104r. Reviewing Classifying Triangles and Finding Angles

Learning Intentions

• To understand that triangles can be classified by their side lengths as scalene, isosceles or equilateral
• To understand that triangles can be classified by their interior angles as acute, right or obtuse
• use the angle sum of a triangle to find unknown angles
• use the exterior angle theorem to find unknown angles

Pre-requisite Summary

• Know that a triangle has $3$ sides, $3$ vertices and $3$ interior angles
• Understand that side lengths can be compared to decide whether sides are equal or different
• Recall that angles are measured in degrees, written as ${}^{\circ }$
• Know how to classify angles as acute, right or obtuse
• Understand that a right angle is ${90}^{\circ }$
• Be able to identify interior and exterior angles on a diagram
• Know that angles on a straight line add to ${180}^{\circ }$
• Be familiar with finding unknown angles using angle facts

Worked Examples

Worked Example 1

Classify each triangle by its side lengths:
a) sides $5$ cm, $5$ cm, $7$ cm
b) sides $6$ cm, $8$ cm, $10$ cm
c) sides $4$ cm, $4$ cm, $4$ cm

Worked Example 2

Classify each triangle by its interior angles:
a) angles ${50}^{\circ },{60}^{\circ },{70}^{\circ }$
b) angles ${90}^{\circ },{35}^{\circ },{55}^{\circ }$
c) angles ${110}^{\circ },{30}^{\circ },{40}^{\circ }$

Worked Example 3

Use the angle sum of a triangle to find the unknown angle:
a) angles are ${48}^{\circ },{67}^{\circ }$ and $x$
b) angles are ${90}^{\circ },{34}^{\circ }$ and $y$
c) angles are ${72}^{\circ },{72}^{\circ }$ and $z$

Worked Example 4

An isosceles triangle has two equal angles of ${41}^{\circ }$. Find the third angle.

Worked Example 5

Use the exterior angle theorem to find the exterior angle:
a) remote interior angles are ${52}^{\circ }$ and ${61}^{\circ }$
b) remote interior angles are ${35}^{\circ }$ and ${78}^{\circ }$

Worked Example 6

Use the exterior angle theorem to find the unknown interior angle:
a) an exterior angle is ${124}^{\circ }$ and one remote interior angle is ${57}^{\circ }$
b) an exterior angle is ${146}^{\circ }$ and one remote interior angle is ${68}^{\circ }$

Problems

Problem 1

Classify each triangle by its side lengths:
a) sides $3$ cm, $3$ cm, $5$ cm
b) sides $7$ cm, $9$ cm, $11$ cm
c) sides $8$ cm, $8$ cm, $8$ cm

Problem 2

Classify each triangle by its interior angles:
a) angles ${45}^{\circ },{65}^{\circ },{70}^{\circ }$
b) angles ${90}^{\circ },{42}^{\circ },{48}^{\circ }$
c) angles ${120}^{\circ },{25}^{\circ },{35}^{\circ }$

Problem 3

Use the angle sum of a triangle to find the unknown angle:
a) angles are ${56}^{\circ },{49}^{\circ }$ and $x$
b) angles are ${90}^{\circ },{47}^{\circ }$ and $y$
c) angles are ${63}^{\circ },{63}^{\circ }$ and $z$

Problem 4

An isosceles triangle has two equal angles of ${38}^{\circ }$. Find the third angle.

Problem 5

Use the exterior angle theorem to find the exterior angle:
a) remote interior angles are ${47}^{\circ }$ and ${66}^{\circ }$
b) remote interior angles are ${29}^{\circ }$ and ${83}^{\circ }$

Problem 6

Use the exterior angle theorem to find the unknown interior angle:
a) an exterior angle is ${132}^{\circ }$ and one remote interior angle is ${59}^{\circ }$
b) an exterior angle is ${151}^{\circ }$ and one remote interior angle is ${73}^{\circ }$

Exercises

Understanding and Fluency

1. Classify each triangle by its side lengths:
   a) sides $4$ cm, $6$ cm, $7$ cm
   b) sides $9$ cm, $9$ cm, $12$ cm
   c) sides $5$ cm, $5$ cm, $5$ cm

