NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Angles on parallel lines

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In a mechanical design for a retractable cherry picker in Canberra, two parallel vertical support beams are connected by a diagonal hydraulic arm that acts as a transversal. The angle formed between the hydraulic arm and the upper support beam on the left side measures 117 degrees.

What is the exact measure, in degrees, of the alternate interior angle formed between the hydraulic arm and the lower support beam on the opposite side of the transversal? Answer as a single exact value.

Concept illustrations

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Adjacent angles on a straight line. The straight angle is 180°. The ray splits it into 65° and 115°, so the two adjacent angles sum to 180°. Drawing is schematic.65°115°65° + 115° = 180°
The straight angle is 180°. The ray splits it into 65° and 115°, so the two adjacent angles sum to 180°. Drawing is schematic.
Triangulating a pentagon. Two diagonals from one vertex split this pentagon into three triangles. Its interior-angle sum is 3 × 180° = 540°.(5 − 2) × 180° = 540°
Two diagonals from one vertex split this pentagon into three triangles. Its interior-angle sum is 3 × 180° = 540°.

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Explanation

Worked solution and answer (click to reveal)

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Worked solution

  1. Two parallel vertical support beams are connected by a diagonal hydraulic arm, which acts as a transversal crossing the parallel lines.
  2. The angle between the hydraulic arm and the upper support beam on the left side is given as 117 degrees.
  3. Since the two support beams are parallel and the hydraulic arm is a transversal, the alternate interior angles formed are equal.
  4. The angle on the upper beam (117°) and the alternate interior angle on the lower beam are on opposite sides of the transversal and inside the parallel lines.
  5. By the property of alternate interior angles formed by a transversal crossing two parallel lines, these angles are congruent.
  6. Therefore, the alternate interior angle on the lower beam is equal in measure to the given angle of 117 degrees.
  7. No conversion or calculation is needed because the angles are directly equal by geometric rule.
  8. The measure of the alternate interior angle is exactly 117 degrees.
  9. This result is consistent with the geometric principle that alternate interior angles are equal when lines are parallel.
  10. The location (Canberra) does not affect the angle measure, as the property holds regardless of location.
  11. The answer is confirmed: the alternate interior angle is 117 degrees.
  12. Final answer: 117 degrees.

Reference answer: 117

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