047i - Finding Unknown Angles in Triangles

Lesson 47 — Applying the Triangle Angle Sum to Find Unknown Angles

Strand: Measurement | Descriptor: AC9M7M05 | Duration: 45 minutes

Learning Intentions

• To apply the triangle angle sum to determine unknown angles.
• To combine the angle sum with other angle facts in multi-step problems.

Success Criteria

I can:

1. Find a missing angle in a triangle, giving a reason.
2. Use the isosceles base-angle property.
3. Combine the angle sum with angles on a straight line and vertically opposite angles.
4. Set out an angle chase with a reason on every line.

Warmup

(6 minutes — angle facts recall, mini whiteboards)

State the size of each angle and name the fact you used.

1. Two angles on a straight line: one is . Find the other.
2. Angles at a point around a full turn: three are , and . Find .
3. Two lines cross; one angle is . Find the vertically opposite angle.
4. A right angle is split into two parts, one . Find the other.

Answers: 1. (angles on a straight line sum to ); 2. (angles at a point sum to ); 3. (vertically opposite angles are equal); 4. (angles in a right angle sum to ).

The three facts to keep visible all lesson, alongside the angle sum:

• Angles on a straight line sum to .
• Angles at a point sum to .
• Vertically opposite angles are equal.

Activities

Activity 1 — Explicit Instruction: Setting out an Angle Chase (12 min)

The convention to enforce: every line of working states a value and a reason. Reasons may be abbreviated as shown.

I do — one step. A triangle has angles , and .

I do — two steps, using a straight line. In a diagram, an exterior angle of sits on a straight line with one interior angle of a triangle. The other interior angles are and .

I do — isosceles. An isosceles triangle has two marked equal sides and an apex angle of . Find the base angles.

Emphasise the marking check. Before using the base-angle property, students must confirm two sides are marked equal, or two angles are marked equal. Assuming isosceles from appearance is the most common error in this topic.

We do: Three diagrams, worked together, each requiring two steps and reasons.

Activity 2 — Independent Practice (12 min)

You do: Find each labelled angle, with a reason for every line.

1. Triangle with angles , , .
2. Right-angled triangle with one other angle ; find the third.
3. Isosceles triangle with apex ; find a base angle.
4. Isosceles triangle with base angle ; find the apex.
5. A triangle sits on a straight line; the exterior angle is and one interior angle is . Find the third interior angle.
6. Two lines cross forming a angle, which is also an angle of a triangle whose second angle is . Find the third.
7. An equilateral triangle has one side extended. Find the exterior angle.
8. A triangle has angles , and . Find and all three angles.

(Answers: 1. ; 2. ; 3. ; 4. ; 5. interior angle , so third ; 6. ; 7. ; 8. , so , giving .)

Activity 3 — Inquiry: the Exterior Angle (12 min)

Pairs. Discovers a genuinely useful shortcut.

Draw a triangle and extend one side to form an exterior angle.

1. Measure the exterior angle.
2. Measure the two interior angles not adjacent to it.
3. Add those two. Compare with the exterior angle.
4. Repeat for three different triangles.
5. What do you conjecture? Can you explain why?

Socratic scaffolding:

Prompt	Purpose
What do you notice from your measurements?	The exterior angle equals the sum of the two opposite interior angles.
Does measuring several triangles prove it?	No — you need a reason.
Call the interior angles , , , with adjacent to the exterior angle . What do you know?	(angle sum) and (straight line).
What can you deduce?	Both expressions equal , so .
Simplify.	Subtract from both sides: .
State the result.	The exterior angle equals the sum of the two opposite interior angles.
Looking back	This is a proof, valid for every triangle — the same move as Lesson 30’s algebra.

Practice with the shortcut:

1. Interior angles and ; find the exterior angle at the third vertex.
2. Exterior angle ; one opposite interior angle is . Find the other.
3. Exterior angle ; the two opposite interior angles are equal. Find each.

(Answers: ; ; each.)

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Find : a triangle has angles , and . Give a reason.
2. An isosceles triangle has an apex angle of . Find a base angle.
3. A triangle’s exterior angle is . One opposite interior angle is . Find the other.
4. A triangle has angles , and . Find .
5. Reasoning. Explain why the exterior angle of a triangle always equals the sum of the two opposite interior angles.

Answers: 1. (angle sum of a triangle); 2. ; 3. ; 4. , so ; 5. The interior angles total , and the exterior angle plus its adjacent interior angle also total . Subtracting the shared adjacent angle leaves the exterior angle equal to the other two interior angles combined.

Common Misconceptions

Misconception	How to pre-empt it
Assuming a triangle is isosceles from its appearance.	Require markings — equal side dashes or equal angle arcs — before using base angles.
Applying the base-angle property to the wrong pair.	The equal angles are those opposite the equal sides. Mark them first.
Confusing the exterior angle with the adjacent interior angle.	Label both on a diagram and note they sum to .
Omitting reasons.	Mark for reasons as well as values. A value with no reason scores nothing.
Adding the exterior angle into the total.	Only the three interior angles sum to .
Stopping after one step in a multi-step chase.	Ask “which angle was actually requested?” before writing the final answer.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A triangle has angles , and . Find the largest angle.

Answer

The angles are , , , so the largest is .

E2 (AMC Junior style). In an isosceles triangle, the apex angle is less than each base angle. Find all three angles.

Answer

Let a base angle be ; the apex is .

Angles: , , .

E3 (Challenge). Two triangles share a vertex, forming vertically opposite angles there. The first triangle has other angles and . The second has one other angle of . Find its third angle.

Answer

The shared angle in the first triangle is . The vertically opposite angle in the second is also . So its third angle is .

E4 (Challenge). In triangle , and angle . Side is extended to . Find angle .

Answer

Base angles: each, so angle . Then angle (angles on a straight line). (Check with the exterior angle rule: ✓)

E5 (Challenge). A triangle has one angle of . The other two are in the ratio . Find them.

Answer

They total and split into parts, so one part is . The angles are and .

Homework

1. Find the missing angle, with a reason: (a) , , (b) , , (c) , , (d) , , .
2. Isosceles triangles — find the requested angle: (a) apex , find a base angle (b) apex , find a base angle (c) base angle , find the apex (d) base angle , find the apex.
3. A triangle’s exterior angle is . One opposite interior angle is . Find the other.
4. A triangle’s two interior angles are and . Find the exterior angle at the third vertex.
5. A triangle has angles , and . Find and all three angles, then classify by angles.
6. A triangle has angles , and . Find all three.
7. In triangle , and angle . Side is extended. Find the exterior angle at .
8. Reasoning. Explain why a triangle cannot have two obtuse angles, using the angle sum.
9. Reasoning. Explain why the exterior angle of a triangle is always greater than either of the two opposite interior angles.
10. Challenge. An isosceles triangle has an apex angle that is twice a base angle. Find all three angles.

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) (b) (c) (d) . Q3 — . Q4 — . Q5 — , so ; angles ; obtuse. Q6 — , so ; angles . Q7 — base angles each, so the exterior angle is . Q8 — two obtuse angles each exceed , totalling more than , leaving nothing for the third. Q9 — the exterior angle equals the sum of the two opposite interior angles, and both are positive, so it exceeds each individually. Q10 — , so ; angles .