3.4 Basic geometric properties and shapes

Achievable Praxis Core: Math (5733)

3. Algebra and geometry

Our Praxis Core: Math course is currently in development and is a work-in-progress.

Basic geometric properties and shapes

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This section covers the basic properties of common two-dimensional shapes. You’ll classify triangles and quadrilaterals by their sides and angles, use Venn diagrams to organize special quadrilaterals, and identify right, acute, obtuse, and straight angles.

Definitions

Triangle: A three-sided polygon; classified by side lengths (scalene, isosceles, equilateral) or by angles (acute, right, obtuse).
Scalene triangle: All three sides have different lengths; all three angles are different.
Isosceles triangle: Two sides (and two angles) are equal.
Equilateral triangle: All three sides and all three angles (each $6 0^{\circ}$ ) are equal.
Quadrilateral: A four-sided polygon; categories include parallelograms, rectangles, squares, rhombuses, trapezoids, and kites.
Parallelogram: Opposite sides parallel and equal in length; opposite angles equal.
Rectangle: A parallelogram with four right angles.
Rhombus: A parallelogram with four equal sides.
Square: A rectangle and rhombus; four right angles and four equal sides.
Trapezoid: At least one pair of opposite sides parallel.
Isosceles trapezoid: A trapezoid with non-parallel sides equal.
Kite: Two distinct pairs of adjacent equal sides.
Angle: The union of two rays with a common endpoint; measured in degrees.

Identifying angles

Angles are classified by their measure in degrees. Being able to recognize angle types quickly helps you classify triangles and solve many geometry problems.

Right angle: exactly $9 0^{\circ}$ ; symbolized with a small square marker.
Acute angle: greater than $0^{\circ}$ and less than $9 0^{\circ}$ .
Obtuse angle: greater than $9 0^{\circ}$ and less than $18 0^{\circ}$ .
Straight angle: exactly $18 0^{\circ}$ ; forms a straight line.

Example: Solve missing angles in a triangle Triangle $A BC$ is isosceles with base angles $A$ and $B$ . The measure of angle $C$ is $4 0^{\circ}$ . Find the measure of angle $A$ .

Because the triangle is isosceles, the base angles are equal:

$A = B$

Use the triangle angle sum:

$A + A + 4 0^{\circ} = 18 0^{\circ}$

$2 A + 4 0^{\circ} = 18 0^{\circ}$

Subtract $4 0^{\circ}$ from both sides:

$2 A = 14 0^{\circ}$

Divide by $2$ :

$A = 7 0^{\circ}$

Answer: $7 0^{\circ}$

Triangles can be classified using either their side lengths or their angle measures. In many problems, you’ll first solve for unknown angles and then use those measures to classify the triangle.

Solving problems with angles in triangles

The interior angles of any triangle always add up to $18 0^{\circ}$ :

$A + B + C = 18 0^{\circ}$

This lets you set up an equation and use algebra to find missing angles, including angles written in terms of variables.

Example: Solve for the angles and classify the triangle A triangle has angles $x$ , $2 x$ , and $3 x$ . Find the angle measures and classify the triangle.

Set up the equation using the angle sum:

$x + 2 x + 3 x = 18 0^{\circ}$

$6 x = 18 0^{\circ}$

$x = 3 0^{\circ}$

Substitute back:

$x = 3 0^{\circ}$

$2 x = 6 0^{\circ}$

$3 x = 9 0^{\circ}$

The triangle contains a $9 0^{\circ}$ angle.

Answer: Right triangle

Example: Solve for $x$ and classify the triangle A triangle has angles $5 0^{\circ}$ , $(2 x + 10)^{\circ}$ , and $(x + 20)^{\circ}$ . Find $x$ and classify the triangle.

(spoiler)

Set up the equation:

$50 + (2 x + 10) + (x + 20) = 180$

$80 + 3 x = 180$

$3 x = 100$

$x = \frac{100}{3} \approx 33. 3^{\circ}$

Angle measures are approximately $5 0^{\circ}$ , $76. 7^{\circ}$ , and $53. 3^{\circ}$ . All angles are less than $9 0^{\circ}$ .

Answer: Acute triangle

When triangles are given by coordinates, you typically classify them using distances (for side lengths) and sometimes slopes (for right angles), rather than measuring angles directly.

To classify a triangle with vertices $(x_{1}, y_{1})$ , $(x_{2}, y_{2})$ , and $(x_{3}, y_{3})$ :

Compute side lengths using the distance formula

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Compare lengths
- All three equal $\Rightarrow$ equilateral
- Exactly two equal $\Rightarrow$ isosceles
- All different $\Rightarrow$ scalene
Check for a right angle using the converse of the Pythagorean theorem

$a^{2} + b^{2} = c^{2}$

where $c$ is the longest side.

For any triangle with side lengths $a \leq b \leq c$ :

Right if $c^{2} = a^{2} + b^{2}$
Acute if $c^{2} < a^{2} + b^{2}$
Obtuse if $c^{2} > a^{2} + b^{2}$

Example: Classify the triangle with vertices $(0, 0)$ , $(4, 0)$ , $(0, 3)$

Compute side lengths:

$A B = 4, A C = 3, BC = 5$

All sides are different, so the triangle is scalene.

