MAT8 Q2W6D4: Generalizations, Assessment, and Reflections on Triangle Inequalities

Learning Goals

By the end of the lesson, you will be able to:

1. Compare two triangles with the same adjacent side pairs using the included angle, and write the correct third-side inequality in MathML (SAS-forward direction): if $AB\cong DE,AC\cong DF,\angle A>\angle D\Rightarrow BC>EF$.
2. Apply the converse comparison (third side controls included angle) to judge which included angle is larger when adjacent pairs are equal, and express the conclusion in MathML: $BC>EF\Rightarrow \angle A>\angle D$.
3. Order angles within a single triangle by ordering sides (SSS reasoning) and justify the ranking using MathML statements, for example $a>b>c\Rightarrow \angle A>\angle B>\angle C$.

Key Ideas & Terms

• Included angle: The angle formed by two given sides meeting at a vertex.
• SAS comparison (forward): With equal adjacent side pairs, a larger included angle implies a longer third side: $\angle A>\angle D\Rightarrow BC>EF$.
• SAS converse: With equal adjacent side pairs, a longer third side implies a larger included angle: $BC>EF\Rightarrow \angle A>\angle D$.
• SSS ordering (within one triangle): Longer side ↔ larger opposite angle; shorter side ↔ smaller opposite angle.
• Triangle inequality (reminder): For any three positive side lengths $a,b,c$, we need $a+b>c,a+c>b,b+c>a$.
• Nondegenerate triangle: All inequalities are strict; equality cases are degenerate and excluded.

Quick Recall / Prior Knowledge

1. Within one triangle: The longest side is opposite which angle?

   Show Answer

   It is opposite the largest angle.

2. SAS forward check: With equal adjacent sides, if $\angle A>\angle D$, what can you conclude about $BC$ vs. $EF?$

   Show Answer

   $BC>EF$.

3. SAS converse check: If with equal adjacent sides we know $EF>BC$, what is true about the included angles?

   Show Answer

   $\angle D>\angle A$.

Explore the Lesson

Comparing Sides and Angles in Triangles

Overview: This section leads you through a discovery path for comparing sides and angles when two triangles have matching adjacent side pairs, and for ordering angles within a single triangle. You will use guiding questions, micro-investigations, and real-life contexts (doors, tripods, frames) to develop and apply two key comparisons: the SAS-forward statement and its converse, plus SSS ordering inside one triangle. All work includes MathML, checkpoints with collapsible answers, and short summaries after each checkpoint.

1) What does the included angle control?

Imagine two rigid arms of fixed lengths joined at a hinge. The opening between the arms is the included angle. If you pull the arms farther apart at the hinge while keeping their lengths fixed, the straight-line distance between the free ends grows. This distance is the third side of a triangle built from the two arms and the invisible segment between their tips.

Guiding question: If the arm lengths are fixed and you increase the included angle, what happens to the third side?

Show Answer

The third side increases.

Mini-summary: With adjacent sides fixed, larger included angle → longer third side.

2) SAS-forward in symbols

Consider triangles $△ABC$ and $△DEFwith\; equal\; adjacent\; side\; pairs:$ AB\cong DE,AC\cong DF$.\; If\; the\; included\; angles\; satisfy$ \angle A>\angle D$,\; then\; you\; can\; conclude$ BC>EF.$$

Checkpoint A: Write the MathML statement that captures the SAS-forward comparison.

Show Answer

$AB\cong DE,AC\cong DF,\angle A>\angle D\Rightarrow BC>EF$.

Mini-summary: Equal adjacent pairs + larger included angle ⇒ longer third side.

3) SAS-converse in symbols

In the same setup with equal adjacent pairs, if you know that one triangle’s third side is longer, then its included angle must also be larger:

$AB\cong DE,AC\cong DF,BC>EF\Rightarrow \angle A>\angle D$.

Checkpoint B: If $EF>BC$, which included angle is larger?

Show Answer

$\angle D>\angle A$.

Mini-summary: With equal adjacent sides, longer third side ⇒ larger included angle.

