Sensim Math · Depth 한국어

4-1 · Angles

Use the 180-degree line to find figure angles

4.MD.C.7 · take · grade 4

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

▶ Practice — 8 problems

Find the measure of angle ㉠.

60° 100° x°
Show solution

Understand

A four-sided figure stands on a straight line. Three of its inside corners are known: top 60 degrees, upper-left 100 degrees, and bottom-right 90 degrees (a right angle on the line). Angle marked is the angle outside the bottom-left corner, between the figure's left side and the line.

Givens
  • The figure is a quadrilateral (4 corners), so its inside angles add to 360 degrees.
  • Top inside angle = 60 degrees.
  • Upper-left inside angle = 100 degrees.
  • Bottom-right corner is a right angle = 90 degrees.
  • The marked angle and the bottom-left inside angle together lie along the straight line, so they add to 180 degrees.
Unknowns
  • The measure of the marked angle (between the left side and the line).
Constraints
  • All four inside angles of the quadrilateral total 360 degrees.
  • A straight angle is 180 degrees.

Plan

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

First subproblem: find the missing fourth inside angle using the 360-degree quadrilateral total. Second subproblem: the marked angle is what is left of the 180-degree straight angle after the inside corner, so subtract.

Execute

#7 Identify Subproblems 4.MD.C.7
The four inside angles of a quadrilateral add to 360 degrees. Subtract the three known ones to get the bottom-left inside angle.
360∘−60∘−100∘−90∘=110∘360^\circ - 60^\circ - 100^\circ - 90^\circ = 110^\circ
The whole turn around a quadrilateral's corners is fixed, so the leftover after removing three corners must be the fourth.
#7 Identify Subproblems 4.MD.C.7
At the bottom-left corner the inside angle (110 degrees) and the marked angle sit side by side along the straight line, so together they make 180 degrees. Subtract to get the marked angle.
180∘−110∘=70∘180^\circ - 110^\circ = 70^\circ
A flat line is a straight angle of 180 degrees; the two angles on it must fill exactly that.
Answer: 70 degrees

Review

60 + 100 + 90 + 110 = 360 degrees, a valid quadrilateral. And 110 + 70 = 180 degrees fills the straight line. The marked angle 70 degrees is acute, which matches a small wedge between a leaning side and the floor.

Draw the diagram (tool 1) and instead first split the quadrilateral mentally: the marked exterior angle equals the sum of the two remote inside angles is not used here, but you can equally check by measuring with a protractor that 70 degrees fits.

Standards · min grade 4

  • 4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Adding and subtracting angle measures with 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, its corners, and the line in the figure.
💡 Find the missing corner with the 360-degree rule, then take it away from the straight line's 180 degrees - just Grade 4 add-and-subtract-angles!