$3^{2} + 4^{2} = 5^{2}$

The triangle satisfies the Pythagorean relationship.

Answer: Scalene right triangle

Diagrams show congruent angles with matching arcs and congruent side lengths with matching tick marks.

Finding plausible third sides of triangles

For any triangle with sides of lengths $x$ , $y$ , and $z$ , the following conditions must all be true:

$x + y > z$
$y + z > x$
$x + z > y$

These are called the triangle inequalities. They guarantee that the sum of any two sides is greater than the remaining side.

Example: If two sides of a triangle measure $5$ and $8$ , what are the possible lengths of the third side?

Let the third side be $x$ .

Apply the triangle inequalities:

$5 + 8 > x ⟹ x < 13$

$5 + x > 8 ⟹ x > 3$

$8 + x > 5$ (always true for $x > 0$ )

Combine the valid bounds.

$3 < x < 13$

Answer: $3 < x < 13$

Example: If two sides of a triangle measure $7$ and $10$ , find the bounds on the third side.

(spoiler)

$7 + 10 > x ⟹ x < 17$
$7 + x > 10 ⟹ x > 3$
$10 + x > 7$ (always true for $x > 0$ )

Answer: $3 < x < 17$

Classifying quadrilaterals

Sidenote

Classifying quadrilaterals

Quadrilateral	Properties	Diagonals
Parallelogram	Opposite sides are parallel and equal	Bisect each other
Rectangle	A parallelogram with all angles equal to $9 0^{\circ}$	Bisect each other and are equal in length
Rhombus	A parallelogram with all sides equal	Bisect each other at right angles
Square	A rectangle and a rhombus (all sides equal, all angles $9 0^{\circ}$ )	Bisect each other, equal in length, and perpendicular
Trapezoid	At least one pair of parallel sides	No general rule
Isosceles trapezoid	A trapezoid with non-parallel sides equal	Equal in length
Kite	Two pairs of adjacent equal sides	Perpendicular; one diagonal bisects the other

Example: Classifying a quadrilateral A quadrilateral has opposite sides parallel and equal in length. All of its angles are $9 0^{\circ}$ , but not all sides are equal. What type of quadrilateral is it?

Opposite sides parallel and equal imply a parallelogram or rectangle.

All angles are $9 0^{\circ}$ , consistent with a rectangle or square.

Not all sides are equal, which eliminates square.

Answer: Rectangle

Example: Classifying a quadrilateral Classify the quadrilateral based on the image below.
Quadrilateral analysis

(spoiler)

How to classify this shape

All four sides $A B, BC, C D, D A$ have tick marks showing they are equal in length.
The diagonals $A C$ and $B D$ intersect at right angles.
A quadrilateral with all four sides equal and diagonals that bisect each other at right angles is a rhombus.

Answer: Rhombus

Special quadrilateral Venn diagram

A Venn diagram helps you see how special quadrilaterals fit inside broader categories:

The large oval represents Quadrilaterals.
Inside, one circle represents Parallelograms, another represents Trapezoids, and a separate circle represents Kites.
Within Parallelograms: overlapping circles represent Rhombi and Rectangles; their overlap represents Squares.
Within Trapezoids: a smaller circle represents Isosceles Trapezoids.

Coordinate geometry and shape classification

On the coordinate plane, you can classify polygons using distances and slopes:

Distances let you compare side lengths.
Slopes help you identify parallel and perpendicular sides.

Together, these tools let you classify triangles and quadrilaterals using coordinates alone.

For any two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , the distance between them is found using the distance formula:

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} .$

Equal distances indicate congruent sides, which is useful for identifying isosceles triangles, rhombi, or squares.

The slope of the line through $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is given by

$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}},$

provided $x_{2} \neq = x_{1}$ . Equal slopes indicate parallel sides, while slopes whose product is $- 1$ indicate perpendicular sides (when both slopes are defined).

By combining distance and slope:

Equal opposite slopes suggest a parallelogram.
Perpendicular adjacent slopes indicate right angles.
Equal side lengths distinguish rhombi and squares from rectangles.

Using slopes and distances to classify quadrilaterals

Parallelogram
- Opposite sides equal: compare distances of $\overline{A B}$ with $\overline{C D}$ and $\overline{BC}$ with $\overline{D A}$ .
- Opposite sides parallel: compare slopes $m_{A B} = m_{C D}$ and $m_{BC} = m_{D A}$ .
Rectangle
- All four right angles: check adjacent sides are perpendicular, i.e. $m_{A B} \cdot m_{BC} = - 1$ .
- Equivalently, parallelogram with one right angle.
Rhombus
- All four sides equal: check all pairwise distances equal.
- Parallelogram with equal sides.
Square
- Both rectangle and rhombus conditions hold.
Isosceles trapezoid
- One pair of opposite sides parallel: $m_{A B} = m_{C D}$ but $m_{BC} \neq = m_{D A}$ .
- Non-parallel sides equal: $d_{BC} = d_{D A}$ .

Example: Classify a quadrilateral by coordinates Given $A (0, 0)$ , $B (4, 0)$ , $C (5, 3)$ , $D (1, 3)$ , determine the type.

Compute distances:

$A B = 4$

$BC = 10$

$C D = 4$

$D A = 10$

Opposite sides are equal.

