Quadrilateral angles sum to 360 degrees

4.MD.C.7 · take · grade 4

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

Using the fact that the four angles of a quadrilateral add up to $360^\circ$ , find the measure of angle $a$ in the figure.

[Figure] A quadrilateral rests at an angle with one vertex sitting on a straight line. Of its four interior angles, the top angle is $100^\circ$ , the left angle is $80^\circ$ , the right angle is $70^\circ$ , and the bottom vertex's interior angle is labeled $b$ . That bottom vertex lies on the straight line, and along the line — from left to right — are a $45^\circ$ angle, the quadrilateral's interior angle $b$ , and angle $a$ , which together form the straight angle $180^\circ$ .

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Understand

A quadrilateral leans with its bottom vertex on a straight line. Three inside angles are 100, 80, and 70 degrees; the bottom inside angle is b. Along the line, from left to right, sit a 45-degree angle, the inside angle b, and angle a, and these three fill the straight angle of 180 degrees. Find angle a.

Givens

Quadrilateral inside angles: top 100, left 80, right 70, bottom = b.
A quadrilateral's four inside angles add to 360 degrees.
On the straight line the three angles 45 degrees, b, and a add to 180 degrees.

Unknowns

The measure of angle a.
(Helper) the bottom inside angle b.

Constraints

Four interior angles total 360 degrees.
Angles along a straight line total 180 degrees.

Plan

#7 Identify Subproblems · also uses: #1 Draw a Diagram

Subproblem 1: find b from the 360-degree quadrilateral rule. Subproblem 2: subtract 45 and b from the straight line's 180 degrees to get a.

Execute

#7 Identify Subproblems 4.MD.C.7

The four inside angles add to 360 degrees, so subtract the three known angles.

b = 360^\circ - 100^\circ - 80^\circ - 70^\circ = 110^\circ

The full set of quadrilateral corners is fixed at 360, so the unknown corner is whatever is left.

#7 Identify Subproblems 4.MD.C.7

The three angles 45 degrees, b = 110 degrees, and a lie along the straight line and add to 180 degrees. Subtract the two known ones.

a = 180^\circ - 45^\circ - 110^\circ = 25^\circ

The flat line is 180 degrees split into three pieces; the last piece is the remainder.

Answer: 25 degrees

Review

Check the quadrilateral: 100 + 80 + 70 + 110 = 360 degrees. Check the line: 45 + 110 + 25 = 180 degrees. Both totals are exact, so a = 25 degrees is consistent.

Work backwards (tool 11): the line piece not taken by 45 and b is 180 - 45 - 110, which directly gives the leftover 25 degrees for a.

Standards · min grade 4

4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Subtracting known angles from the 360-degree and 180-degree totals.
4.G.A.1 Draw points, lines, line segments, rays, angles, and identify in figures — Reading the quadrilateral's corners and the angles on the line.

💡 First the 360-degree rule gives the missing corner, then the 180-degree straight line gives angle a - two friendly Grade 4 subtractions!