Interior and Exterior Angles of Triangles and Polygons

Learning Intentions

• To know that the sum of interior angles in a triangle is ${180}^{\circ }$
• To know what an exterior angle is
• find exterior angles in triangles using supplementary angles
• To understand that the angle sum of a polygon can be determined by decomposing into triangles

Pre-requisite Summary

• Understand that a triangle has $3$ interior angles
• Know that angles on a straight line add to ${180}^{\circ }$
• Be able to identify adjacent angles in a diagram
• Understand that an exterior angle is formed outside a shape when a side is extended
• Be able to find a missing angle in a triangle using known angle facts
• Understand that a polygon can be split into triangles by drawing diagonals from one vertex
• Know that each triangle contributes ${180}^{\circ }$ to an interior angle sum

Worked Examples

Worked Example 1

a) Find the missing angle in a triangle with angles ${63}^{\circ }$ and ${48}^{\circ }$.
b) Explain why the interior angles add to ${180}^{\circ }$.

Worked Example 2

A triangle has interior angles ${35}^{\circ }$, $x$, and ${92}^{\circ }$.
a) Write an equation using the triangle angle sum.
b) Find $x$.

Worked Example 3

A side of a triangle is extended to form an exterior angle. The adjacent interior angle is ${118}^{\circ }$.
a) Explain what an exterior angle is.
b) Find the exterior angle using supplementary angles.

Worked Example 4

A triangle has an interior angle of ${47}^{\circ }$ next to an exterior angle labelled $y$.
a) Write an equation using supplementary angles.
b) Find $y$.

Worked Example 5

A quadrilateral is decomposed into triangles by drawing one diagonal.
a) State how many triangles are formed.
b) Find the interior angle sum of the quadrilateral.

Worked Example 6

A pentagon is decomposed into triangles by drawing diagonals from one vertex.
a) State how many triangles are formed.
b) Find the interior angle sum of the pentagon.
c) Explain the rule used.

Problems

Problem 1

a) Find the missing angle in a triangle with angles ${54}^{\circ }$ and ${71}^{\circ }$.
b) Explain why the interior angles add to ${180}^{\circ }$.

Problem 2

A triangle has interior angles ${28}^{\circ }$, $x$, and ${104}^{\circ }$.
a) Write an equation using the triangle angle sum.
b) Find $x$.

Problem 3

A side of a triangle is extended to form an exterior angle. The adjacent interior angle is ${126}^{\circ }$.
a) Explain what an exterior angle is.
b) Find the exterior angle using supplementary angles.

Problem 4

A triangle has an interior angle of ${59}^{\circ }$ next to an exterior angle labelled $y$.
a) Write an equation using supplementary angles.
b) Find $y$.

Problem 5

A quadrilateral is decomposed into triangles by drawing one diagonal.
a) State how many triangles are formed.
b) Find the interior angle sum of the quadrilateral.

Problem 6

A hexagon is decomposed into triangles by drawing diagonals from one vertex.
a) State how many triangles are formed.
b) Find the interior angle sum of the hexagon.
c) Explain the rule used.

Exercises

Understanding and Fluency

1. Find the missing angle in each triangle:
   a) ${40}^{\circ },{65}^{\circ },x$
   b) ${72}^{\circ },{51}^{\circ },x$
   c) ${90}^{\circ },{36}^{\circ },x$

2. Find the missing angle in each triangle:
   a) ${33}^{\circ },{79}^{\circ },x$
   b) ${105}^{\circ },{42}^{\circ },x$
   c) ${58}^{\circ },{58}^{\circ },x$

3. Find each exterior angle using supplementary angles:
   a) interior angle ${120}^{\circ }$
   b) interior angle ${47}^{\circ }$
   c) interior angle ${88}^{\circ }$

4. Find each exterior angle using supplementary angles:
   a) interior angle ${135}^{\circ }$
   b) interior angle ${62}^{\circ }$
   c) interior angle ${91}^{\circ }$

5. A triangle has an interior angle and an adjacent exterior angle. Find the exterior angle:
   a) interior angle ${39}^{\circ }$
   b) interior angle ${114}^{\circ }$
   c) interior angle ${76}^{\circ }$

6. Decompose each polygon into triangles and find its interior angle sum:
   a) quadrilateral
   b) pentagon
   c) hexagon

7. Decompose each polygon into triangles and find its interior angle sum:
   a) heptagon
   b) octagon
   c) nonagon

8. Mixed practice:
   a) Find the third angle of a triangle with angles ${84}^{\circ }$ and ${49}^{\circ }$
   b) Find the exterior angle next to an interior angle of ${103}^{\circ }$
   c) Find the interior angle sum of a pentagon

Reasoning

9. Explain why the interior angles of a triangle always add to ${180}^{\circ }$.

10. A student says an exterior angle is any angle outside a polygon. Explain what is missing from this description.

11. Explain why an interior angle and its adjacent exterior angle in a triangle are supplementary.

12. A student says a pentagon can be decomposed into $5$ triangles from one vertex. Explain the mistake.

13. Explain why the interior angle sum of a hexagon is greater than that of a quadrilateral.

14. A student finds an exterior angle by subtracting from ${360}^{\circ }$. Explain why this is incorrect in this context.

Problem-solving

15. A triangular road sign has angles ${48}^{\circ }$, ${67}^{\circ }$, and $x$. Find $x$.

16. A triangle has an interior angle of ${125}^{\circ }$ next to an exterior angle. Find the exterior angle.

17. A builder draws a pentagonal tile and wants to know the sum of its interior angles. Decompose it into triangles and find the sum.

18. A hexagonal window frame is split into triangles from one vertex. How many triangles are formed and what is the sum of the interior angles?

19. A triangle has interior angles $2$ equal angles and one angle of ${44}^{\circ }$. Find the two equal angles.

20. A quadrilateral and a pentagon are both decomposed into triangles. Compare their interior angle sums.

Potential Misunderstandings

• Students may think the interior angles of any polygon add to ${180}^{\circ }$ rather than only a triangle
• Students may confuse an exterior angle with a non-adjacent outside angle
• Students may forget that the exterior angle considered here forms a straight line with the interior angle
• Students may subtract from ${360}^{\circ }$ instead of ${180}^{\circ }$ when finding an adjacent exterior angle
• Students may count the number of triangles formed in a polygon incorrectly
• Students may think the number of triangles in a decomposed polygon is the same as the number of sides
• Students may forget that diagonals for decomposition are drawn from one vertex to non-adjacent vertices
• Students may find the number of triangles correctly but not multiply by ${180}^{\circ }$ to get the interior angle sum

Next: 064. Line and Rotational Symmetry