Compute slopes:

$m_{A B} = 0$ , $m_{C D} = 0$

$m_{BC} = 3$ , $m_{D A} = 3$

Opposite sides are parallel.

Adjacent slopes are not perpendicular.

All sides are not equal.

Answer: Parallelogram (non-rectangular, non-rhombus)

Basic geometric properties and shapes

Definitions

Triangle: A three-sided polygon; classified by side lengths (scalene, isosceles, equilateral) or by angles (acute, right, obtuse).
Scalene triangle: All three sides have different lengths; all three angles are different.
Isosceles triangle: Two sides (and two angles) are equal.
Equilateral triangle: All three sides and all three angles (each $6 0^{\circ}$ ) are equal.
Quadrilateral: A four-sided polygon; categories include parallelograms, rectangles, squares, rhombuses, trapezoids, and kites.
Parallelogram: Opposite sides parallel and equal in length; opposite angles equal.
Rectangle: A parallelogram with four right angles.
Rhombus: A parallelogram with four equal sides.
Square: A rectangle and rhombus; four right angles and four equal sides.
Trapezoid: At least one pair of opposite sides parallel.
Isosceles trapezoid: A trapezoid with non-parallel sides equal.
Kite: Two distinct pairs of adjacent equal sides.
Angle: The union of two rays with a common endpoint; measured in degrees.

Identifying angles

Angles are classified by their measure in degrees. Being able to recognize angle types quickly helps you classify triangles and solve many geometry problems.

Right angle: exactly $9 0^{\circ}$ ; symbolized with a small square marker.
Acute angle: greater than $0^{\circ}$ and less than $9 0^{\circ}$ .
Obtuse angle: greater than $9 0^{\circ}$ and less than $18 0^{\circ}$ .
Straight angle: exactly $18 0^{\circ}$ ; forms a straight line.

Example: Solve missing angles in a triangle Triangle $A BC$ is isosceles with base angles $A$ and $B$ . The measure of angle $C$ is $4 0^{\circ}$ . Find the measure of angle $A$ .

Because the triangle is isosceles, the base angles are equal:

$A = B$

Use the triangle angle sum:

$A + A + 4 0^{\circ} = 18 0^{\circ}$

$2 A + 4 0^{\circ} = 18 0^{\circ}$

Subtract $4 0^{\circ}$ from both sides:

$2 A = 14 0^{\circ}$

Divide by $2$ :

$A = 7 0^{\circ}$

Answer: $7 0^{\circ}$

Triangles can be classified using either their side lengths or their angle measures. In many problems, you’ll first solve for unknown angles and then use those measures to classify the triangle.

Solving problems with angles in triangles

The interior angles of any triangle always add up to $18 0^{\circ}$ :

$A + B + C = 18 0^{\circ}$

This lets you set up an equation and use algebra to find missing angles, including angles written in terms of variables.

Example: Solve for the angles and classify the triangle A triangle has angles $x$ , $2 x$ , and $3 x$ . Find the angle measures and classify the triangle.

Set up the equation using the angle sum:

$x + 2 x + 3 x = 18 0^{\circ}$

$6 x = 18 0^{\circ}$

$x = 3 0^{\circ}$

Substitute back:

$x = 3 0^{\circ}$

$2 x = 6 0^{\circ}$

$3 x = 9 0^{\circ}$

The triangle contains a $9 0^{\circ}$ angle.

Answer: Right triangle

Example: Solve for $x$ and classify the triangle A triangle has angles $5 0^{\circ}$ , $(2 x + 10)^{\circ}$ , and $(x + 20)^{\circ}$ . Find $x$ and classify the triangle.

(spoiler)

Set up the equation:

$50 + (2 x + 10) + (x + 20) = 180$

$80 + 3 x = 180$

$3 x = 100$

$x = \frac{100}{3} \approx 33. 3^{\circ}$

Angle measures are approximately $5 0^{\circ}$ , $76. 7^{\circ}$ , and $53. 3^{\circ}$ . All angles are less than $9 0^{\circ}$ .

Answer: Acute triangle

When triangles are given by coordinates, you typically classify them using distances (for side lengths) and sometimes slopes (for right angles), rather than measuring angles directly.

To classify a triangle with vertices $(x_{1}, y_{1})$ , $(x_{2}, y_{2})$ , and $(x_{3}, y_{3})$ :

Compute side lengths using the distance formula

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Compare lengths
- All three equal $\Rightarrow$ equilateral
- Exactly two equal $\Rightarrow$ isosceles
- All different $\Rightarrow$ scalene
Check for a right angle using the converse of the Pythagorean theorem

$a^{2} + b^{2} = c^{2}$

where $c$ is the longest side.

For any triangle with side lengths $a \leq b \leq c$ :

Right if $c^{2} = a^{2} + b^{2}$
Acute if $c^{2} < a^{2} + b^{2}$
Obtuse if $c^{2} > a^{2} + b^{2}$

Example: Classify the triangle with vertices $(0, 0)$ , $(4, 0)$ , $(0, 3)$

Compute side lengths:

$A B = 4, A C = 3, BC = 5$

All sides are different, so the triangle is scalene.

$3^{2} + 4^{2} = 5^{2}$

The triangle satisfies the Pythagorean relationship.

Answer: Scalene right triangle

Diagrams show congruent angles with matching arcs and congruent side lengths with matching tick marks.

