12.2 Triangles, Polygons, Quadrilaterals & the Pythagorean Theorem
Key Takeaways
- Triangles are classified by side lengths (scalene, isosceles, equilateral) and interior angle measures (acute, right, obtuse, equiangular); the Triangle Angle Sum Theorem dictates that interior angles always sum to 180°.
- The Exterior Angle Theorem states that any exterior angle of a triangle equals the sum of its two remote interior angles; the Triangle Inequality Theorem mandates that the sum of any two sides must exceed the third side (|a - b| < c < a + b).
- The Pythagorean Theorem (a² + b² = c²) applies exclusively to right triangles, with common primitive triples including 3-4-5, 5-12-13, 8-15-17, and 7-24-25, alongside special right triangles (45°-45°-90° with ratio x : x : x√2 and 30°-60°-90° with ratio x : x√3 : 2x).
- Quadrilaterals follow a strict hierarchy: parallelograms possess opposite congruent parallel sides and bisecting diagonals; rectangles add four right angles and congruent diagonals; rhombuses add four congruent sides and perpendicular diagonals; squares embody all properties of both rectangles and rhombuses.
- For any convex n-gon, interior angles sum to (n - 2) × 180°, each interior angle in a regular n-gon equals ((n - 2) × 180°)/n, exterior angles always sum to 360°, and total diagonals equal n(n - 3)/2.
Triangles, Polygons, Quadrilaterals & the Pythagorean Theorem
Quick Answer: On the WEST-B (Objective 0015), polygon questions require mastery of key formulas and theorems: Triangle Angle Sum ($180^\circ$), Exterior Angle Theorem ($m\angle\text{ext} = m\angle A + m\angle B$), Triangle Inequality ($a + b > c$), and the Pythagorean Theorem ($a^2 + b^2 = c^2$). Recognize standard triples ($3-4-5, 5-12-13, 8-15-17, 7-24-25$) and special right triangles ($45^\circ-45^\circ-90^\circ$ with sides $x, x, x\sqrt{2}$; $30^\circ-60^\circ-90^\circ$ with sides $x, x\sqrt{3}, 2x$). For any $n$-sided convex polygon, Interior Angle Sum $= (n-2) \times 180^\circ$, Exterior Angle Sum $= 360^\circ$, and Number of Diagonals $= \frac{n(n-3)}{2}$. When triangles are similar ($\sim$) with scale factor $k$, perimeter ratio is $k$ and area ratio is $k^2$.
1. Triangle Classifications & Fundamental Theorems
Triangles are three-sided polygons ($n=3$) classified by their side relationships and interior angle measures.
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| TRIANGLE CLASSIFICATION MATRIX |
| |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | CLASSIFICATION BY SIDE| DEFINING PROPERTY | ANGLE IMPLICATION | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Scalene Triangle | All 3 sides different length| All 3 interior angles have diff deg | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Isosceles Triangle | At least 2 congruent sides | Base angles opposite equal sides ≅ | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Equilateral Triangle | All 3 sides congruent | Equiangular (all 3 angles = 60°) | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | CLASSIFICATION BY ANGLE DEFINING PROPERTY | SIDE IMPLICATION | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Acute Triangle | All 3 angles < 90° | a² + b² > c² (where c = longest) | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Right Triangle | Exactly one angle = 90° | a² + b² = c² (Pythagorean Theorem) | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Obtuse Triangle | Exactly one angle > 90° | a² + b² < c² (where c = longest) | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Equiangular Triangle | All 3 angles = 60° | Must be equilateral | |
| +-----------------------+-----------------------------+-------------------------------------+ |
+---------------------------------------------------------------------------------------------------+
Essential Triangle Theorems
-
Triangle Angle Sum Theorem: The sum of the interior angle measures of any triangle in a plane is always $180^\circ$:
-
Exterior Angle Theorem: The measure of an exterior angle of a triangle equals the sum of the measures of the two non-adjacent (remote) interior angles:
Exterior Angle Diagram
A
/\
/ \
/ \
/ \
B /________\ C _____ D (Exterior Ray)
\ /
\ /
∠ACD = ∠A + ∠B
-
Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side:
- Side Range Formula: If two side lengths $a$ and $b$ are known (with $a \le b$), the third side $c$ must fall strictly between their difference and their sum:
-
Side-Angle Inequality: In any triangle, the longest side is always opposite the largest angle, and the shortest side is always opposite the smallest angle.
