Reasoning · Grade 4-1 Triangles and Angles

Problem

Angles from overlapping right-triangle set squares

Two set squares overlap with their bases on the same horizontal line. The left one's slanted edge meets the base at 30 degrees. The right one's slanted edge meets the base at 45 degrees. Find angle 1 where the two slanted edges cross.
45° 30° 1
Your answer
How to solve
Strategy Draw a Diagram — The overlapping set squares look tangled, so the first job is to see past the shading and pick out just the lines that matter: the base line and the two slanted edges. Those three lines by themselves make one plain triangle whose two base angles are the given 30° and 45°. Once that hidden triangle is separated out as a subproblem, the answer is two short subtractions.
1STEP 1

Strip the picture down to three lines

Reduce the picture to two slants and the base.

2STEP 2

Fill in the two base angles of that triangle

That triangle's base angles are 45 and 30 degrees.

∠ PLR = 45°, ∠ PRL = 30°
3STEP 3

Find the angle at the crossing point inside the triangle

The apex angle is 180 − 75 = 105 degrees.

∠ LPR = 180° - (45° + 30°) = 180° - 75° = 105°
4STEP 4

Swing from the downward angle to the leftward angle

Angle 1 is its neighbour: 180 − 105 = 75 degrees.

① = 180° - 105° = 75°
Answer
75 degrees
180 − 105 = 75
75° is acute, and the arc in the figure is clearly narrower than a right angle, so the size matches the picture. It is also worth noticing that ① is the exterior angle of triangle LPR at P, and 45° + 30° = 75° is exactly the sum of the two far-away interior angles — the exterior angle rule confirms the answer in one line. Finally, the four angles at the crossing point are 105°, 75°, 105°, 75°, which add to 360° as angles around a point must.
Takeaway

Rub out every line that does not touch the marked angle — under two overlapping set squares there was just one ordinary triangle hiding all along.

  • Strip the picture down to three lines
  • Fill in the two base angles of that triangle
  • Find the angle at the crossing point inside the triangle
  • Swing from the downward angle to the leftward angle