Finding plausible third sides of triangles

For any triangle with sides of lengths $x$ , $y$ , and $z$ , the following conditions must all be true:

$x + y > z$
$y + z > x$
$x + z > y$

These are called the triangle inequalities. They guarantee that the sum of any two sides is greater than the remaining side.

Example: If two sides of a triangle measure $5$ and $8$ , what are the possible lengths of the third side?

Let the third side be $x$ .

Apply the triangle inequalities:

$5 + 8 > x ⟹ x < 13$

$5 + x > 8 ⟹ x > 3$

$8 + x > 5$ (always true for $x > 0$ )

Combine the valid bounds.

$3 < x < 13$

Answer: $3 < x < 13$

Example: If two sides of a triangle measure $7$ and $10$ , find the bounds on the third side.

(spoiler)

$7 + 10 > x ⟹ x < 17$
$7 + x > 10 ⟹ x > 3$
$10 + x > 7$ (always true for $x > 0$ )

Answer: $3 < x < 17$

Classifying quadrilaterals

Sidenote

Classifying quadrilaterals

Quadrilateral	Properties	Diagonals
Parallelogram	Opposite sides are parallel and equal	Bisect each other
Rectangle	A parallelogram with all angles equal to $9 0^{\circ}$	Bisect each other and are equal in length
Rhombus	A parallelogram with all sides equal	Bisect each other at right angles
Square	A rectangle and a rhombus (all sides equal, all angles $9 0^{\circ}$ )	Bisect each other, equal in length, and perpendicular
Trapezoid	At least one pair of parallel sides	No general rule
Isosceles trapezoid	A trapezoid with non-parallel sides equal	Equal in length
Kite	Two pairs of adjacent equal sides	Perpendicular; one diagonal bisects the other

Example: Classifying a quadrilateral A quadrilateral has opposite sides parallel and equal in length. All of its angles are $9 0^{\circ}$ , but not all sides are equal. What type of quadrilateral is it?

Opposite sides parallel and equal imply a parallelogram or rectangle.

All angles are $9 0^{\circ}$ , consistent with a rectangle or square.

Not all sides are equal, which eliminates square.

Answer: Rectangle

Example: Classifying a quadrilateral Classify the quadrilateral based on the image below.

(spoiler)

How to classify this shape

All four sides $A B, BC, C D, D A$ have tick marks showing they are equal in length.
The diagonals $A C$ and $B D$ intersect at right angles.
A quadrilateral with all four sides equal and diagonals that bisect each other at right angles is a rhombus.

Answer: Rhombus

Special quadrilateral Venn diagram

A Venn diagram helps you see how special quadrilaterals fit inside broader categories:

The large oval represents Quadrilaterals.
Inside, one circle represents Parallelograms, another represents Trapezoids, and a separate circle represents Kites.
Within Parallelograms: overlapping circles represent Rhombi and Rectangles; their overlap represents Squares.
Within Trapezoids: a smaller circle represents Isosceles Trapezoids.

Coordinate geometry and shape classification

On the coordinate plane, you can classify polygons using distances and slopes:

Distances let you compare side lengths.
Slopes help you identify parallel and perpendicular sides.

Together, these tools let you classify triangles and quadrilaterals using coordinates alone.

For any two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , the distance between them is found using the distance formula:

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} .$

Equal distances indicate congruent sides, which is useful for identifying isosceles triangles, rhombi, or squares.

The slope of the line through $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is given by

$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}},$

provided $x_{2} \neq = x_{1}$ . Equal slopes indicate parallel sides, while slopes whose product is $- 1$ indicate perpendicular sides (when both slopes are defined).

By combining distance and slope:

Equal opposite slopes suggest a parallelogram.
Perpendicular adjacent slopes indicate right angles.
Equal side lengths distinguish rhombi and squares from rectangles.

Using slopes and distances to classify quadrilaterals

Parallelogram
- Opposite sides equal: compare distances of $\overline{A B}$ with $\overline{C D}$ and $\overline{BC}$ with $\overline{D A}$ .
- Opposite sides parallel: compare slopes $m_{A B} = m_{C D}$ and $m_{BC} = m_{D A}$ .
Rectangle
- All four right angles: check adjacent sides are perpendicular, i.e. $m_{A B} \cdot m_{BC} = - 1$ .
- Equivalently, parallelogram with one right angle.
Rhombus
- All four sides equal: check all pairwise distances equal.
- Parallelogram with equal sides.
Square
- Both rectangle and rhombus conditions hold.
Isosceles trapezoid
- One pair of opposite sides parallel: $m_{A B} = m_{C D}$ but $m_{BC} \neq = m_{D A}$ .
- Non-parallel sides equal: $d_{BC} = d_{D A}$ .

Example: Classify a quadrilateral by coordinates Given $A (0, 0)$ , $B (4, 0)$ , $C (5, 3)$ , $D (1, 3)$ , determine the type.

Compute distances:

$A B = 4$

$BC = 10$

$C D = 4$

$D A = 10$

Opposite sides are equal.

Compute slopes:

$m_{A B} = 0$ , $m_{C D} = 0$

$m_{BC} = 3$ , $m_{D A} = 3$

Opposite sides are parallel.

Adjacent slopes are not perpendicular.