4) Why non-included angles can mislead

Suppose two triangles share the same adjacent side pairs, but you are given a non-included angle measurement (an angle not formed by those two sides). Changing a non-included angle may or may not alter the third side in a predictable way, because it does not directly control the opening between the two fixed sides. Therefore, non-included angle comparisons are not valid for third-side conclusions.

Checkpoint C (concept): Can a larger non-included angle guarantee a longer third side when adjacent pairs are equal?

Show Answer

No. Only the included angle controls the third side under equal adjacent pairs.

Mini-summary: Compare included angles, not non-included angles.

5) SSS ordering inside a single triangle

Within a triangle, ordering sides automatically orders angles: the longest side faces the largest angle, the shortest side faces the smallest angle.

Checkpoint D: If $a>b>c$, order the angles.

Show Answer

$\angle A>\angle B>\angle C$.

Mini-summary: Longer side ↔ larger opposite angle.

6) Real-life scenario: Doors and gaps

Two identical doors open from the same hinge lengths. One is opened to $30°,\; the\; other\; to$ 75°.\; With\; the\; door\; edges\; acting\; as\; fixed-length\; sides\; about\; a\; hinge\; (the\; included\; angle),\; the\; straight-line\; gap\; across\; the\; doorway\; acts\; as\; the\; third\; side.$$

Guiding question: Which doorway produces a larger straight-line gap?

Show Answer

The door opened to $75°creates\; the\; larger\; gap\; (longer\; third\; side).$

Mini-summary: A larger included angle widens the gap.

7) Worked micro-investigation: Frame designs

Two frames have equal struts $22cm\; and$ 35cm.\; Design\; A\; chooses\; third\; side$ x=50cm;\; Design\; B\; chooses\; third\; side$ y=30cm.\; Which\; included\; angle\; is\; larger?$$$$

Show Answer

$x>y\Rightarrow \angle A>\angle B(with\; equal\; adjacent\; pairs).$

Mini-summary: Bigger third side under equal adjacent pairs means wider included angle.

8) Inside-one-triangle ordering practice

In triangle $△MNO$, suppose side lengths are $7.5cm,$ 10.1cm,\; and$ 12.2cm.\; Order\; the\; angles\; from\; largest\; to\; smallest.$$$

Show Answer

$\angle N>\angle O>\angle M.$

Mini-summary: Largest angle sits opposite the longest side.

9) Contrapositive lens

You can also reason by contrapositive for SAS-forward: with equal adjacent pairs, if $BC\le EF$, then $\angle A\le \angle D.\; This\; perspective\; helps\; when\; data\; come\; in\; the\; opposite\; direction\; (e.g.,\; you\; know\; that\; one\; third\; side\; is\; not\; larger).$

Checkpoint E: Complete the contrapositive statement in MathML.

Show Answer

$BC\le EF\Rightarrow \angle A\le \angle D$ (adjacent pairs equal).

Mini-summary: Contrapositive preserves truth; it can be easier to apply.

10) Spotting invalid comparisons

When two triangles do not have equal adjacent side pairs, you cannot use SAS-forward/converse directly. Likewise, when the angle provided is not the included angle, SAS does not apply. Always verify the required conditions before making a comparison statement.

Checkpoint F: Which information must you confirm before using SAS-forward?

Show Answer

That the two adjacent side pairs are equal and that the angles being compared are the included angles between those sides.

Mini-summary: Check conditions first, then conclude.

11) Application set: construction and design

In many applications (roof trusses, camera rigs, deployable frames), two struts have fixed lengths and the included angle is changed by a joint. Choosing a crossbar length or target opening angle is equivalent to selecting one outcome of the SAS relationship. Engineers rely on the monotonic relationship: as the included angle increases, the tip-to-tip distance increases.

Guiding question: If a design requires a maximum gap, should the included angle be minimized or maximized?

Show Answer

Maximized.

Mini-summary: To widen a fixed-arm frame, increase the included angle.

