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Lesson 18 ~25 min Unit 3 · Geometry +85 XP

Geometric Reasoning — Multi-step Problems

Many geometry problems can't be solved in one step. You combine angle facts from triangles, quadrilaterals, parallel lines and polygons — one reason at a time — until you reach the answer. Each step needs a written REASON.

Today's hook: A triangle sits inside a parallelogram. One angle is $40^{\circ}$, another is unknown. You need TWO different angle facts to crack it. Game on.
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Think First
warm-up

Without looking anything up, list every angle fact you remember. Triangle angle sum, straight line, vertically opposite, co-interior, anything — just dump them onto the page. We're going to combine them today.

Record your answer in your workbook.
1
What "Geometric Reasoning" Means
+5 XP

Geometric reasoning means using known angle facts as the REASON for each step of working. In multi-step problems you may use 2, 3 or even 4 different facts in a row. Each step should be in this form:

angle value $=$ ...   (reason)

The reason is just as important as the number — it's how you earn the marks.

You'll meet phrases like "$\angle$ sum of $\triangle$", "alternate $\angle$s on parallel lines", "co-interior $\angle$s, $AB \parallel CD$", "$\angle$ sum of quad". These short reasons are EXACTLY what marker is looking for. Every angle you find should sit next to a reason.

Step format $\angle ABC = 70^{\circ}$    (alt ∠s, AB || CD) $\angle BCD = 110^{\circ}$    (co-int ∠s, AB || CD) $\angle x = 40^{\circ}$    (∠ sum of $\triangle$) Always: value — reason — check.
Step = value $+$ reason
Find what you know
Look at the diagram — what's given, what's marked, what can you fill in instantly?
Pick a fact
Choose ONE angle fact that uses what you know.
Write the reason
No reason = no mark. Always state which fact you used.
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What You'll Master
objectives

Know

  • Angle sum of triangle = $180^{\circ}$
  • Angle sum of quadrilateral = $360^{\circ}$
  • Parallel-line angles: alternate, co-interior, corresponding
  • Polygon angle sum: $(n-2) \times 180^{\circ}$

Understand

  • Why each step needs a justification
  • Why one diagram can use several facts at once
  • How to choose the most efficient fact

Can Do

  • Plan a multi-step solution before writing
  • Write each step with value AND reason
  • Solve problems that combine triangle + parallel + quadrilateral facts
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Reasons / Words You Need
vocabulary
∠ sum of $\triangle$Three interior angles of a triangle add to $180^{\circ}$.
∠ sum of quadFour interior angles of a quadrilateral add to $360^{\circ}$.
Alt ∠s, $AB \parallel CD$Alternate angles on parallel lines are equal.
Co-int ∠s, $AB \parallel CD$Co-interior angles add to $180^{\circ}$.
Corr ∠s, $AB \parallel CD$Corresponding angles on parallel lines are equal.
∠s on str lineTwo angles on a straight line add to $180^{\circ}$.
Vert opp ∠sVertically opposite angles are equal.
Ext ∠ of $\triangle$Exterior angle = sum of the two opposite interior angles.
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The 4-Step Plan
+5 XP

For any multi-step problem, follow this:

  1. Label. Mark every known angle on the diagram. Give every unknown a letter.
  2. Plan. Find a chain of facts that gets from "known" to "wanted". 2 or 3 steps is normal.
  3. Solve. One angle per line: value $=$ ... (reason).
  4. Check. Do the angles add up? Does the answer look sensible?

Example: a triangle inside a pair of parallel lines. Step 1: alt $\angle$s give one inside angle of the triangle. Step 2: another angle is found from corresponding angles. Step 3: triangle angle sum finds the last angle. Each step is short, each step has a reason.