All sides are not equal.

Answer: Parallelogram (non-rectangular, non-rhombus)

Achievable Praxis Core: Math (5733)

3. Algebra and geometry

Our Praxis Core: Math course is currently in development and is a work-in-progress.

Basic geometric properties and shapes

10 min read

Font

Discuss

Feedback

Definitions

Triangle: A three-sided polygon; classified by side lengths (scalene, isosceles, equilateral) or by angles (acute, right, obtuse).
Scalene triangle: All three sides have different lengths; all three angles are different.
Isosceles triangle: Two sides (and two angles) are equal.
Equilateral triangle: All three sides and all three angles (each $6 0^{\circ}$ ) are equal.
Quadrilateral: A four-sided polygon; categories include parallelograms, rectangles, squares, rhombuses, trapezoids, and kites.
Parallelogram: Opposite sides parallel and equal in length; opposite angles equal.
Rectangle: A parallelogram with four right angles.
Rhombus: A parallelogram with four equal sides.
Square: A rectangle and rhombus; four right angles and four equal sides.
Trapezoid: At least one pair of opposite sides parallel.
Isosceles trapezoid: A trapezoid with non-parallel sides equal.
Kite: Two distinct pairs of adjacent equal sides.
Angle: The union of two rays with a common endpoint; measured in degrees.

Identifying angles

Angles are classified by their measure in degrees. Being able to recognize angle types quickly helps you classify triangles and solve many geometry problems.

Right angle: exactly $9 0^{\circ}$ ; symbolized with a small square marker.
Acute angle: greater than $0^{\circ}$ and less than $9 0^{\circ}$ .
Obtuse angle: greater than $9 0^{\circ}$ and less than $18 0^{\circ}$ .
Straight angle: exactly $18 0^{\circ}$ ; forms a straight line.

Example: Solve missing angles in a triangle Triangle $A BC$ is isosceles with base angles $A$ and $B$ . The measure of angle $C$ is $4 0^{\circ}$ . Find the measure of angle $A$ .

Because the triangle is isosceles, the base angles are equal:

$A = B$

Use the triangle angle sum:

$A + A + 4 0^{\circ} = 18 0^{\circ}$

$2 A + 4 0^{\circ} = 18 0^{\circ}$

Subtract $4 0^{\circ}$ from both sides:

$2 A = 14 0^{\circ}$

Divide by $2$ :

$A = 7 0^{\circ}$

Answer: $7 0^{\circ}$

Triangles can be classified using either their side lengths or their angle measures. In many problems, you’ll first solve for unknown angles and then use those measures to classify the triangle.

Solving problems with angles in triangles

The interior angles of any triangle always add up to $18 0^{\circ}$ :

$A + B + C = 18 0^{\circ}$

This lets you set up an equation and use algebra to find missing angles, including angles written in terms of variables.

Example: Solve for the angles and classify the triangle A triangle has angles $x$ , $2 x$ , and $3 x$ . Find the angle measures and classify the triangle.

Set up the equation using the angle sum:

$x + 2 x + 3 x = 18 0^{\circ}$

$6 x = 18 0^{\circ}$

$x = 3 0^{\circ}$

Substitute back:

$x = 3 0^{\circ}$

$2 x = 6 0^{\circ}$

$3 x = 9 0^{\circ}$

The triangle contains a $9 0^{\circ}$ angle.

Answer: Right triangle

Example: Solve for $x$ and classify the triangle A triangle has angles $5 0^{\circ}$ , $(2 x + 10)^{\circ}$ , and $(x + 20)^{\circ}$ . Find $x$ and classify the triangle.

(spoiler)

Set up the equation:

$50 + (2 x + 10) + (x + 20) = 180$

$80 + 3 x = 180$

$3 x = 100$

$x = \frac{100}{3} \approx 33. 3^{\circ}$

Angle measures are approximately $5 0^{\circ}$ , $76. 7^{\circ}$ , and $53. 3^{\circ}$ . All angles are less than $9 0^{\circ}$ .

Answer: Acute triangle

When triangles are given by coordinates, you typically classify them using distances (for side lengths) and sometimes slopes (for right angles), rather than measuring angles directly.

To classify a triangle with vertices $(x_{1}, y_{1})$ , $(x_{2}, y_{2})$ , and $(x_{3}, y_{3})$ :

Compute side lengths using the distance formula

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Compare lengths
- All three equal $\Rightarrow$ equilateral
- Exactly two equal $\Rightarrow$ isosceles
- All different $\Rightarrow$ scalene
Check for a right angle using the converse of the Pythagorean theorem

$a^{2} + b^{2} = c^{2}$

where $c$ is the longest side.

For any triangle with side lengths $a \leq b \leq c$ :

Right if $c^{2} = a^{2} + b^{2}$
Acute if $c^{2} < a^{2} + b^{2}$
Obtuse if $c^{2} > a^{2} + b^{2}$

Example: Classify the triangle with vertices $(0, 0)$ , $(4, 0)$ , $(0, 3)$

Compute side lengths:

$A B = 4, A C = 3, BC = 5$

All sides are different, so the triangle is scalene.

$3^{2} + 4^{2} = 5^{2}$

The triangle satisfies the Pythagorean relationship.

