Angle Sum of a Quadrilateral

\( A+B+C+D=360^{\circ} \)
Geometry GCSE
Question 11 of 20
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\( In ABCD, A = 4x°, B = (2x + 10)°, C = (x + 20)°, D = (3x − 10)°. Find x. \)

Tips: use ^ for powers, sqrt() for roots, and type pi for π.
Hint (H)
Add and set equal to 360°.

Explanation

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Four interior angles add to 360°.

Worked examples

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  1. \( In quadrilateral ABCD, A=92°, B=108°, C=87°. Find D. \)
    1. \( Angle sum in a quadrilateral: A + B + C + D = 360°. \)
    2. \( So D = 360° − (92° + 108° + 87°) = 360° − 287° = 73°. \)
    Answer: 73°
  2. \( In a quadrilateral, A=100°, B=70°, C=95°. Find D. \)
    1. \( Use A + B + C + D = 360°. \)
    2. \( D = 360° − (100° + 70° + 95°) = 95°. \)
    Answer: 95°
  3. \( In quadrilateral ABCD, A=(2x+10)°, B=(3x−5)°, C=95°, D=85°. Find x. \)
    1. \( Sum to 360°: (2x+10) + (3x−5) + 95 + 85 = 360. \)
    2. \( 5x + 185 = 360 ⇒ 5x = 175 ⇒ x = 35. \)
    Answer: \( x = 35 \)
  4. A right-angled corner in a quadrilateral is 90°, and the other two angles are 128° and 74°. Find the fourth angle.
    1. \( Total = 360°. \)
    2. \( Fourth angle = 360° − (90° + 128° + 74°) = 68°. \)
    Answer: 68°
  5. A rectangle is a quadrilateral with all angles equal. What is each angle?
    1. \( A + B + C + D = 360° and all four are equal. \)
    2. \( Each angle = 360° ÷ 4 = 90°. \)
    Answer: 90° each
  6. \( In quadrilateral ABCD, A=75°, B=90°, C=130°. Find D. \)
    1. \( Use A + B + C + D = 360°. \)
    2. \( D = 360° − (75° + 90° + 130°) = 65°. \)
    Answer: 65°
  7. \( In quadrilateral ABCD, A=(3x+12)°, B=(2x+8)°, C=(4x+30)°, D=(x+10)°. Find x. \)
    1. \( Sum to 360°: (3x+12)+(2x+8)+(4x+30)+(x+10)=360. \)
    2. \( 10x + 60 = 360 ⇒ x = 30. \)
    Answer: \( x = 30 \)
  8. A parallelogram has one interior angle 112°. Find the other three angles.
    1. Adjacent angles in a parallelogram are supplementary (sum 180°).
    2. So the angle next to 112° is 68°.
    3. Opposite angles are equal, so the remaining two are 112° and 68°.
    4. \( Check with the quadrilateral sum: 112°+68°+112°+68°=360°. \)
    Answer: Angles: 112°, 68°, 112°, 68°
  9. \( Even if ABCD is concave, the interior angles still sum to 360°. If A=40°, B=120°, C=150°, find D. \)
    1. Angle sum property holds for any quadrilateral (convex or concave).
    2. \( D = 360° − (40° + 120° + 150°) = 50°. \)
    Answer: 50°
  10. \( In quadrilateral ABCD, A=2x+10°, B=3x°, C=4x°, D=80°. Find x. \)
    1. \( Sum to 360°: (2x+10) + 3x + 4x + 80 = 360. \)
    2. \( 9x + 90 = 360 ⇒ 9x = 270 ⇒ x = 30. \)
    Answer: \( x = 30 \)