049i - Problem Solving with Angle Sums

Lesson 49 — Problem Solving and Consolidation: Angle Sums

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

Learning Intentions

• To solve multi-step problems using triangle and polygon angle sums.
• To combine angle facts and justify each step of a solution.

Success Criteria

I can:

1. Select the appropriate angle fact for each step of a problem.
2. Work through a multi-step angle chase with a reason on every line.
3. Form and solve an equation to find an unknown angle.
4. Check that my answer is consistent with the whole diagram.

Warmup

(6 minutes — fact recall relay, mini whiteboards)

State the fact that lets you find each unknown.

1. Two angles on a straight line, one is .
2. A triangle with angles , and .
3. A pentagon’s interior angle sum.
4. Each interior angle of a regular hexagon.
5. Two intersecting lines, one angle ; find the vertically opposite angle.
6. An isosceles triangle with apex .

Answers: 1. (straight line); 2. (triangle angle sum); 3. (from ); 4. ; 5. (vertically opposite); 6. base angles each.

Activities

Activity 1 — Multi-step Angle Chases (14 min)

Pairs. Every line needs a value and a reason.

Problem 1. In triangle , and angle . Side is extended to . Find angle .

Problem 2. A quadrilateral has angles , , and . Find all four and classify the largest.

Problem 3. Two triangles meet at a point, forming vertically opposite angles. Triangle 1 has angles , and the shared angle. Triangle 2 has the vertically opposite angle plus one angle of . Find its third angle.

Problem 4. A regular pentagon and a regular hexagon share one full side. Find the angle between them at a shared vertex, on the outside.

Problem 5. In a hexagon, four angles are each and the remaining two are equal. Find them.

Socratic scaffolding for Problem 4:

Prompt	Purpose
Understand: what is being asked?	The gap angle outside, where the two shapes meet at a vertex.
What is a regular pentagon’s interior angle?	.
And a regular hexagon’s?	.
At the shared vertex, what surrounds the point?	The pentagon’s angle, the hexagon’s angle, and the gap.
What must they total?	— angles at a point.
Carry it out	.
Looking back	Is the answer sensible? The gap is larger than either shape’s angle, which fits the picture. ✓

Answers:

1. Base angles , so angle ; angle (straight line). (Exterior angle check: ✓)
2. , so ; angles ; the largest is obtuse.
3. Shared angle ; vertically opposite is also ; third angle .
4. .
5. , so each is .

Activity 2 — Forming Equations from Angle Problems (12 min)

Links Lessons 35 and 48.

I do. A triangle’s angles are , and . Find all three.

Angles: , , . Check: ✓

You do:

1. A triangle has angles , and .
2. A triangle has angles , and .
3. A quadrilateral has angles , , and .
4. A pentagon has angles , , , and .
5. An isosceles triangle has apex and base angles each.

(Answers: 1. ; . 2. ; . 3. , so ; . 4. , so ; — impossible, since a interior angle means the “vertex” is a straight line. See the discussion below. 5. , so ; angles .)

Discussion of Q4. The algebra is correct but the answer is geometrically impossible — a polygon cannot have a interior angle, because the two sides at that vertex would lie in a straight line, making it a quadrilateral instead. This is a valuable outcome: always check that an algebraic answer makes geometric sense. Ask students what ratio would work.

Discussion of Q5. The answer is not a whole number of degrees. That is perfectly legitimate — angles need not be integers. Contrast with Q4, where the problem is genuine impossibility rather than untidiness.

Activity 3 — Inquiry: Angles in a Star (10 min)

Pairs.

Draw a five-pointed star by joining every second vertex of a regular pentagon.

1. Measure one point angle of the star.
2. What do all five point angles total?
3. Can you explain the result without measuring?