Answer: Scalene right triangle

Diagrams show congruent angles with matching arcs and congruent side lengths with matching tick marks.

Finding plausible third sides of triangles

For any triangle with sides of lengths $x$ , $y$ , and $z$ , the following conditions must all be true:

$x + y > z$
$y + z > x$
$x + z > y$

These are called the triangle inequalities. They guarantee that the sum of any two sides is greater than the remaining side.

Example: If two sides of a triangle measure $5$ and $8$ , what are the possible lengths of the third side?

Let the third side be $x$ .

Apply the triangle inequalities:

$5 + 8 > x ⟹ x < 13$

$5 + x > 8 ⟹ x > 3$

$8 + x > 5$ (always true for $x > 0$ )

Combine the valid bounds.

$3 < x < 13$

Answer: $3 < x < 13$

Example: If two sides of a triangle measure $7$ and $10$ , find the bounds on the third side.

(spoiler)

$7 + 10 > x ⟹ x < 17$
$7 + x > 10 ⟹ x > 3$
$10 + x > 7$ (always true for $x > 0$ )

Answer: $3 < x < 17$

Classifying quadrilaterals

Sidenote

Classifying quadrilaterals

Quadrilateral	Properties	Diagonals
Parallelogram	Opposite sides are parallel and equal	Bisect each other
Rectangle	A parallelogram with all angles equal to $9 0^{\circ}$	Bisect each other and are equal in length
Rhombus	A parallelogram with all sides equal	Bisect each other at right angles
Square	A rectangle and a rhombus (all sides equal, all angles $9 0^{\circ}$ )	Bisect each other, equal in length, and perpendicular
Trapezoid	At least one pair of parallel sides	No general rule
Isosceles trapezoid	A trapezoid with non-parallel sides equal	Equal in length
Kite	Two pairs of adjacent equal sides	Perpendicular; one diagonal bisects the other

Example: Classifying a quadrilateral A quadrilateral has opposite sides parallel and equal in length. All of its angles are $9 0^{\circ}$ , but not all sides are equal. What type of quadrilateral is it?

Opposite sides parallel and equal imply a parallelogram or rectangle.

All angles are $9 0^{\circ}$ , consistent with a rectangle or square.

Not all sides are equal, which eliminates square.

Answer: Rectangle

Example: Classifying a quadrilateral Classify the quadrilateral based on the image below.
Quadrilateral analysis

(spoiler)

How to classify this shape

All four sides $A B, BC, C D, D A$ have tick marks showing they are equal in length.
The diagonals $A C$ and $B D$ intersect at right angles.
A quadrilateral with all four sides equal and diagonals that bisect each other at right angles is a rhombus.

Answer: Rhombus

Special quadrilateral Venn diagram

A Venn diagram helps you see how special quadrilaterals fit inside broader categories:

The large oval represents Quadrilaterals.
Inside, one circle represents Parallelograms, another represents Trapezoids, and a separate circle represents Kites.
Within Parallelograms: overlapping circles represent Rhombi and Rectangles; their overlap represents Squares.
Within Trapezoids: a smaller circle represents Isosceles Trapezoids.

Coordinate geometry and shape classification

On the coordinate plane, you can classify polygons using distances and slopes:

Distances let you compare side lengths.
Slopes help you identify parallel and perpendicular sides.

Together, these tools let you classify triangles and quadrilaterals using coordinates alone.

For any two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , the distance between them is found using the distance formula:

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} .$

Equal distances indicate congruent sides, which is useful for identifying isosceles triangles, rhombi, or squares.

The slope of the line through $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is given by

$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}},$

provided $x_{2} \neq = x_{1}$ . Equal slopes indicate parallel sides, while slopes whose product is $- 1$ indicate perpendicular sides (when both slopes are defined).

By combining distance and slope:

Equal opposite slopes suggest a parallelogram.
Perpendicular adjacent slopes indicate right angles.
Equal side lengths distinguish rhombi and squares from rectangles.

Using slopes and distances to classify quadrilaterals

Parallelogram
- Opposite sides equal: compare distances of $\overline{A B}$ with $\overline{C D}$ and $\overline{BC}$ with $\overline{D A}$ .
- Opposite sides parallel: compare slopes $m_{A B} = m_{C D}$ and $m_{BC} = m_{D A}$ .
Rectangle
- All four right angles: check adjacent sides are perpendicular, i.e. $m_{A B} \cdot m_{BC} = - 1$ .
- Equivalently, parallelogram with one right angle.
Rhombus
- All four sides equal: check all pairwise distances equal.
- Parallelogram with equal sides.
Square
- Both rectangle and rhombus conditions hold.
Isosceles trapezoid
- One pair of opposite sides parallel: $m_{A B} = m_{C D}$ but $m_{BC} \neq = m_{D A}$ .
- Non-parallel sides equal: $d_{BC} = d_{D A}$ .

Example: Classify a quadrilateral by coordinates Given $A (0, 0)$ , $B (4, 0)$ , $C (5, 3)$ , $D (1, 3)$ , determine the type.

Compute distances:

$A B = 4$

$BC = 10$

$C D = 4$

$D A = 10$

Opposite sides are equal.

Compute slopes:

$m_{A B} = 0$ , $m_{C D} = 0$

$m_{BC} = 3$ , $m_{D A} = 3$

Opposite sides are parallel.