2. The Pythagorean Theorem & Special Right Triangles
In any right triangle with perpendicular legs $a$ and $b$ and hypotenuse $c$ (opposite the $90^\circ$ angle):
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| HIGH-FREQUENCY PYTHAGOREAN TRIPLES & MULTIPLES |
| |
| +-------------------+----------------------------+------------------------------------------+ |
| | BASE TRIPLE | COMMON SCALAR MULTIPLES | APPLICATION CONTEXT | |
| +-------------------+----------------------------+------------------------------------------+ |
| | 3 - 4 - 5 | 6-8-10, 9-12-15, 12-16-20 | Standard test ladder & shadow problems | |
| | | 15-20-25, 30-40-50 | Scaled right triangles | |
| +-------------------+----------------------------+------------------------------------------+ |
| | 5 - 12 - 13 | 10-24-26, 15-36-39 | Diagonal distances & ramp problems | |
| +-------------------+----------------------------+------------------------------------------+ |
| | 8 - 15 - 17 | 16-30-34 | Perimeter & coordinate distances | |
| +-------------------+----------------------------+------------------------------------------+ |
| | 7 - 24 - 25 | 14-48-50 | High-yield right triangle test items | |
| +-------------------+----------------------------+------------------------------------------+ |
| | 9 - 40 - 41 | --- | Advanced hypotenuse calculations | |
| +-------------------+----------------------------+------------------------------------------+ |
+---------------------------------------------------------------------------------------------------+
Special Right Triangles
45°-45°-90° (Isosceles Right) 30°-60°-90° (Half Equilateral)
/| /|
/ | / |
/ | / |
x√2 / | x 2x / | x√3 (Long Leg)
/ | / | (opp 60°)
/ | / |
/45° | /30° |
/_______| /_______|
x x (Short Leg, opp 30°)
-
$45^\circ-45^\circ-90^\circ$ Triangle (Isosceles Right Triangle):
- Side Ratio: $1 : 1 : \sqrt{2}$
- Legs: $a = b = x$
- Hypotenuse: $c = x\sqrt{2}$
- Reverse: Given hypotenuse $c$, each leg $= \frac{c}{\sqrt{2}} = \frac{c\sqrt{2}}{2}$. (Commonly found as the diagonal of a square: $d = s\sqrt{2}$).
-
$30^\circ-60^\circ-90^\circ$ Triangle:
- Side Ratio: $1 : \sqrt{3} : 2$
- Short Leg (opposite $30^\circ$): $x$
- Long Leg (opposite $60^\circ$): $x\sqrt{3}$
- Hypotenuse (opposite $90^\circ$): $2x$
- Memory Hook: Hypotenuse is exactly twice the short leg; long leg is short leg times $\sqrt{3}$.
3. Quadrilateral Hierarchy & Properties
A quadrilateral is a four-sided polygon whose four interior angles sum to $(4-2)\times 180^\circ = 360^\circ$.
QUADRILATERAL HIERARCHY
Quadrilateral
(Sum = 360°)
/ \
/ \
Trapezoid Parallelogram
(1 pair ||) (2 pairs ||)
/ / \
/ / \
Isosceles Trapezoid Rectangle Rhombus
(legs ≅, diags ≅) (4 right ∠) (4 sides ≅)
\ /
\ /
Square
(Regular Quadrilateral)
Quadrilateral Property Reference Table
| Shape | Parallel Sides | Equal Sides | Angle Properties | Diagonal Properties |
|---|---|---|---|---|
| Trapezoid | 1 pair | None required | Consecutive angles between bases supplementary | Diagonals intersect |
| Isosceles Trapezoid | 1 pair | Non-parallel legs equal | Base angles equal ($m\angle A = m\angle B$) | Diagonals are congruent ($d_1 = d_2$) |
| Parallelogram | 2 pairs opposite | Opposite sides equal | Opposite angles equal; Consecutive supplementary | Diagonals bisect each other |
| Rectangle | 2 pairs opposite | Opposite sides equal | Four right angles ($90^\circ$) | Diagonals congruent & bisect each other |
| Rhombus | 2 pairs opposite | All 4 sides equal | Opposite angles equal; Consecutive supplementary | Diagonals perpendicular ($\perp$) & bisect vertex angles |
| Square | 2 pairs opposite | All 4 sides equal | Four right angles ($90^\circ$) | Diagonals congruent, perpendicular, and bisect angles ($45^\circ$) |
4. Polygon Formulas & Regular Polygons
An $n$-sided polygon ($n$-gon) has $n$ vertices and $n$ interior angles.