12) Synthesis task: Both directions in one scenario

Two triangles have equal adjacent side pairs $12cm\; and$ 18cm.\; Triangle\; X’s\; included\; angle\; is$ 41°and\; triangle\; Y’s\; included\; angle\; is$ 53°.\; Conclude\; the\; third-side\; comparison\; (forward).\; Then\; invert\; the\; statement\; (converse)\; to\; explain\; which\; included\; angle\; is\; larger\; if\; you\; were\; only\; told\; that\; Y’s\; third\; side\; exceeds\; X’s.$$$$

Show Answer

Forward: $\angle Y>\angle X\Rightarrow s{Y}_{\text{third}}>s{X}_{\text{third}}$. Converse: $s{Y}_{\text{third}}>s{X}_{\text{third}}\Rightarrow \angle Y>\angle X$.

Mini-summary: The two directions mirror each other under equal adjacent pairs.

13) Checklist and common pitfalls

• Verify equality of the two adjacent side pairs before using SAS.
• Confirm that the angle referred to is the included angle.
• Use strict inequalities for nondegenerate triangles.
• Do not conclude from a non-included angle.

Show Answer

Example correction: If you mistakenly used a non-included angle, restate the problem in terms of the included angle or gather the needed data.

Mini-summary: A short pre-check prevents invalid inferences.

14) References

• Geometry texts on triangle inequalities and side–angle relationships.
• Design notes from basic statics and linkages illustrating hinge-angle effects.

Example in Action

Five worked examples followed by five “Now You Try” tasks. Reveal the solution only after attempting each item. All mathematical statements are encoded in MathML and answers are inside collapsible panels.

Worked Example 1 — SAS Forward (Included angle controls the third side)

Two triangles share adjacent side lengths $12$ cm and $18$ cm. The included angles are $41°$ and $53°$. Which triangle has the longer third side?

Show Answer

$53°>41°\Rightarrow s{\mathrm{third}}_{\text{53°}}>s{\mathrm{third}}_{\text{41°}}$.

Worked Example 2 — SAS Converse (Third side controls the included angle)

Two frames are built with arms $10cm\; and$ 16cm.\; If\; one\; third\; side\; is$ 22cm\; and\; the\; other\; is$ 20cm,\; which\; included\; angle\; is\; larger?$$$$

Show Answer

$22>20\Rightarrow \angle \text{with 22}>\angle \text{with 20}$.

Worked Example 3 — SSS Within One Triangle (Order angles by sides)

In triangle $△MNO$, sides are $7.5,10.1,12.2$ cm. Order the angles from largest to smallest.

Show Answer

$\angle N>\angle O>\angle M$.

Worked Example 4 — Range First, Then Compare (Design bounds + SAS)

Two designs use the same adjacent arms $22cm\; and$ 35cm.\; Design\; A\; chooses\; the\; maximum\; feasible\; third\; side\; in$ [14,50].\; Design\; B\; chooses\; the\; minimum\; feasible\; third\; side\; in$ [30,55].\; Whose\; included\; angle\; is\; larger?$$$$

Show Answer

Design A (uses $t=50)\; has\; the\; larger\; included\; angle\; than\; Design\; B\; (uses$ t=30).$$

Worked Example 5 — Beware the Non-Included Angle

Two tripods have legs $9cm\; and$ 14cm\; meeting\; at\; a\; joint.\; A\; designer\; compares\; a\; non-included\; angle\; and\; claims\; tripod\; A’s\; third\; leg\; is\; longer\; because\; that\; non-included\; angle\; is\; larger.\; Explain\; the\; error\; and\; give\; the\; correct\; method.$$

Show Answer

Non-included angles do not determine the third side for fixed adjacent pairs. Compare included angles; larger included angle ⇒ longer third side.

Now You Try — 5 Tasks

1. Two triangles have equal adjacent sides $9$ cm and $15$ cm. If included angles are $36°$ and $49°$, which triangle has the longer third side?
   Show Answer

   The triangle with $49°$ has the longer third side.

2. Equal adjacent sides $11$ cm and $20$ cm. If one third side is $25$ and the other is $23,\; which\; included\; angle\; is\; larger?Show\; Answer$

   The triangle with third side $25has\; the\; larger\; included\; angle.$

3. In one triangle with sides $8,13,15cm,\; order\; the\; angles\; from\; largest\; to\; smallest.Show\; Answer$

   $\angle \mathrm{opp(15)}>\angle \mathrm{opp(13)}>\angle \mathrm{opp(8)}$.