Socratic scaffolding:

Prompt	Purpose
What did you measure for one point?	About .
So what is the total for five points?	About — suspiciously familiar.
Now find a reason. What shape is each point of the star?	An isosceles triangle sitting on a side of the inner pentagon.
What is the inner pentagon’s interior angle?	.
So what are the base angles of each point triangle?	The base angles are exterior to the inner pentagon: each.
Find the point angle.	.
Total for five points.	.
Looking back	A neat result, derived rather than measured — and the measurement confirms it.

Answers: 1. ; 2. ; 3. As derived above.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. A triangle has angles , and . Find all three.
2. A hexagon has five angles of , , , and . Find the sixth.
3. An isosceles triangle has apex . One side is extended. Find the exterior angle at a base vertex.
4. A regular octagon and a square share a side. Find the angle between them at a shared vertex.
5. Reasoning. A student calculates a pentagon’s angles as and . Is this possible? Explain.

Answers: 1. , so ; angles ; 2. ; 3. base angles each, so the exterior angle is ; 4. ; 5. The total is , not , so it is wrong. (A interior angle is itself possible in a concave pentagon — the error is the total, not the reflex angle.)

Common Misconceptions

Misconception	How to pre-empt it
Using for a triangle or for a quadrilateral.	Always state and compute before substituting.
Accepting an algebraically correct but geometrically impossible answer.	Activity 2 Q4 confronts this directly. Always sense-check against the diagram.
Rejecting a non-integer angle as an error.	Activity 2 Q5 shows these are legitimate. Distinguish “untidy” from “impossible”.
Omitting reasons in a multi-step chase.	Mark for reasons. A bare value earns nothing.
Confusing interior and exterior angles at a vertex.	Label both and note they sum to .
Assuming shapes are regular when the problem does not say so.	Only divide by when regularity is stated or marked.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A regular polygon has an exterior angle of . How many sides has it?

Answer

Exterior angles of any polygon total , so sides.

E2 (AMC Junior style). In a quadrilateral, three angles are equal and the fourth is more than each of them. Find all four.

Answer

Angles: , , , .

E3 (Challenge). A regular hexagon, a square and an equilateral triangle all meet at a single point with no gaps. Verify that this is possible.

Answer

That leaves a gap, so these three alone do not fill the point. Adding a second square gives ✓ — so hexagon, square, triangle, square works. A useful lesson in checking rather than assuming.

E4 (Challenge). Find the sum of the five point angles of a five-pointed star drawn inside a regular pentagon.

Answer

Each point angle is , so the total is . (Derived in Activity 3.)

E5 (Challenge). In triangle , angle is twice angle , and angle is more than angle . Find all three.

Answer

Let angle ; then and .

Angles: , , . (Check: ✓)

Homework

1. Find all angles: (a) a triangle with , , (b) a triangle with , , (c) a quadrilateral with , , , .
2. A pentagon has four angles of , , and . Find the fifth.
3. A hexagon has three angles of and three equal unknown angles. Find each.
4. An isosceles triangle has an apex of . Find (a) each base angle (b) the exterior angle at a base vertex.
5. A regular pentagon and a regular hexagon share a side. Find the angle between them at a shared vertex.
6. A regular polygon has an exterior angle of . Find (a) the number of sides (b) each interior angle.
7. A triangle’s exterior angle is ; the two opposite interior angles are in the ratio . Find them.
8. Reasoning. Explain why a polygon cannot have an interior angle of exactly .
9. Reasoning. Explain why the exterior angles of any polygon always total .
10. Challenge. In a pentagon, the angles are , , , and . Find all five and state whether the pentagon is convex.

Answers: Q1 — (a) (b) ; (c) ; . Q2 — . Q3 — , so each is . Q4 — (a) (b) . Q5 — . Q6 — (a) (b) . Q7 — parts total , so one part is ; the angles are about and . Q8 — the two sides at that vertex would form a straight line, so it would not be a vertex at all — the shape would have one fewer side. Q9 — walking once around the polygon turns you through one full revolution, and each exterior angle is one of those turns. Q10 — , so ; angles ; all are less than , so it is convex.