Triangle between parallel lines 50° 70° x x = 180 - 50 - 70 = 60° (∠ sum of $\triangle$)
Plan: known facts $\to$ chain $\to$ answer
Mark the diagram
Pencil in every value as you go — you'll see chains form.
One angle per line
Don't bundle multiple facts into one step — the marker can't follow it.
Cite the reason
Even if the calculation is obvious, write the named fact you used.
Book notes · 4-step plan
  • Label diagram; give unknowns letters.
  • Plan a chain of 2–3 known facts.
  • One angle per line, with reason.
Which reason fits: in a triangle two angles are $50^{\circ}$ and $80^{\circ}$, so the third is $50^{\circ}$ because:
5
Parallel Lines + Triangles
+5 XP

The most common multi-step problem mixes a triangle with a pair of parallel lines. The chain often looks like this:

  1. Use alternate or corresponding angles to bring an angle DOWN into the triangle.
  2. Use $\angle$ sum of $\triangle$ to find the last angle.

Example: $AB \parallel CD$. A transversal makes a $40^{\circ}$ angle with $AB$. A triangle drops below $CD$ with one side along the transversal and a base angle of $70^{\circ}$. Step 1: the alternate angle gives $40^{\circ}$ inside the triangle. Step 2: $x = 180^{\circ} - 40^{\circ} - 70^{\circ} = 70^{\circ}$.

Parallel + triangle combo 40° ? 40° (alt) 70°
$x = 180 - 40 - 70 = 70^{\circ}$
Find the "Z"
Alternate angles form a "Z" shape between two parallel lines.
Or the "F"
Corresponding angles form an "F".
Or the "C"
Co-interior angles form a "C" — they add to $180^{\circ}$.
Book notes · Parallel + triangle
  • Z = alternate angles (equal).
  • F = corresponding angles (equal).
  • C = co-interior angles (add to $180^{\circ}$).
True or false?

Co-interior angles between two parallel lines add to $180^{\circ}$.

6
Polygons and Long Chains
+5 XP

For any polygon with $n$ sides, the interior angle sum is:

$\text{Angle sum} = (n - 2) \times 180^{\circ}$

For a regular polygon (all angles equal), each interior angle is the sum divided by $n$. In multi-step problems you often combine this with the angle sum of a triangle.

Examples of angle sums:
• Triangle ($n=3$): $180^{\circ}$
• Quadrilateral ($n=4$): $360^{\circ}$
• Pentagon ($n=5$): $540^{\circ}$
• Hexagon ($n=6$): $720^{\circ}$
For a regular hexagon, each angle is $720 \div 6 = 120^{\circ}$.

Regular hexagon — each angle 120° 120° Sum = (6-2) × 180 = 720°  |  each = 720/6 = 120°
Each angle (regular polygon) $= \dfrac{(n-2) \times 180}{n}$
Sum first, then divide
For regular polygons, find the SUM first, then split equally.
Combine with triangle
Often a polygon angle plus a triangle finds the last angle in 2 steps.
Cite the formula
Write the reason "$\angle$ sum of polygon $=(n-2)\times180$".
Book notes · Polygon angle sum
  • Angle sum $= (n - 2) \times 180^{\circ}$.
  • Regular: each angle $= \frac{(n-2) \times 180}{n}$.
  • Often combined with triangle / parallel-line facts.
A pentagon has interior angle sum equal to __________ degrees.