4. Triangle X: $7,15,18;\; Triangle\; Y:$ 7,15,19.\; Which\; has\; the\; larger\; largest\; angle?Show\; Answer$$

   Triangle Y, since $19>18.$

5. Equal arms $27cm\; and$ 34cm.\; Catalog\; integers$ [12,50].\; Which\; value\; gives\; the\; widest\; included\; angle?Show\; Answer$$$

   $t=50.$

Try It Out

1. SAS Forward: Two triangles have equal adjacent side pairs $AB=DE=12$ cm and $AC=DF=17cm.\; If$ \angle A=38°and$ \angle D=51°,\; compare$ BCand$ EF.Show\; Answer$$$$$

   $\angle D>\angle A\Rightarrow EF>BC$.

2. SAS Converse: Equal adjacent arms $PQ=UV=10cm\; and$ PR=UW=19cm.\; If$ QR=23>VW=21,\; which\; included\; angle\; is\; larger?Show\; Answer$$$

   $QR>VW\Rightarrow \angle P>\angle U$.

3. SSS Within One Triangle: In $△MNO,\; sides$ m=5,$ n=11,$ o=12.\; Order\; the\; angles.Show\; Answer$$$$

   $\angle O>\angle N>\angle M$.

4. Cross-Triangle Largest Angle: Triangle X: $7,15,18.\; Triangle\; Y:$ 7,15,19.\; Which\; triangle\; has\; the\; larger\; largest\; angle?Show\; Answer$$

   Triangle Y.

5. Feasible Range + Angle Width: Equal arms $19cm\; and$ 27cm.\; Find\; the\; open\; interval\; for\; third\; side$ t$;\; decide\; which\; value\; gives\; a\; wider\; included\; angle:$ t=21or$ t=34.Show\; Answer$$$$

   Range: $|27-19|<t<46$ ⇒ $8<t<46.\; Wider\; at$ t=34.$$

6. Angle Must Be Included: Two tripods use equal adjacent legs; you are given non-included angles. Can you compare third legs?
   Show Answer

   No. Compare included angles only.

7. Structured SAS Justification: With $AB=DE=25,$ AC=DF=31,\; and$ \angle A>\angle D,\; conclude.Show\; Answer$$$

   $BC>EF$.

8. SSS Ordering with Equality: Sides $13,$ 13,$ 9.\; State\; the\; angle\; order.Show\; Answer$$$

   Two largest angles are equal; the smallest angle is opposite $9.$

9. From Words to Symbols: Translate: “With equal adjacent side pairs, the triangle with the longer third side has the larger included angle.”
   Show Answer

   $AB\cong DE,AC\cong DF,BC>EF\Rightarrow \angle A>\angle D$.

10. Cross-Design Decision: Arms $22cm\; and$ 35;\; pick\; between$ x=48and$ y=29for\; a\; wider\; included\; angle.Show\; Answer$$$$

    Choose $x=48(larger\; third\; side).$

Check Yourself

1. SAS Forward: Equal adjacent sides $13and$ 20cm;\; included\; angles$ 48°and$ 52°.\; Which\; triangle\; has\; the\; longer\; third\; side?\; State\; the\; inequality.Show\; Answer$$$$

   $52°>48°\Rightarrow s{\mathrm{third}}_{\text{52°}}>s{\mathrm{third}}_{\text{48°}}$.

2. SAS Converse: Equal adjacent sides $8and$ 15cm;\; third\; sides$ 19and$ 17.\; Which\; included\; angle\; is\; larger?Show\; Answer$$$$

   The one opposite third side $19.$

3. SSS Ordering: Sides $7,10,12cm.\; Order\; the\; angles\; from\; largest\; to\; smallest.Show\; Answer$

   $\angle \mathrm{opp(12)}>\angle \mathrm{opp(10)}>\angle \mathrm{opp(7)}$.

4. Across Two Triangles: X has $6,11,17;\; Y\; has$ 6,11,16.\; Which\; has\; the\; larger\; largest\; angle?Show\; Answer$$

   Triangle X.