Adjacent slopes are not perpendicular.

All sides are not equal.

Answer: Parallelogram (non-rectangular, non-rhombus)

Basic geometric properties and shapes

Definitions

Triangle: A three-sided polygon; classified by side lengths (scalene, isosceles, equilateral) or by angles (acute, right, obtuse).
Scalene triangle: All three sides have different lengths; all three angles are different.
Isosceles triangle: Two sides (and two angles) are equal.
Equilateral triangle: All three sides and all three angles (each $6 0^{\circ}$ ) are equal.
Quadrilateral: A four-sided polygon; categories include parallelograms, rectangles, squares, rhombuses, trapezoids, and kites.
Parallelogram: Opposite sides parallel and equal in length; opposite angles equal.
Rectangle: A parallelogram with four right angles.
Rhombus: A parallelogram with four equal sides.
Square: A rectangle and rhombus; four right angles and four equal sides.
Trapezoid: At least one pair of opposite sides parallel.
Isosceles trapezoid: A trapezoid with non-parallel sides equal.
Kite: Two distinct pairs of adjacent equal sides.
Angle: The union of two rays with a common endpoint; measured in degrees.

Identifying angles

Angles are classified by their measure in degrees. Being able to recognize angle types quickly helps you classify triangles and solve many geometry problems.

Right angle: exactly $9 0^{\circ}$ ; symbolized with a small square marker.
Acute angle: greater than $0^{\circ}$ and less than $9 0^{\circ}$ .
Obtuse angle: greater than $9 0^{\circ}$ and less than $18 0^{\circ}$ .
Straight angle: exactly $18 0^{\circ}$ ; forms a straight line.

Example: Solve missing angles in a triangle Triangle $A BC$ is isosceles with base angles $A$ and $B$ . The measure of angle $C$ is $4 0^{\circ}$ . Find the measure of angle $A$ .

Because the triangle is isosceles, the base angles are equal:

$A = B$

Use the triangle angle sum:

$A + A + 4 0^{\circ} = 18 0^{\circ}$

$2 A + 4 0^{\circ} = 18 0^{\circ}$

Subtract $4 0^{\circ}$ from both sides:

$2 A = 14 0^{\circ}$

Divide by $2$ :

$A = 7 0^{\circ}$

Answer: $7 0^{\circ}$

Triangles can be classified using either their side lengths or their angle measures. In many problems, you’ll first solve for unknown angles and then use those measures to classify the triangle.

Solving problems with angles in triangles

The interior angles of any triangle always add up to $18 0^{\circ}$ :

$A + B + C = 18 0^{\circ}$

This lets you set up an equation and use algebra to find missing angles, including angles written in terms of variables.

Example: Solve for the angles and classify the triangle A triangle has angles $x$ , $2 x$ , and $3 x$ . Find the angle measures and classify the triangle.

Set up the equation using the angle sum:

$x + 2 x + 3 x = 18 0^{\circ}$

$6 x = 18 0^{\circ}$

$x = 3 0^{\circ}$

Substitute back:

$x = 3 0^{\circ}$

$2 x = 6 0^{\circ}$

$3 x = 9 0^{\circ}$

The triangle contains a $9 0^{\circ}$ angle.

Answer: Right triangle

Example: Solve for $x$ and classify the triangle A triangle has angles $5 0^{\circ}$ , $(2 x + 10)^{\circ}$ , and $(x + 20)^{\circ}$ . Find $x$ and classify the triangle.

(spoiler)

Set up the equation:

$50 + (2 x + 10) + (x + 20) = 180$

$80 + 3 x = 180$

$3 x = 100$

$x = \frac{100}{3} \approx 33. 3^{\circ}$

Angle measures are approximately $5 0^{\circ}$ , $76. 7^{\circ}$ , and $53. 3^{\circ}$ . All angles are less than $9 0^{\circ}$ .

Answer: Acute triangle

When triangles are given by coordinates, you typically classify them using distances (for side lengths) and sometimes slopes (for right angles), rather than measuring angles directly.

To classify a triangle with vertices $(x_{1}, y_{1})$ , $(x_{2}, y_{2})$ , and $(x_{3}, y_{3})$ :

Compute side lengths using the distance formula

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$
Compare lengths
- All three equal $\Rightarrow$ equilateral
- Exactly two equal $\Rightarrow$ isosceles
- All different $\Rightarrow$ scalene
Check for a right angle using the converse of the Pythagorean theorem

$a^{2} + b^{2} = c^{2}$

where $c$ is the longest side.

For any triangle with side lengths $a \leq b \leq c$ :

Right if $c^{2} = a^{2} + b^{2}$
Acute if $c^{2} < a^{2} + b^{2}$
Obtuse if $c^{2} > a^{2} + b^{2}$

Example: Classify the triangle with vertices $(0, 0)$ , $(4, 0)$ , $(0, 3)$

Compute side lengths:

$A B = 4, A C = 3, BC = 5$

All sides are different, so the triangle is scalene.

$3^{2} + 4^{2} = 5^{2}$

The triangle satisfies the Pythagorean relationship.

Answer: Scalene right triangle

Diagrams show congruent angles with matching arcs and congruent side lengths with matching tick marks.

