12.2 Triangles, Polygons, Quadrilaterals & the Pythagorean Theorem

Q: 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?

8 feet. Initially, the ladder forms a right triangle with hypotenuse c = 25 ft and base b1 = 7 ft. The initial wall height is h1 = √(25² - 7²) = √(625 - 49) = √576 = 24 ft (7-24-25 triple). When the top slips down 4 ft, the new wall height is h2 = 24 - 4 = 20 ft. The new base distance is b2 = √(25² - 20²) = √(625 - 400) = √225 = 15 ft (15-20-25 triple). The base slid outward by b2 - b1 = 15 - 7 = 8 feet.

Q: 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?

First, find the number of sides n. The exterior angle is E = 180° - 150° = 30°. Since the sum of exterior angles is 360°, n = 360° / 30° = 12 sides (a dodecagon). Next, apply the diagonal formula D = n(n - 3) / 2: D = 12(12 - 3) / 2 = 12(9) / 2 = 108 / 2 = 54 diagonals.

Q: 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?

12 cm. The ratio of the areas of two similar figures equals the square of the linear scale factor: Area(DEF) / Area(ABC) = 72 / 32 = 9 / 4. Taking the square root gives the linear scale factor k = √(9/4) = 3/2 = 1.5. Therefore, the corresponding side DE = k × AB = 1.5 × 8 cm = 12 cm.

Q: 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?

38 cm. Let the third side be x. By the Triangle Inequality Theorem, |14 - 9| 5 28 < P < 46. Among the given options, only 38 cm lies strictly between 28 cm and 46 cm (28 cm and 46 cm are boundary values and cannot form a valid triangle).

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.

Last updated: August 2026

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.

+---------------------------------------------------------------------------------------------------+
|                                   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 $180^\circ$ :
$m\angle A + m\angle B + m\angle C = 180^\circ$
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:
$m\angle\text{ext} = m\angle A + m\angle B$

                          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:
$a + b > c, \quad a + c > b, \quad b + c > a$
- 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: $|b - a| < c < a + b$
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):

a^2 + b^2 = c^2

+---------------------------------------------------------------------------------------------------+
|                         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.

+---------------------------------------------------------------------------------------------------+
|                                   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

+---------------------------------------------------------------------------------------------------+
|                                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 ( $a^2 + b^2 = c^2$ ): $h_1^2 + 7^2 = 25^2 \implies h_1^2 + 49 = 625 \implies h_1^2 = 576 \implies h_1 = 24\text{ ft}$ (Recognize the primitive 7-24-25 triple).
Calculate new height after top slips: $h_2 = 24 - 4 = 20\text{ ft}$
Calculate new base distance $b_2$ with ladder length $c = 25\text{ ft}$ : $b_2^2 + 20^2 = 25^2 \implies b_2^2 + 400 = 625 \implies b_2^2 = 225 \implies b_2 = 15\text{ ft}$ (Recognize the 3-4-5 triple scaled by 5: 15-20-25).
Calculate outward slide distance: $\Delta b = b_2 - b_1 = 15\text{ ft} - 7\text{ ft} = 8\text{ ft}$ 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 $E$ : $E = 180^\circ - I = 180^\circ - 150^\circ = 30^\circ$
Find the number of sides $n$ : $n = \frac{360^\circ}{E} = \frac{360^\circ}{30^\circ} = 12\text{ sides (Dodecagon)}$
Calculate the total number of diagonals: $D = \frac{n(n - 3)}{2} = \frac{12(12 - 3)}{2} = \frac{12 \times 9}{2} = \frac{108}{2} = 54\text{ diagonals}$

Problem 3: Similar Triangles and Area Ratios

Problem: Triangles $\triangle ABC$ and $\triangle DEF$ 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: $\frac{\text{Area}(\triangle DEF)}{\text{Area}(\triangle ABC)} = \frac{72}{32} = \frac{9}{4}$
Find the linear scale factor $k$ : $k = \sqrt{\frac{\text{Area}(\triangle DEF)}{\text{Area}(\triangle ABC)}} = \sqrt{\frac{9}{4}} = \frac{3}{2} = 1.5$
Solve for corresponding side $DE$ : $DE = k \times AB = \frac{3}{2} \times 8\text{ cm} = 12\text{ cm}$

Loading diagram...

Quadrilateral Classification Flowchart

Test Your Knowledge

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?

4 feet

8 feet

15 feet

18 feet

Test Your Knowledge

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?

108

Test Your Knowledge

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?

9.6 cm

10.5 cm

18 cm

12 cm

Test Your Knowledge

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?

38 cm

28 cm

46 cm

52 cm

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