Small angles combine into larger angles

4.MD.C.74.MD.C.5 · take · grade 4

Archetype: Angle Facts in a Figure · step in a 13-type progression

Find how many acute angles, large and small, can be found in the figure.

Show solution

Understand

Several rays come out of one point on a straight line, making 5 small acute angles in a row labeled 1-5. By joining neighboring small angles we get bigger angles too. We must count every acute angle (small and large) that appears in the figure.

Givens

5 small angles in a row, labeled 1, 2, 3, 4, 5, sit between neighboring rays above the base line.
Each of the 5 small angles is acute (less than 90 degrees).
The 5 small angles together fill the half-turn above the line, so they add up to 180 degrees.
Neighboring small angles can be combined into a single larger angle.

Unknowns

How many of the angles formed (single or combined) are acute.

Constraints

Only acute angles (strictly less than 90 degrees) are counted.
A combined angle must use neighboring small angles with no gaps.

Plan

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

List every angle you can build from consecutive small angles in an orderly way (length 1, then 2, then 3...). Sort them into groups by how many small angles they contain, then keep only the ones small enough to still be acute.

Execute

#2 Make a Systematic List 4.MD.C.5

The 5 angles 1, 2, 3, 4, 5 each stand alone. The problem tells us all 5 are acute, so all 5 single angles count.

5 \text{ single acute angles}

An angle is the opening between two rays; each gap between neighboring rays is one such angle.

#2 Make a Systematic List 4.MD.C.7

Joining two neighboring small angles gives the pairs (1+2), (2+3), (3+4), (4+5): 4 larger angles. Since the 5 small angles fill 180 degrees and each is small, two neighbors still open less than 90 degrees, so each of these 4 is acute.

(1{+}2),(2{+}3),(3{+}4),(4{+}5) \rightarrow 4 \text{ acute angles}

Angle measure adds: two angles laid side by side make one angle whose size is their sum.

#7 Identify Subproblems 4.MD.C.7

Three-in-a-row angles are (1+2+3), (2+3+4), (3+4+5): 3 of them. Four-in-a-row are (1+2+3+4), (2+3+4+5): 2 of them. All five together is 1 angle equal to the whole 180-degree straight angle. As we combine more small angles the opening grows past 90 degrees, so these longer combinations are no longer acute and are not counted.

3 + 2 + 1 = 6 \text{ angles that are 90}^\circ \text{ or more}

Keep adding pieces of the 180-degree half-turn and the angle eventually reaches a right angle and beyond, so it stops being acute.

#2 Make a Systematic List 4.MD.C.7

Acute angles are the 5 singles plus the 4 pairs.

5 + 4 = 9

Just total up the groups we kept.

Answer: 9 acute angles

Review

There are 15 angles in all (5 + 4 + 3 + 2 + 1). The longer 6 combinations open toward and past a right angle, so it makes sense that only the 9 shortest ones stay acute. 9 is less than 15, as it must be.

Guess and check (tool 6): sketch the rays roughly equally spaced (each small angle about 36 degrees). Singles ~36 degrees and pairs ~72 degrees are under 90 degrees, but triples ~108 degrees are over 90 degrees, confirming 5 + 4 = 9 acute angles.

Standards · min grade 4

4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Identifying each opening between neighboring rays as an angle.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Combining neighboring small angles and judging whether the total stays acute.

💡 List the angles by how many pieces they use, then keep only the ones still smaller than a right angle - that is Grade 4 angle sense you already have!