Set up an equation with one unknown angle

4.MD.C.7 · take · grade 4

Archetype: Find Two Unknowns from Sum and Difference · step in a 8-type progression

Angle $\angle BOC$ is $30^\circ$ larger than angle $\angle COD$ . Using the figure, find the measure of angle $\angle COD$ .

[Figure] Straight line $AD$ passes through point $O$ , and two rays $OB$ and $OC$ extend upward from $O$ . Starting from the left end of the line, the angles $\angle AOB = 40^\circ$ , $\angle BOC$ , and $\angle COD$ lie in order, so the three angles together form the straight angle $180^\circ$ .

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Understand

On straight line AD through point O, rays OB and OC go upward. From the left, the angles AOB = 40 degrees, BOC, and COD line up and together make the straight angle of 180 degrees. Also angle BOC is 30 degrees bigger than angle COD. Find angle COD.

Givens

Angle AOB = 40 degrees.
Angles AOB, BOC, COD sit in order along straight line AD, so they add to 180 degrees.
Angle BOC = angle COD + 30 degrees.

Unknowns

The measure of angle COD.

Constraints

Angles on a straight line add to 180 degrees.
BOC is exactly 30 degrees larger than COD.

Plan

#6 Guess and Check · also uses: #7 Identify Subproblems

First peel off the known 40 degrees to find how much BOC and COD share. Then, since BOC is 30 more than COD, take that extra 30 off the shared total and split the rest into the two equal parts; guess-and-check confirms COD.

Execute

#7 Identify Subproblems 4.MD.C.7

The three angles fill the 180-degree straight line, so BOC plus COD is the part left after the 40-degree angle.

\angle BOC + \angle COD = 180^\circ - 40^\circ = 140^\circ

A flat line is 180 degrees; remove the known piece and the other two pieces share the rest.

#6 Guess and Check 4.MD.C.7

BOC is COD plus 30. If we set aside that extra 30 degrees from the 140, the leftover 110 degrees splits into two equal COD-sized parts. Half of 110 is the measure of COD.

(140^\circ - 30^\circ) \div 2 = 110^\circ \div 2 = 55^\circ

Two parts differ by 30; take the extra away so they match, share equally, and that equal part is the smaller angle.

Answer: 55 degrees

Review

If COD = 55 degrees, then BOC = 85 degrees. Check the line: 40 + 85 + 55 = 180 degrees, and 85 is exactly 30 more than 55. Everything fits.

Convert to algebra (tool 13): let COD = x, then BOC = x + 30, and 40 + (x + 30) + x = 180, giving 2x = 110, x = 55 degrees.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Using the 180-degree straight-line total and the 30-degree difference to find COD.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the rays OB, OC and the line AD through point O.

💡 Take the known angle off the 180-degree line, set aside the extra 30, then share what is left in two - all Grade 4 angle adding and subtracting!