Finding plausible third sides of triangles

For any triangle with sides of lengths $x$ , $y$ , and $z$ , the following conditions must all be true:

$x + y > z$
$y + z > x$
$x + z > y$

These are called the triangle inequalities. They guarantee that the sum of any two sides is greater than the remaining side.

Example: If two sides of a triangle measure $5$ and $8$ , what are the possible lengths of the third side?

Let the third side be $x$ .

Apply the triangle inequalities:

$5 + 8 > x ⟹ x < 13$

$5 + x > 8 ⟹ x > 3$

$8 + x > 5$ (always true for $x > 0$ )

Combine the valid bounds.

$3 < x < 13$

Answer: $3 < x < 13$

Example: If two sides of a triangle measure $7$ and $10$ , find the bounds on the third side.

(spoiler)

$7 + 10 > x ⟹ x < 17$
$7 + x > 10 ⟹ x > 3$
$10 + x > 7$ (always true for $x > 0$ )

Answer: $3 < x < 17$

Classifying quadrilaterals

Sidenote

Classifying quadrilaterals

Quadrilateral	Properties	Diagonals
Parallelogram	Opposite sides are parallel and equal	Bisect each other
Rectangle	A parallelogram with all angles equal to $9 0^{\circ}$	Bisect each other and are equal in length
Rhombus	A parallelogram with all sides equal	Bisect each other at right angles
Square	A rectangle and a rhombus (all sides equal, all angles $9 0^{\circ}$ )	Bisect each other, equal in length, and perpendicular
Trapezoid	At least one pair of parallel sides	No general rule
Isosceles trapezoid	A trapezoid with non-parallel sides equal	Equal in length
Kite	Two pairs of adjacent equal sides	Perpendicular; one diagonal bisects the other

Example: Classifying a quadrilateral A quadrilateral has opposite sides parallel and equal in length. All of its angles are $9 0^{\circ}$ , but not all sides are equal. What type of quadrilateral is it?

Opposite sides parallel and equal imply a parallelogram or rectangle.

All angles are $9 0^{\circ}$ , consistent with a rectangle or square.

Not all sides are equal, which eliminates square.

Answer: Rectangle

Example: Classifying a quadrilateral Classify the quadrilateral based on the image below.

(spoiler)

How to classify this shape

All four sides $A B, BC, C D, D A$ have tick marks showing they are equal in length.
The diagonals $A C$ and $B D$ intersect at right angles.
A quadrilateral with all four sides equal and diagonals that bisect each other at right angles is a rhombus.

Answer: Rhombus

Special quadrilateral Venn diagram

A Venn diagram helps you see how special quadrilaterals fit inside broader categories:

The large oval represents Quadrilaterals.
Inside, one circle represents Parallelograms, another represents Trapezoids, and a separate circle represents Kites.
Within Parallelograms: overlapping circles represent Rhombi and Rectangles; their overlap represents Squares.
Within Trapezoids: a smaller circle represents Isosceles Trapezoids.

Coordinate geometry and shape classification

On the coordinate plane, you can classify polygons using distances and slopes:

Distances let you compare side lengths.
Slopes help you identify parallel and perpendicular sides.

Together, these tools let you classify triangles and quadrilaterals using coordinates alone.

For any two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , the distance between them is found using the distance formula:

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} .$

Equal distances indicate congruent sides, which is useful for identifying isosceles triangles, rhombi, or squares.

The slope of the line through $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is given by

$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}},$

provided $x_{2} \neq = x_{1}$ . Equal slopes indicate parallel sides, while slopes whose product is $- 1$ indicate perpendicular sides (when both slopes are defined).

By combining distance and slope:

Equal opposite slopes suggest a parallelogram.
Perpendicular adjacent slopes indicate right angles.
Equal side lengths distinguish rhombi and squares from rectangles.

Using slopes and distances to classify quadrilaterals

Parallelogram
- Opposite sides equal: compare distances of $\overline{A B}$ with $\overline{C D}$ and $\overline{BC}$ with $\overline{D A}$ .
- Opposite sides parallel: compare slopes $m_{A B} = m_{C D}$ and $m_{BC} = m_{D A}$ .
Rectangle
- All four right angles: check adjacent sides are perpendicular, i.e. $m_{A B} \cdot m_{BC} = - 1$ .
- Equivalently, parallelogram with one right angle.
Rhombus
- All four sides equal: check all pairwise distances equal.
- Parallelogram with equal sides.
Square
- Both rectangle and rhombus conditions hold.
Isosceles trapezoid
- One pair of opposite sides parallel: $m_{A B} = m_{C D}$ but $m_{BC} \neq = m_{D A}$ .
- Non-parallel sides equal: $d_{BC} = d_{D A}$ .

Example: Classify a quadrilateral by coordinates Given $A (0, 0)$ , $B (4, 0)$ , $C (5, 3)$ , $D (1, 3)$ , determine the type.

Compute distances:

$A B = 4$

$BC = 10$

$C D = 4$

$D A = 10$

Opposite sides are equal.

Compute slopes:

$m_{A B} = 0$ , $m_{C D} = 0$

$m_{BC} = 3$ , $m_{D A} = 3$

Opposite sides are parallel.

Adjacent slopes are not perpendicular.

All sides are not equal.

Answer: Parallelogram (non-rectangular, non-rhombus)