Watch Me Solve It · 3 examples

Watch Me Solve It · Two-step parallel + triangle
+15 XP per step
Q1
PROBLEM
Two parallel lines are crossed by a transversal making an angle of $65^{\circ}$ on top. A triangle below has the same transversal as one side and a second angle of $80^{\circ}$. Find $x$, the third angle of the triangle.
  1. 1
    Use alternate angles
    Inside the triangle: angle $= 65^{\circ}$ (alt ∠s, lines parallel)
  2. 2
    Apply triangle angle sum
    $x + 65 + 80 = 180$ (∠ sum of $\triangle$)
  3. 3
    Solve
    $x = 180 - 65 - 80 = 35^{\circ}$
    Two reasons stacked: alternate angles + triangle sum.
Answer$x = 35^{\circ}$.
Watch Me Solve It · Quadrilateral with diagonals
+15 XP per step
Q2
PROBLEM
In quadrilateral $ABCD$, $\angle A = 95^{\circ}$, $\angle B = 110^{\circ}$, $\angle C = 75^{\circ}$. Find $\angle D$, then find the angle the diagonal $BD$ makes with side $DA$, given that $\angle ABD = 50^{\circ}$.
  1. 1
    Use quadrilateral angle sum
    $\angle D = 360 - 95 - 110 - 75 = 80^{\circ}$ (∠ sum of quad)
  2. 2
    Use triangle $ABD$
    In $\triangle ABD$: $\angle A + \angle ABD + \angle ADB = 180$ (∠ sum of $\triangle$)
  3. 3
    Solve
    $\angle ADB = 180 - 95 - 50 = 35^{\circ}$
    Two facts combined: quad sum + triangle sum.
Answer$\angle D = 80^{\circ}$, $\angle ADB = 35^{\circ}$.
Watch Me Solve It · Regular pentagon angle
+15 XP per step
Q3
PROBLEM
Find one interior angle of a regular pentagon, then find the angle between a diagonal from one vertex and the adjacent side (the pentagon is regular and symmetric, so we get an isosceles triangle).
  1. 1
    Pentagon angle sum
    Sum $= (5 - 2) \times 180 = 540^{\circ}$. Each angle $= \frac{540}{5} = 108^{\circ}$.
  2. 2
    Isosceles triangle
    Pulling a diagonal from one vertex creates an isosceles triangle with apex $108^{\circ}$.
  3. 3
    Base angles
    Each base angle $= \frac{180 - 108}{2} = 36^{\circ}$ (∠ sum of $\triangle$, isosceles).
    So diagonal meets the side at $36^{\circ}$.
AnswerPentagon angle $= 108^{\circ}$, diagonal meets side at $36^{\circ}$.

Common Pitfalls

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Common Pitfalls
heads-up
Skipping the reason
Writing "$x = 60^{\circ}$" without saying WHY loses marks even if the answer is correct.
Fix: Add a short named reason after every step: e.g. ($\angle$ sum of $\triangle$).
Using the wrong parallel-line fact
Calling co-interior angles "equal" (they ADD to $180^{\circ}$) or alternate angles "supplementary" (they're EQUAL).
Fix: Z = equal; F = equal; C = adds to $180^{\circ}$.
Forgetting to plan
Diving straight into algebra without seeing which chain of facts to use can land you in a dead end.
Fix: Mark known values on the diagram first; let the chain reveal itself.
Copy Into Your Books

Key facts

  • $\triangle$ sum $= 180^{\circ}$
  • Quad sum $= 360^{\circ}$
  • Polygon sum $= (n-2) \times 180^{\circ}$

Parallel lines

  • Alt ∠s equal (Z)
  • Corresponding ∠s equal (F)
  • Co-int ∠s add to $180^{\circ}$ (C)

Other angle facts

  • Vertically opp ∠s equal
  • ∠s on str line add to $180^{\circ}$
  • Ext ∠ $=$ sum of 2 opp interior

Writing reasons

  • Value $=$ ... (reason)
  • One step per line
  • Cite the named fact

How are you completing this lesson?

Brain Trainer · 4 problems

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Brain Trainer · Multi-step Reasoning
4 problems

Four quick drills combining angle facts.

  1. 1 A triangle has angles $45^{\circ}$, $x$, $2x$. Find $x$.

    $45 + x + 2x = 180$, so $3x = 135$.$x = 45^{\circ}$

  2. 2 A regular octagon. Find one interior angle.

    Sum $= 6 \times 180 = 1080$. Each $= 1080/8$.$135^{\circ}$

  3. 3 Parallel lines crossed by transversal. Co-interior pair: one is $70^{\circ}$. Find the other.

    $180 - 70$.$110^{\circ}$

  4. 4 Quadrilateral with $\angle A = 90$, $\angle B = 80$, $\angle C = x$, $\angle D = 2x$. Find $x$.

    $90 + 80 + 3x = 360$.$x = 63.33...^{\circ} \approx 63.3^{\circ}$
Complete in your workbook.

