MAT8 Q2W7D4: SAS Inequality (Hinge) & Triangle Inequality

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

1. Decide which of two triangles has the longer third side or larger included angle when two corresponding sides are equal, justifying with SAS-inequality reasoning in at least 4 comparison items with ≥ 85% accuracy.
   $Given:\mathrm{AB}=\mathrm{DE},\mathrm{AC}=\mathrm{DF},\angle A>\angle D\Rightarrow \mathrm{BC}>\mathrm{EF}$
2. Use the triangle inequality to determine the possible range of a third side given two side lengths, and to test validity of three side lengths, in at least 6 tasks with complete inequality work shown.
   $|a-b|<c<a+b$
3. Solve real-world design-navigation problems that require comparing an included angle or an opposite side using SAS-inequality logic and triangle inequality, providing clear written justification in at least 3 word problems.

• Included angle - the angle formed by two given sides that share a common endpoint. Example: for sides $\mathrm{AB}$ and $\mathrm{AC}$, the included angle is $\angle A$.
• SAS Inequality (hinge) - with two pairs of corresponding equal sides, the triangle with the larger included angle has the longer opposite third side, and conversely the triangle with the longer opposite third side has the larger included angle.
  $\mathrm{AB}=\mathrm{DE},\mathrm{AC}=\mathrm{DF},\angle A>\angle D\Rightarrow \mathrm{BC}>\mathrm{EF}$
• Triangle inequality - in any triangle, the sum of the lengths of any two sides is greater than the length of the third side.
  $|a-b|<c<a+b$
• Third-side range - all possible values for the unknown side that satisfy the triangle inequality.
• Converse comparison - if two sides are equal pairwise and the third side is longer in one triangle, then the included angle opposite it is larger in that triangle.

A) Identify an included angle - In triangle $JKM$, for sides $\mathrm{JK}$ and $\mathrm{JM}$, name the included angle.

Show Answer

The included angle is formed at the shared endpoint $J$, so it is $\angle J$.

B) Quick triangle-validity test - Do lengths $5$, $7$, $12$ form a triangle? Use the three strict inequalities.

Show Answer

$5+7=12$ is not strictly greater, so it fails. Not a triangle.

C) Same two sides, different included angles - Two triangles have $\mathrm{AB}=\mathrm{DE}$, $\mathrm{AC}=\mathrm{DF}$. If $\angle A=62°$ and $\angle D=73°$, which third side is longer: $\mathrm{BC}$ or $\mathrm{EF}$?

Show Answer

Larger included angle gives longer opposite side. $\mathrm{EF}>\mathrm{BC}$.

D) Third-side range warm-up - If $a=8$ and $b=13$, find the allowable range for $c$.

Show Answer

$|8-13|<c<8+13$ gives $5<c<21$.

1) Why today’s ideas matter

Imagine two folding frames made from the same pair of metal bars joined by a hinge. You open one a little wider than the other. Which frame has the longer base? You do not need a tape measure - just reasoning. Today, you will learn two powerful comparison tools that turn such questions into quick, logical decisions: a within-triangle rule that limits which lengths can form a triangle and tells you the full range of a missing side, and a between-triangles rule that compares opposite sides when two corresponding sides are fixed and the included angle changes, and conversely compares the included angles when the third sides change.

• Use the exterior-free inequality for within-triangle range decisions.
• Use side-angle hinge comparisons to rank opposite sides across two triangles with the same adjacent legs.
• Prove or explain with short, clear justifications.

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2) Triangle inequality and third-side range

For side lengths $a$, $b$, $c$ within a single triangle:

$$a+b>c,b+c>a,c+a>b$$

This compresses to a third-side range:

$$|a-b|<c<a+b$$

Quick Demo A - Valid or not: Can segments $4$, $7$, $11$ form a triangle?

Show Answer

Check $4+7>11$? It equals, not greater. Not a triangle.

Quick Demo B - Third-side range: If $a=8$, $b=13$, then $5<c<21$.

Show Answer

Calculation: $|8-13|<c<8+13$ gives the open interval.

Mini-summary: When two sides are fixed, the third side must fall strictly between their difference and their sum. Equality creates a straight, degenerate case.

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3) SAS hinge: side vs included angle across two triangles

Statement: If triangles $\Delta ABC$ and $\Delta DEF$ satisfy $\mathrm{AB}=\mathrm{DE},\mathrm{AC}=\mathrm{DF},\angle A>\angle D$, then $\mathrm{BC}>\mathrm{EF}$. Conversely, if $\mathrm{BC}>\mathrm{EF}$, then $\angle A>\angle D$.

Guiding Check 1: With $\mathrm{AB}=\mathrm{DE}$, $\mathrm{AC}=\mathrm{DF}$ and $\angle A<\angle D$, compare $\mathrm{BC}$ and $\mathrm{EF}$.

Show Answer

Smaller included angle produces shorter opposite side. Therefore $\mathrm{BC}<\mathrm{EF}$.

Guiding Check 2: If the spans satisfy $\mathrm{BC}=\mathrm{EF}$ under the same leg pairs, what about the included angles?

