Grade 4 Quadrilaterals and Parallel Lines

What this problem type covers

Nineteen problem types across five levels: three lines meeting at a point with two of them perpendicular, the distance across three parallel lines, two inside angles that add to 180, a bend between two parallels, three squares joined in a step, four rectangles making a square with a square hole, a parallelogram with an isosceles triangle on its side, a parallelogram with a triangle on its base, a parallelogram joined to a rhombus, a bent line between parallels with a perpendicular dropped in, two perpendiculars from a point inside a parallelogram, two lines crossing between parallels, folded rectangular paper, a trapezoid with two isosceles triangles, a zigzag with two turns, counting the rectangles that contain a star, counting the sizes of square a dot array allows, folded rhombus paper, and the difference between two angles a triangle makes across parallels.

Example problems

One example from each difficulty band, with worked solutions.

These problems are generated automatically. The answers are derived as each problem is built and are checked by tests, but mistakes are still possible — please tell us if you find one.

1. 1 · ★1
   Lines 가, 나 and 다 are all parallel. How far apart are 가 and 다?
   
   • ①35 cm
   • ②39 cm
   • ③31 cm
   • ④43 cm
   • ⑤11 cm
   Show solution

   ③31 cm

   1. The distance between two parallel lines is measured along a perpendicular.
   2. 가 to 나 is 10 cm and 나 to 다 is 21 cm.
   3. So 가 to 다 is 10 + 21 = 31 cm.
   4. A length measured at a slant is not the distance between the lines.
2. 2 · ★2
   Four identical rectangles are joined without overlapping. What is the difference between the perimeter of the outer square and that of the inner one?
   
   • ①12 cm
   • ②96 cm
   • ③72 cm
   • ④48 cm
   • ⑤24 cm
   Show solution

   ④48 cm

   1. The outer square has sides of 6 + 12 cm, so its perimeter is 72 cm.
   2. The inner square has sides of 12 − 6 cm, so its perimeter is 24 cm.
   3. The difference is 72 − 24 = 48 cm.
3. 3 · ★3
   Lines 가 and 나 are parallel. How big is ㉮?
   
   • ①67 degrees
   • ②35 degrees
   • ③113 degrees
   • ④99 degrees
   • ⑤58 degrees
   Show solution

   ②35 degrees

   1. Drop a perpendicular from one line to the other to make a quadrilateral.
   2. A straight line is 180, so the angle beside the perpendicular is 90 − 32.
   3. A quadrilateral closes on 360, so the remaining angle is 360 − (90 − 32) − 90 − 67.
   4. A straight line is 180, so ㉮ is 67 − 32 = 35 degrees.
4. 4 · ★4
   ㄱㄹ is parallel to ㄴㄷ, and ㄱㄴ, ㄱㅁ and ㄱㄹ are all the same length. How big is ㉮?
   
   • ①64 degrees
   • ②108 degrees
   • ③54 degrees
   • ④55 degrees
   • ⑤72 degrees
   Show solution

   ③54 degrees

   1. ㄱㄴㅁ is isosceles, so angle ㄱㅁㄴ is 72 degrees as well.
   2. Alternate angles are equal, so angle ㄹㄱㅁ is 72 too.
   3. ㄱㅁㄹ is isosceles as well, so ㉮ is (180 − 72) ÷ 2 = 54.
   4. A straight line is 180, so angle ㄹㅁㄷ follows and ㉯ is 55 degrees.
5. 5 · ★5
   A rhombus of paper is folded as shown. How big is angle ㄱㄴㄹ?
   
   • ①31 degrees
   • ②66 degrees
   • ③45 degrees
   • ④21 degrees
   • ⑤69 degrees
   Show solution

   ④21 degrees

   1. Neighbouring angles of a rhombus add to 180, so the folded corner is 180 − 111 = 69.
   2. The angles of a triangle add to 180, so the remaining angle is 180 − 69 − 66.
   3. A folded angle equals the angle it covers, so that angle appears twice.
   4. Taking both away from 111 leaves angle ㄱㄴㄹ as 21 degrees.

Where people usually go wrong

• Measuring between parallel lines along a slant rather than along a perpendicular.
• Answering with the larger of two angles that add to 180.
• Using 180 degrees where a full turn of 360 is needed.
• Not halving after the folded angle and the angle it covers are found to be equal.
• Taking a folded angle away once when it appears on both sides of the crease.
• Adding the angle carried across a parallel instead of taking it off.
• Counting only across, or only down, when counting rectangles round a star.
• Missing the tilted squares in a dot array.