5. Third-Side Interval: With sides $14and$ 23cm,\; find\; the\; open\; interval\; for$ t$.Show\; Answer$$

   $9<t<37.$

6. Catalog Decision: Equal arms $18and$ 26cm.\; Compare\; designs$ t=21and$ t=33for\; included-angle\; width.Show\; Answer$$$$

   $33>21\Rightarrow \; wider\; at$ t=33.$$

7. Non-Included Angle Trap: Is comparing non-included angles valid under equal adjacent pairs?
   Show Answer

   No.

8. Reverse Inequality: With equal adjacent sides, if $\angle A<\angle D,\; compare$ BCand$ EF.Show\; Answer$$$

   $BC<EF.$

9. Complete the Statement: With equal adjacent pairs and $\angle A>\angle D,\; conclude\; \_\_\_\_.Show\; Answer$

   $BC>EF.$

10. Allow Equality: If $a\ge b>cin\; one\; triangle,\; state\; the\; angle\; order.Show\; Answer$

    $\angle A\ge \angle B>\angle C(equality\; only\; when$ a=b).$$

11. SAS Equality Case: If $\angle A=\angle Dwith\; equal\; adjacent\; pairs,\; compare$ BCand$ EF.Show\; Answer$$$

    They are equal.

12. Doors and Hinges: Openings $30°vs.$ 75°.\; Which\; gap\; is\; wider?Show\; Answer$$

    The $75°opening.$

13. Symbolic Completion: Fill the blank: $AB\cong DE,AC\cong DF,BC>EF\Rightarrow \_\_\_\_$.
    Show Answer

    $\angle A>\angle D.$

Go Further

1. Design Studio: Pick the Widest Hinge

   Two designs use adjacent arms $24cm\; and$ 37cm\; with\; catalog\; choices$ \{20,26,32,38,45,50\}for\; the\; third\; side.\; Which\; creates\; the\; largest\; included\; angle?$$$

   Show Answer

   $t=50,\; since\; with\; equal\; adjacent\; sides\; a\; longer\; third\; side\; gives\; a\; larger\; included\; angle.$

2. Counterexample Builder: Non-Included Angle

   Two tripods use equal legs $10cm\; and$ 17cm.\; Show\; why\; comparing\; non-included\; angles\; cannot\; determine\; the\; longer\; third\; leg.$$

   Show Answer

   Rearranging the same non-included angles near different sides can flip which third leg is longer; only the included angle controls the spread for fixed adjacent legs.

3. Across-Triangle Reasoning

   Triangle X: $9,12,16.\; Triangle\; Y:$ 9,12,15.\; Decide\; which\; has\; the\; larger\; largest\; angle\; using\; SSS\; logic\; only.$$

   Show Answer

   Triangle X, since $16>15.$

4. Proof Skeleton

   Complete: with equal adjacent pairs and $\angle A>\angle D,\; conclude\; the\; third-side\; inequality\; and\; name\; the\; direction.$

   Show Answer

   $BC>EF(forward\; direction).$

5. Contrapositive Lens

   Rewrite SAS-forward as a contrapositive and apply it: if $BC\le EF,\; what\; is\; true\; of\; the\; included\; angles?$

   Show Answer

   $\angle A\le \angle D.$

My Reflection

Notebook Task: Title your page Comparing Sides and Angles – Reflection (Day 4) and respond to the prompts. Reveal sample ideas after you write.

3–2–1 Reflection

1. 3 things I learned today
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   • __________________________________________________________
   • __________________________________________________________
2. 2 places I can apply this
   • __________________________________________________________
   • __________________________________________________________
3. 1 question I still have

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Show Sample Ideas

Examples only:

1. Equal adjacent pairs + larger included angle ⇒ longer third side: $AB\cong DE,AC\cong DF,\angle A>\angle D\Rightarrow BC>EF$.
2. Within one triangle, longest side ↔ largest angle: if $a>b>c$, then $\angle A>\angle B>\angle C.$
3. Triangle inequality reminder: $a+b>c,a+c>b,b+c>a.$

Commitment Line

One small action I will do next session:

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