find their union. Imagine one ray at zero degrees, another at twenty degrees, another at fifty degrees and another at seventy degrees. The zero-to-fifty angle measures fifty degrees and the twenty-to-seventy angle also measures fifty. They overlap from twenty to fifty. Adding their measures counts that region twice; the total zero-to-seventy opening is seventy degrees, not one hundred.

Connect the diagram to a situation. A hinged panel turns 30° from its starting direction, then another 45° in the same direction. Its total turn is 75°. If it then turns back 20°, its final opening from the start is 55°. The direction of each action matters. Mark the successive directions and identify the opening asked for before writing an equation.

Worked missing part: A straight angle is divided into 70° and c degrees. Because the parts fill the 180° opening without overlap, 70 + c = 180 and c = 110°. Check that seventy plus one hundred ten rebuilds the straight angle.

Worked three-part angle: An opening totals 140°. Two parts are 25° and 60°. Together they account for 85°, so the remaining part is 140 − 85 = 55°. Draw all three around the same vertex and label the complete opening as well as its parts.

Guided practice: Draw a right angle and add a ray inside it. Measure both parts with a protractor, compare their sum with ninety and discuss small tool-reading differences. Then use exact labeled diagrams to solve missing-angle equations without relying on approximate drawing measurements.

Independent evidence: Solve an unknown part in a right angle, an unknown part in an obtuse angle and a three-part problem. Supply equations and addition checks. Explain why an overlapping pair cannot be summed directly, and distinguish an exact given label from an approximate visual estimate.