Show Answer

Equal opposite spans imply equal included angles: $\angle A=\angle D$.

Mini-summary: With two corresponding equal sides fixed, a more open hinge gives a longer opposite side, and a longer opposite side signals a more open hinge.

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4) Blend the tools

Use triangle inequality to filter feasible third sides inside one triangle, then use the hinge comparison to rank designs across two triangles with the same legs.

Scenario Blend: Legs are $1.2$ m and $1.6$. Range for base $\mathrm{BC}$: $0.4<\mathrm{BC}<2.8$. Compare two openings, $40°$ vs $65°$: the larger angle yields the longer base.

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The candidate base $0.3$ m is invalid because it violates $0.4<\mathrm{BC}<2.8$. With valid designs, $65°$ gives the longer base.

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5) Real-world modeling

Construction - Two beams meet at a top joint; the base bay is opposite the joint. With the same beam lengths, opening the joint increases bay width.

Show Answer

Making the included angle extremely small risks too small an opposite span and approaches a degenerate configuration.

Navigation - With fixed legs $1.4$ km and $2.0$, a larger turn yields a longer start-to-finish distance.

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If the straight distance falls short, increase the included angle to increase the opposite span.

Sports - Two players keep fixed distances from a pivot. Opening the pivot angle widens the passing lane opposite the pivot.

Mini-summary: When two corresponding sides are fixed, opening the hinge lengthens the opposite side. The reverse is also true.

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6) Common pitfalls

• Using a non-included angle in hinge reasoning.
• Allowing equality in triangle inequality. Equality collapses the triangle.
• Comparing triangles with different legs by angles alone.

Show Answer

The angle must be the included angle between the two fixed sides because the opposite span is controlled by that angle. Non-included angles do not determine the span across those sides.

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7) References

• Big Ideas Math - Inequalities in one triangle; SAS hinge comparisons.
• CK-12 - Triangle inequality and third-side bounds.
• MathBitsNotebook - SAS inequality (hinge) demonstrations and practice.
• Open geometry notes - Comparison theorems within and between triangles.

Example 1 - SAS forward: compare third sides from included angles

Two triangles have equal corresponding legs $\mathrm{AB}=\mathrm{DE}$, $\mathrm{AC}=\mathrm{DF}$. Their included angles are $\angle A=58°$ and $\angle D=71°$. Decide which opposite third side is longer.

Show Answer

With the same two adjacent sides, the triangle with the larger included angle has the longer opposite third side. Since $71°>58°$, we get $\mathrm{EF}>\mathrm{BC}$.

Example 2 - SAS converse: compare included angles from third sides

Two triangles have $\mathrm{AB}=\mathrm{DE}$, $\mathrm{AC}=\mathrm{DF}$. The measured third sides are $\mathrm{BC}=12$ and $\mathrm{EF}=13$. Which included angle is larger?

Show Answer

Longer opposite third side implies larger included angle when adjacent side pairs are equal. Thus $\angle D>\angle A$.

Example 3 - Triangle inequality: third-side range and candidates

A triangle has $a=5.5$, $b=8$. Find the allowable range for $c$. Check $c=13$ and $c=12.9$.

Show Answer

Range $2.5<c<13.5$. Both 13 and 12.9 lie in the open interval. Possible.

Example 4 - Mixed design: filter by range, then compare by hinge

Legs $1.5$ m and $2.2$. Give the base range and decide which opening, $40°$ or $75°$, yields a longer base. Is $b=0.6$ feasible?

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Range $0.7<b<3.7$. With the same legs, $75°$ gives the longer base. A base of 0.6 m violates the lower bound, so not feasible.

Example 5 - Edge case awareness

Can segments $4$, $7$, $11$ form a triangle? If not, describe the shape and how to fix it.

Show Answer

$4+7=11$ gives a degenerate straight segment. Increase one of the shorter sides slightly or decrease the largest slightly to restore a strict inequality.

1. SAS forward - legs equal in pairs and $\angle D=67°$, $\angle A=52°$. Compare $\mathrm{EF}$ and $\mathrm{BC}$.
   Show Answer

   $\mathrm{EF}>\mathrm{BC}$ because the larger included angle faces the longer side.
2. SAS converse - legs equal in pairs and $\mathrm{BC}>\mathrm{EF}$. Compare $\angle A$ and $\angle D$.
   Show Answer

   $\angle A>\angle D$.
3. Third-side range - $\mathrm{PQ}=7.8$, $\mathrm{PR}=12.1$. Find $\mathrm{QR}$.
   Show Answer

   $4.3<\mathrm{QR}<19.9$.
4. Validity test - 6, 9, 14.
   Show Answer

   Valid because $6+9>14$.
5. Range then hinge - legs 1.4 m and 2.3 m. Range for base and which angle 42° or 88° gives longer base?
   Show Answer

   Range $0.9<b<3.7$. The 88° setting produces the longer base.
6. Achievability - legs 0.9 km and 1.7 km. Target straight distance 2.55 km?
   Show Answer