2. Classify each triangle by its interior angles:
   a) ${40}^{\circ },{60}^{\circ },{80}^{\circ }$
   b) ${90}^{\circ },{30}^{\circ },{60}^{\circ }$
   c) ${100}^{\circ },{50}^{\circ },{30}^{\circ }$

3. State whether each triangle is scalene, isosceles or equilateral:
   a) all sides different
   b) two sides equal
   c) all sides equal

4. State whether each triangle is acute, right or obtuse:
   a) all angles less than ${90}^{\circ }$
   b) one angle equal to ${90}^{\circ }$
   c) one angle greater than ${90}^{\circ }$

5. Use the angle sum of a triangle to find each unknown angle:
   a) ${61}^{\circ },{52}^{\circ },x$
   b) ${90}^{\circ },{26}^{\circ },y$
   c) ${74}^{\circ },{74}^{\circ },z$

6. Find the missing angle in each triangle:
   a) ${33}^{\circ },{88}^{\circ },a$
   b) ${90}^{\circ },{51}^{\circ },b$
   c) ${67}^{\circ },{67}^{\circ },c$

7. Use the exterior angle theorem to find each exterior angle:
   a) remote interior angles ${45}^{\circ }$ and ${58}^{\circ }$
   b) remote interior angles ${72}^{\circ }$ and ${39}^{\circ }$
   c) remote interior angles ${64}^{\circ }$ and ${64}^{\circ }$

8. Use the exterior angle theorem to find each unknown interior angle:
   a) exterior angle ${117}^{\circ }$, one remote interior angle ${53}^{\circ }$
   b) exterior angle ${138}^{\circ }$, one remote interior angle ${61}^{\circ }$
   c) exterior angle ${126}^{\circ }$, one remote interior angle ${48}^{\circ }$

Reasoning

9. Explain why an equilateral triangle must also be isosceles.

10. A student says that a triangle with angles ${70}^{\circ },{70}^{\circ },{40}^{\circ }$ is scalene because the angles are not all equal. Explain the mistake.

11. Noah says a triangle can be both right and obtuse. Is he correct? Explain.

12. Explain why the interior angles of a triangle cannot include two obtuse angles.

13. A student says the exterior angle of a triangle is found by adding all three interior angles. Describe the error.

Problem-solving

14. A triangle has two equal sides and one angle of ${90}^{\circ }$. How could this triangle be classified by side lengths and by angles?

15. A triangular sign has interior angles of ${48}^{\circ }$ and ${73}^{\circ }$. Find the third angle and classify the triangle by angles.

16. An exterior angle of a triangle is ${125}^{\circ }$. One remote interior angle is ${54}^{\circ }$. Find the other remote interior angle.

17. A triangle has side lengths $6$ cm, $6$ cm and $10$ cm, and one interior angle is ${112}^{\circ }$. Classify it by side lengths and by angles.

18. The exterior angle at one vertex of a triangle is ${140}^{\circ }$. The two remote interior angles are equal. Find both of those interior angles.

19. A triangle is isosceles and one of its equal angles is ${37}^{\circ }$. Find all three interior angles.

Potential Misunderstandings

• Students may think an isosceles triangle must have exactly two equal sides, rather than at least two equal sides
• Students may think an equilateral triangle is not also isosceles
• Students may confuse classification by side lengths with classification by angles
• Students may think a triangle can be both right and obtuse
• Students may forget that the interior angles of a triangle add to ${180}^{\circ }$
• Students may make arithmetic errors when subtracting from ${180}^{\circ }$
• Students may forget that equal sides in an isosceles triangle have equal opposite angles
• Students may confuse an exterior angle with an interior angle
• Students may think the exterior angle theorem uses the two adjacent interior angles instead of the two remote interior angles
• Students may forget that an exterior angle is equal to the sum of the two remote interior angles
• Students may rely only on the appearance of a diagram instead of using the given angle facts