105. Quadrilaterals and Their Angle Properties

Learning Intentions

• To know the meaning of the terms convex and non-convex
• classify quadrilaterals as parallelograms, rectangles, rhombuses, squares, kites and/or trapeziums
• use the angle sum of a quadrilateral to find unknown angles
• To understand that properties of angles in parallel lines can be used to find unknown angles in trapeziums and parallelograms

Pre-requisite Summary

• Know that a quadrilateral is a polygon with $4$ sides
• Recall that interior angles are the angles inside a shape
• Know that the interior angles of parallel lines can form corresponding, alternate and co-interior angle pairs
• Know that co-interior angles on parallel lines add to ${180}^{\circ }$
• Be able to identify equal sides, right angles and parallel sides from a diagram
• Recall that the angle sum of a quadrilateral is ${360}^{\circ }$
• Know that a convex shape has all interior angles less than ${180}^{\circ }$
• Know that a non-convex shape has at least one interior angle greater than ${180}^{\circ }$

Worked Examples

Worked Example 1

State whether each quadrilateral is convex or non-convex:
a) a quadrilateral with all interior angles less than ${180}^{\circ }$
b) a quadrilateral with one interior angle of ${220}^{\circ }$

Worked Example 2

Classify each quadrilateral:
a) a quadrilateral with two pairs of parallel sides
b) a quadrilateral with four right angles and opposite sides equal
c) a quadrilateral with four equal sides and four right angles

Worked Example 3

Classify each quadrilateral:
a) a quadrilateral with four equal sides but not all angles right angles
b) a quadrilateral with two pairs of adjacent equal sides
c) a quadrilateral with one pair of parallel sides

Worked Example 4

Use the angle sum of a quadrilateral to find the unknown angle:
a) angles are ${85}^{\circ }$, ${110}^{\circ }$, ${92}^{\circ }$ and $x$
b) angles are ${90}^{\circ }$, ${90}^{\circ }$, ${118}^{\circ }$ and $y$

Worked Example 5

A parallelogram has one interior angle of ${68}^{\circ }$. Find:
a) an adjacent interior angle
b) the opposite interior angle

Worked Example 6

A trapezium has parallel sides. Two interior angles on the same side of a transversal are $x$ and ${123}^{\circ }$. Find $x$.

Problems

Problem 1

State whether each quadrilateral is convex or non-convex:
a) a quadrilateral with interior angles ${90}^{\circ }$, ${95}^{\circ }$, ${85}^{\circ }$ and ${90}^{\circ }$
b) a quadrilateral with one interior angle of ${190}^{\circ }$

Problem 2

Classify each quadrilateral:
a) a quadrilateral with two pairs of parallel sides
b) a quadrilateral with four right angles and all sides equal
c) a quadrilateral with four right angles

Problem 3

Classify each quadrilateral:
a) a quadrilateral with four equal sides
b) a quadrilateral with two pairs of adjacent equal sides
c) a quadrilateral with exactly one pair of parallel sides

Problem 4

Use the angle sum of a quadrilateral to find the unknown angle:
a) angles are ${102}^{\circ }$, ${76}^{\circ }$, ${94}^{\circ }$ and $x$
b) angles are ${90}^{\circ }$, ${73}^{\circ }$, ${90}^{\circ }$ and $y$

Problem 5

A parallelogram has one interior angle of ${112}^{\circ }$. Find:
a) an adjacent interior angle
b) the opposite interior angle

Problem 6

A trapezium has parallel sides. Two interior angles on the same side of a transversal are $m$ and ${97}^{\circ }$. Find $m$.

Exercises

Understanding and Fluency

1. State whether each quadrilateral is convex or non-convex:
   a) all interior angles are less than ${180}^{\circ }$
   b) one interior angle is ${205}^{\circ }$
   c) one interior angle is reflex

2. Classify each quadrilateral:
   a) two pairs of parallel sides
   b) four right angles
   c) four equal sides
   d) two pairs of adjacent equal sides

3. Name the most specific quadrilateral:
   a) four equal sides and four right angles
   b) one pair of parallel sides
   c) opposite sides parallel and equal
   d) four equal sides, no right angle stated

4. Use the angle sum of a quadrilateral to find each unknown angle:
   a) ${70}^{\circ }$, ${95}^{\circ }$, ${105}^{\circ }$, $x$
   b) ${88}^{\circ }$, ${92}^{\circ }$, ${91}^{\circ }$, $y$
   c) ${90}^{\circ }$, ${90}^{\circ }$, ${90}^{\circ }$, $z$

5. Find the missing angle in each quadrilateral:
   a) ${120}^{\circ }$, ${80}^{\circ }$, ${70}^{\circ }$, $a$
   b) ${96}^{\circ }$, ${104}^{\circ }$, ${83}^{\circ }$, $b$
   c) ${110}^{\circ }$, ${110}^{\circ }$, ${70}^{\circ }$, $c$

6. In a parallelogram, one angle is ${58}^{\circ }$. Find:
   a) the opposite angle
   b) an adjacent angle
   c) the other adjacent angle

7. In a parallelogram, one angle is ${126}^{\circ }$. Find:
   a) the opposite angle
   b) an adjacent angle
   c) all four angles

8. In a trapezium, consecutive interior angles between the parallel sides are co-interior. Find each unknown:
   a) $d+{116}^{\circ }={180}^{\circ }$
   b) $e+{74}^{\circ }={180}^{\circ }$
   c) $f+{139}^{\circ }={180}^{\circ }$

Reasoning

9. Explain why a square can also be called a rectangle and a rhombus.

10. A student says that every trapezium is a parallelogram because both shapes have parallel sides. Explain the mistake.

11. Noah says that a rhombus must have four right angles. Is he correct? Explain.

12. Explain why adjacent angles in a parallelogram are supplementary.

13. A student says a quadrilateral with one interior angle greater than ${180}^{\circ }$ is convex. Describe the error.

Problem-solving

14. A quadrilateral has interior angles ${85}^{\circ }$, ${95}^{\circ }$, ${90}^{\circ }$ and $x$. Find $x$ and state whether the quadrilateral is convex or non-convex.

15. A window frame is shaped like a rectangle. Three interior angles are right angles. Find the fourth angle.

16. In a parallelogram, one interior angle is ${72}^{\circ }$. Find the other three angles.

17. A trapezium has one pair of parallel sides. One angle on a transversal is ${108}^{\circ }$. Find the co-interior angle on the same side.

18. A quadrilateral has side properties showing two pairs of adjacent equal sides. Classify the quadrilateral and state whether this information guarantees any parallel sides.

19. A shape has four equal sides and one interior angle of ${65}^{\circ }$. Classify the quadrilateral and find the other three interior angles.

Potential Misunderstandings

• Students may think convex means symmetrical, rather than having all interior angles less than ${180}^{\circ }$
• Students may think any quadrilateral with parallel sides is a parallelogram
• Students may confuse a kite with a rhombus
• Students may think a rectangle must have all sides equal
• Students may think a rhombus must have four right angles
• Students may not realise that a square belongs to more than one quadrilateral family
• Students may forget that the interior angles of a quadrilateral add to ${360}^{\circ }$
• Students may confuse opposite angles with adjacent angles in a parallelogram
• Students may forget that adjacent angles in a parallelogram add to ${180}^{\circ }$
• Students may not recognise co-interior angles inside trapeziums when parallel sides are present
• Students may rely on the appearance of a diagram instead of the stated properties
• Students may think non-convex quadrilaterals are impossible because they “bend inward”