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 Solve unknown angles
• use the exterior angle theorem to find unknown angles

Pre-requisite Summary

• Know that a triangle has sides, vertices and 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
• Know how to Identify angles as acute, right or obtuse
• Understand that a right angle is
• Be able to Identify interior and exterior angles on a diagram
• Know that angles on a straight line add to
• Be familiar with finding unknown angles Use angle facts

Worked Examples

Worked Example 1

Classify each triangle by its side lengths:

a) sides cm, cm, cm

b) sides cm, cm, cm

c) sides cm, cm, cm

Worked Example 2

Classify each triangle by its interior angles:

a) angles

b) angles

c) angles

Worked Example 3

Use the angle sum of a triangle to find the unknown angle:

a) angles are and

b) angles are and

c) angles are and

Worked Example 4

An isosceles triangle has two equal angles of . Find the third angle.

Worked Example 5

Use the exterior angle theorem to find the exterior angle:

a) remote interior angles are and

b) remote interior angles are and

Worked Example 6

Use the exterior angle theorem to find the unknown interior angle:

a) an exterior angle is and one remote interior angle is

b) an exterior angle is and one remote interior angle is

Problems

Problem 1

Classify each triangle by its side lengths:

a) sides cm, cm, cm

b) sides cm, cm, cm

c) sides cm, cm, cm

Problem 2

Classify each triangle by its interior angles:

a) angles

b) angles

c) angles

Problem 3

Use the angle sum of a triangle to find the unknown angle:

a) angles are and

b) angles are and

c) angles are and

Problem 4

An isosceles triangle has two equal angles of . Find the third angle.

Problem 5

Use the exterior angle theorem to find the exterior angle:

a) remote interior angles are and

b) remote interior angles are and

Problem 6

Use the exterior angle theorem to find the unknown interior angle:

a) an exterior angle is and one remote interior angle is

b) an exterior angle is and one remote interior angle is

Exercises

Understanding and Fluency

Exercise 1.

Classify each triangle by its side lengths:

a) sides cm, cm, cm

b) sides cm, cm, cm

c) sides cm, cm, cm

Exercise 2.

Classify each triangle by its interior angles:

a)

b)

c)

Exercise 3.

State whether each triangle is scalene, isosceles or equilateral:

a) all sides different

b) two sides equal

c) all sides equal

Exercise 4.

State whether each triangle is acute, right or obtuse:

a) all angles less than

b) one angle equal to

c) one angle greater than

Exercise 5.

Use the angle sum of a triangle to find each unknown angle:

a)

b)

c)

Exercise 6.

Find the missing angle in each triangle:

a)

b)

c)

Exercise 7.

Use the exterior angle theorem to find each exterior angle:

a) remote interior angles and

b) remote interior angles and

c) remote interior angles and

Exercise 8.

Use the exterior angle theorem to find each unknown interior angle:

a) exterior angle , one remote interior angle

b) exterior angle , one remote interior angle

c) exterior angle , one remote interior angle

Reasoning

Exercise 9.

Explain why an equilateral triangle must also be isosceles.

Exercise 10.

A student says that a triangle with angles is scalene because the angles are not all equal. Explain the mistake.

Exercise 11.

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

Exercise 12.

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

Exercise 13.

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

Problem-solving

Exercise 14.

A triangle has two equal sides and one angle of . How could this triangle be classified by side lengths and by angles?

Exercise 15.

A triangular sign has interior angles of and . Find the third angle and classify the triangle by angles.

Exercise 16.

An exterior angle of a triangle is . One remote interior angle is . Find the other remote interior angle.

Exercise 17.

A triangle has side lengths cm, cm and cm, and one interior angle is . Classify it by side lengths and by angles.

Exercise 18.

The exterior angle at one vertex of a triangle is . The two remote interior angles are equal. Find both of those interior angles.

Exercise 19.

A triangle is isosceles and one of its equal angles is . 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
• Students may make arithmetic errors when subtracting from
• 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

Next: 105. Quadrilaterals and Their Angle Properties