   Range $0.8<d<2.6$. 2.55 km is feasible by opening the angle sufficiently.
7. Rank bases by angle - 25°, 60°, 95°, 120°.
   Show Answer

   Order: 25° < 60° < 95° < 120°.
8. Not enough information - only $\mathrm{AB}=\mathrm{DE}$, and $\mathrm{BC}>\mathrm{EF}$. Can we conclude $\angle A>\angle D$?
   Show Answer

   No. Two corresponding side pairs must be equal to use the hinge comparison.
9. Integer values - sides 8 and 15. List integer third sides.
   Show Answer

   All integers 8 through 22 inclusive satisfy $7<x<23$.
10. Identify valid sets - (3,8,12), (5,9,13), (6.5,6.5,13.1).
    Show Answer

    (3,8,12) not valid, (5,9,13) valid, (6.5,6.5,13.1) not valid.

1. Legs equal in pairs. $\angle D=68°$, $\angle A=41°$. Compare $\mathrm{EF}$ and $\mathrm{BC}$.
   Show Answer

   $\mathrm{EF}>\mathrm{BC}$.
2. Converse hinge. $\mathrm{BC}=10.2$, $\mathrm{EF}=9.8$ with equal legs.
   Show Answer

   $\angle A>\angle D$.
3. Validity of 7, 9, 17.
   Show Answer

   Not valid since $7+9>17is\; false.$
4. Range for $\mathrm{QR}$ with $\mathrm{PQ}=8.5$, $\mathrm{PR}=13.2$.
   Show Answer

   $4.7<\mathrm{QR}<21.7$.
5. Order opposite sides by angle - 32°, 74°, 91°, 119°.
   Show Answer

   32° < 74° < 91° < 119°.
6. Feasibility with legs 1.1 and 1.9 km for 3.2 km straight distance.
   Show Answer

   Not achievable because $0.8<d<3.0$.
7. How many integers for third side if sides are 9 and 14?
   Show Answer

   17 values: 6 through 22 inclusive.
8. Can 4.2, 4.2, 8.4 form a triangle?
   Show Answer

   No. Equality creates a degenerate segment.
9. Base X longer than base Y with equal legs. Which hinge is larger?
   Show Answer

   Design X has the larger included angle.
10. Legs 2.0 m and 3.1 m. Range for base and which opening 38° or 82° is longer?
    Show Answer

    Range $1.1<b<5.1$. The 82° opening yields a longer base.
11. Choose all valid sets: (6,7,14), (5,9,15), (10,12,21).
    Show Answer

    Only (10,12,21) is valid.
12. Complete the third inequality: $a+b>c,b+c>a,?$.
    Show Answer

    $c+a>b$.
13. Why does equality fail in triangle inequality?
    Show Answer

    Equality is collinear and encloses no area. It is not a triangle.
14. Parameter interval for $z$ if $x=12$, $y=20$.
    Show Answer

    $8<z<32$.
15. Why must the hinge angle be included?
    Show Answer

    Because the opposite span is controlled by the angle between the fixed sides; non-included angles do not control that span.

Extension 1 - Threshold design

Equal legs 2.4 m each. Base must be at least 3.5 m. Is this possible, and should the included angle be small, moderate, or large?

Show Answer

Range $0<\mathrm{BC}<4.8$, so 3.5 m is possible. The included angle should be large to reach a larger opposite side.

Extension 2 - Converse hinge from field measurement

Same legs; measured bases satisfy $\mathrm{BC}>\mathrm{EF}$. What about the hinge angles?

Show Answer

$\angle A>\angle D$ by the converse hinge comparison.

Extension 3 - Max possible third side under a cap

With $a=9$, $b=14$, state the range for $c$. If a cap is $20$ cm, does it cover all possible triangles?

Show Answer

$5<c<23$. A 20 cm cap excludes values in $($$20,23$$)$, so it does not cover all triangles.

Extension 4 - Route planning

Legs 1.3 km and 2.6 km. Can you cross a 3.7 km river by adjusting only the turn?

Show Answer

Range $1.3<d<3.9$, so yes by opening the turn sufficiently.

Extension 5 - Indirect reasoning in a spec

Client claims: same legs and equal hinge angles but different base lengths. Is this possible?

Show Answer

No. With equal legs and equal included angles, opposite bases must match. The claim leads to contradiction.

Write in your notebook. Choose one prompt.

Option A - 3-2-1

• 3 insights about how the included angle controls the opposite side when two adjacent sides are fixed.
• 2 decisions you can now make faster in real situations and why.
• 1 question you still have about using ranges or the hinge idea.

Option B - Checklist

• I can compute a valid third-side interval using the difference-and-sum form.
• I can determine which base is longer when two corresponding sides are equal in both designs.
• I can use a converse argument when angles are unknown but opposite sides are measured.
• I can reject edge cases that only meet equality in the triangle inequality.

Optional sketch - Draw two triangles with the same pair of adjacent sides. Label a smaller included angle in one and a larger included angle in the other. Note how the opposite third side compares.