Quick Check · 5 questions

1
A triangle has angles $60^{\circ}$, $50^{\circ}$ and $x$. Find $x$.
+10 XP
2
Co-interior angles between two parallel lines:
+10 XP
3
The sum of interior angles of a pentagon is:
+10 XP
4
Alternate angles on parallel lines are:
+10 XP
5
A quadrilateral has angles $100^{\circ}, 90^{\circ}, 75^{\circ}$ and $x$. Find $x$.
+10 XP

Show Your Working · 3 questions

Show Your Working
9 marks total
Apply Easy 3 MARKS

Q6. Two parallel lines are crossed by a transversal. One angle is $75^{\circ}$.
(a) Find the alternate angle (state reason).
(b) Find the co-interior angle (state reason).
(c) Find the corresponding angle (state reason).

Answer in your workbook.
Apply Medium 3 MARKS

Q7. A triangle inside two parallel lines has one angle of $40^{\circ}$ between a side and the upper parallel line (forming an alternate-angles pair) and another angle of $65^{\circ}$ on the lower line (corresponding pair).
(a) Find the corresponding angle inside the triangle (with reason).
(b) Find the alternate angle inside the triangle (with reason).
(c) Use the triangle angle sum to find the third angle.

Answer in your workbook.
Reason Hard 3 MARKS

Q8. A regular polygon has interior angle $150^{\circ}$.
(a) Set up an equation using $(n-2) \times 180 = n \times 150$.
(b) Solve for $n$.
(c) Name the polygon.

Answer in your workbook.
Comprehensive Answers

Quick Check

1. C — $x = 70^{\circ}$.

2. B — Co-interior angles add to $180^{\circ}$.

3. D — Pentagon sum $= 540^{\circ}$.

4. A — Alternate angles are equal.

5. B — $360 - 100 - 90 - 75 = 95^{\circ}$.

Show Your Working Model Answers

Q6 (3 marks): (a) Alt $= 75^{\circ}$ (alt ∠s, lines parallel) [1]. (b) Co-int $= 180 - 75 = 105^{\circ}$ (co-int ∠s, lines parallel) [1]. (c) Corr $= 75^{\circ}$ (corr ∠s, lines parallel) [1].

Q7 (3 marks): (a) $65^{\circ}$ (corr ∠s, lines parallel) [1]. (b) $40^{\circ}$ (alt ∠s, lines parallel) [1]. (c) Third angle $= 180 - 40 - 65 = 75^{\circ}$ (∠ sum of $\triangle$) [1].

Q8 (3 marks): (a) $(n - 2) \times 180 = 150n$ [1]. (b) $180n - 360 = 150n \Rightarrow 30n = 360 \Rightarrow n = 12$ [1]. (c) Regular $12$-gon (dodecagon) [1].

Stretch Challenge · +25 XP, +10 coins

The Folded Triangle

A rectangular piece of paper is folded so that one corner lands on the opposite side, forming a crease. The crease cuts off a triangle. Suppose the angle between the crease and the bottom edge is $35^{\circ}$. (a) Why are the two triangles formed by the fold congruent? (b) Find the angle the crease makes with the right edge of the original rectangle. (c) Find every other angle in the folded figure (use multiple reasons).

Reveal solution

(a) Folding maps one triangle onto the other — sides and angles are preserved (reflection is a congruence). (b) The other base angle of the right-triangle: $180 - 90 - 35 = 55^{\circ}$ (∠ sum of $\triangle$, right angle at corner). (c) Crease makes $35^{\circ}$ at one end, $55^{\circ}$ at the other; reflected triangle has matching angles; full set: $35^{\circ}, 55^{\circ}, 90^{\circ}$ and their reflected partners.

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Quick Review

Triangle

Interior angles sum to $180^{\circ}$.

Quadrilateral

Interior angles sum to $360^{\circ}$.

Polygon

Sum $= (n-2) \times 180^{\circ}$.

Parallel facts

Alt $=$, corr $=$, co-int add to $180^{\circ}$.

Write reasons

Always: value $=$ ... (named fact).

Plan first

Mark known angles; find chain to unknown.

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