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| POLYGON FORMULAS REFERENCE |
| |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | FORMULA NAME | MATHEMATICAL EXPRESSION | APPLICATION NOTE | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Interior Angle Sum | S = (n - 2) × 180° | Valid for ANY convex n-gon | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | One Interior Angle | I = ((n - 2) × 180°) / n | ONLY for REGULAR (equiangular) n-gon| |
| | | or I = 180° - (360° / n) | | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Exterior Angle Sum | Always = 360° | One exterior angle per vertex | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | One Exterior Angle | E = 360° / n | ONLY for REGULAR n-gon | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Number of Diagonals | D = (n(n - 3)) / 2 | Total distinct interior diagonals | |
| +-----------------------+-----------------------------+-------------------------------------+ |
+---------------------------------------------------------------------------------------------------+
Common Regular Polygons Values
| Polygon Name | Sides ($n$) | Sum of Interior Angles | One Interior Angle (Regular) | One Exterior Angle | Total Diagonals |
|---|---|---|---|---|---|
| Triangle | 3 | $180^\circ$ | $60^\circ$ | $120^\circ$ | 0 |
| Quadrilateral | 4 | $360^\circ$ | $90^\circ$ | $90^\circ$ | 2 |
| Pentagon | 5 | $540^\circ$ | $108^\circ$ | $72^\circ$ | 5 |
| Hexagon | 6 | $720^\circ$ | $120^\circ$ | $60^\circ$ | 9 |
| Heptagon | 7 | $900^\circ$ | $\approx 128.57^\circ$ | $\approx 51.43^\circ$ | 14 |
| Octagon | 8 | $1080^\circ$ | $135^\circ$ | $45^\circ$ | 20 |
| Decagon | 10 | $1440^\circ$ | $144^\circ$ | $36^\circ$ | 35 |
| Dodecagon | 12 | $1800^\circ$ | $150^\circ$ | $30^\circ$ | 54 |
5. Triangle Congruence vs. Similarity
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| CONGRUENCE VS. SIMILARITY COMPARISON |
| |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | PROPERTY | CONGRUENCE (≅) | SIMILARITY (~) | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Definition | Same size AND same shape | Same shape, proportional size | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Corresponding Angles | Strictly Equal (m∠A = m∠D) | Strictly Equal (m∠A = m∠D) | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Corresponding Sides | Strictly Equal (AB = DE) | Proportional: DE/AB = EF/BC = k | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Valid Postulates | SSS, SAS, ASA, AAS, HL | AA (Angle-Angle), SAS ~, SSS ~ | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Non-Valid Postulates | AAA (similarity only!), SSA | A single angle pair is insufficient | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Ratio of Perimeters | 1 : 1 | Scale Factor = k | |
| +-----------------------+-----------------------------+-------------------------------------+ |
| | Ratio of Areas | 1 : 1 | Scale Factor Squared = k² | |
| +-----------------------+-----------------------------+-------------------------------------+ |
+---------------------------------------------------------------------------------------------------+
6. Step-by-Step Worked Problems & Derivations
Problem 1: Ladder Sliding Down a Wall (Pythagorean Application)
Problem: A 25-foot ladder is leaning against a vertical wall, with its base resting 7 feet from the bottom of the wall. If the top of the ladder slips down 4 feet, how many feet does the base of the ladder slide outward away from the wall?
Step-by-Step Solution:
- Model initial right triangle:
- Hypotenuse $c = 25\text{ ft}$, Base leg $b_1 = 7\text{ ft}$, Initial height $= h_1$.
- Apply Pythagorean Theorem ($a^2 + b^2 = c^2$): (Recognize the primitive 7-24-25 triple).
- Calculate new height after top slips:
- Calculate new base distance $b_2$ with ladder length $c = 25\text{ ft}$: (Recognize the 3-4-5 triple scaled by 5: 15-20-25).
- Calculate outward slide distance: The ladder base slides 8 feet outward.
Problem 2: Regular Polygon Angles and Diagonals
Problem: Each interior angle of a regular convex polygon measures $150^\circ$. Determine the number of sides $n$ and the total number of distinct diagonals in this polygon.
Step-by-Step Solution:
- Find one exterior angle $E$:
- Find the number of sides $n$:
- Calculate the total number of diagonals:
Problem 3: Similar Triangles and Area Ratios
Problem: Triangles $\triangle ABC$ and $ riangle DEF$ are similar ($\triangle ABC \sim \triangle DEF$). The area of $\triangle ABC$ is $32\text{ cm}^2$ and the area of $\triangle DEF$ is $72\text{ cm}^2$. If side $AB = 8\text{ cm}$, what is the length of corresponding side $DE$?
Step-by-Step Solution:
- Set up the ratio of areas:
- Find the linear scale factor $k$:
- Solve for corresponding side $DE$:
A 25-foot ladder is placed against a vertical building with its base 7 feet from the foundation. If the top of the ladder slips down 4 feet along the wall, how many feet will the base of the ladder slide outward away from the building?
Each interior angle of a regular convex polygon measures 150°. What is the total number of distinct diagonals that can be drawn inside this polygon?
Triangle ABC is similar to Triangle DEF (△ABC ~ △DEF). If the area of △ABC is 32 cm², the area of △DEF is 72 cm², and side AB = 8 cm, what is the length of the corresponding side DE?
A triangle has two side lengths measuring 9 cm and 14 cm. According to the Triangle Inequality Theorem, which of the following could be the perimeter of the triangle?