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: Exactly two sides are equal; the two angles opposite those sides are also 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: Exactly one pair of opposite sides parallel.
Isosceles trapezoid: A trapezoid with non-parallel sides equal.
Kite: Two distinct pairs of consecutive (adjacent) sides that are equal in length, with no pair of opposite sides equal.
Angle: The union of two rays with a common endpoint; measured in degrees.

Identifying angles

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

Right angle: exactly $9 0^{\circ}$ ; marked in diagrams with a small square.
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.

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 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: $7 0^{\circ}$

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

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.

When triangles are given by coordinates, you typically classify them using distances (for side lengths) and 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 a triangle with sides $4$ , $5$ , and $6$

Assign $a = 4$ , $b = 5$ , $c = 6$ (longest side).

Compare $c^{2}$ with $a^{2} + b^{2}$ :

$c^{2} = 36, a^{2} + b^{2} = 16 + 25 = 41$

Since $c^{2} < a^{2} + b^{2}$ , the triangle is acute.

Answer: Acute triangle

Sidenote

Diagram notation

In triangle diagrams, congruent angles are marked with matching arcs and congruent sides are marked with matching tick marks.

Finding plausible third sides of triangles

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

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

These are 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: $3 < x < 13$

Common pitfalls

Triangle inequality boundaries are excluded. In the example above, $x = 3$ or $x = 13$ would make the three points fall on a straight line - a degenerate triangle with zero area, not a valid triangle. The bounds are strict: $3 < x < 13$ .

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	Exactly 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

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.

For any two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ :

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$

Equal slopes indicate parallel sides; slopes with a product of $- 1$ indicate perpendicular sides. A quick reference for common shapes:

Parallelogram: opposite sides parallel ( $m_{A B} = m_{C D}$ , $m_{BC} = m_{D A}$ ) and equal in length.
Rectangle: parallelogram with perpendicular adjacent sides ( $m_{A B} \cdot m_{BC} = - 1$ ).
Rhombus: parallelogram with all four sides equal.
Square: both rectangle and rhombus conditions hold.
Isosceles trapezoid: exactly one pair of parallel sides, with non-parallel sides equal in length.
Kite: two pairs of consecutive equal sides, no pair of opposite sides equal.

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

Special quadrilateral Venn diagram

The diagram below shows how special quadrilaterals nest inside broader categories - for example, every square is both a rectangle and a rhombus, and every rectangle is a parallelogram.

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: Exactly two sides are equal; the two angles opposite those sides are also 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: Exactly one pair of opposite sides parallel.
Isosceles trapezoid: A trapezoid with non-parallel sides equal.
Kite: Two distinct pairs of consecutive (adjacent) sides that are equal in length, with no pair of opposite sides equal.
Angle: The union of two rays with a common endpoint; measured in degrees.

Identifying angles

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

Right angle: exactly $9 0^{\circ}$ ; marked in diagrams with a small square.
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.

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 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: $7 0^{\circ}$

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

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.

When triangles are given by coordinates, you typically classify them using distances (for side lengths) and 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 a triangle with sides $4$ , $5$ , and $6$

Assign $a = 4$ , $b = 5$ , $c = 6$ (longest side).

Compare $c^{2}$ with $a^{2} + b^{2}$ :

$c^{2} = 36, a^{2} + b^{2} = 16 + 25 = 41$

Since $c^{2} < a^{2} + b^{2}$ , the triangle is acute.

Answer: Acute triangle

Sidenote

Diagram notation

In triangle diagrams, congruent angles are marked with matching arcs and congruent sides are marked with matching tick marks.

Finding plausible third sides of triangles

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

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

These are 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: $3 < x < 13$

Common pitfalls

Triangle inequality boundaries are excluded. In the example above, $x = 3$ or $x = 13$ would make the three points fall on a straight line - a degenerate triangle with zero area, not a valid triangle. The bounds are strict: $3 < x < 13$ .

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	Exactly 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

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.

For any two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ :

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$

Equal slopes indicate parallel sides; slopes with a product of $- 1$ indicate perpendicular sides. A quick reference for common shapes:

Parallelogram: opposite sides parallel ( $m_{A B} = m_{C D}$ , $m_{BC} = m_{D A}$ ) and equal in length.
Rectangle: parallelogram with perpendicular adjacent sides ( $m_{A B} \cdot m_{BC} = - 1$ ).
Rhombus: parallelogram with all four sides equal.
Square: both rectangle and rhombus conditions hold.
Isosceles trapezoid: exactly one pair of parallel sides, with non-parallel sides equal in length.
Kite: two pairs of consecutive equal sides, no pair of opposite sides equal.

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

Special quadrilateral Venn diagram

The diagram below shows how special quadrilaterals nest inside broader categories - for example, every square is both a rectangle and a rhombus, and every rectangle is a parallelogram.

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

8 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: Exactly two sides are equal; the two angles opposite those sides are also 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: Exactly one pair of opposite sides parallel.
Isosceles trapezoid: A trapezoid with non-parallel sides equal.
Kite: Two distinct pairs of consecutive (adjacent) sides that are equal in length, with no pair of opposite sides equal.
Angle: The union of two rays with a common endpoint; measured in degrees.

Identifying angles

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

Right angle: exactly $9 0^{\circ}$ ; marked in diagrams with a small square.
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.

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 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: $7 0^{\circ}$

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

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.

When triangles are given by coordinates, you typically classify them using distances (for side lengths) and 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 a triangle with sides $4$ , $5$ , and $6$

Assign $a = 4$ , $b = 5$ , $c = 6$ (longest side).

Compare $c^{2}$ with $a^{2} + b^{2}$ :

$c^{2} = 36, a^{2} + b^{2} = 16 + 25 = 41$

Since $c^{2} < a^{2} + b^{2}$ , the triangle is acute.

Answer: Acute triangle

Sidenote

Diagram notation

In triangle diagrams, congruent angles are marked with matching arcs and congruent sides are marked with matching tick marks.

Finding plausible third sides of triangles

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

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

These are 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: $3 < x < 13$

Common pitfalls

Triangle inequality boundaries are excluded. In the example above, $x = 3$ or $x = 13$ would make the three points fall on a straight line - a degenerate triangle with zero area, not a valid triangle. The bounds are strict: $3 < x < 13$ .

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	Exactly 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

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.

For any two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ :

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$

Equal slopes indicate parallel sides; slopes with a product of $- 1$ indicate perpendicular sides. A quick reference for common shapes:

Parallelogram: opposite sides parallel ( $m_{A B} = m_{C D}$ , $m_{BC} = m_{D A}$ ) and equal in length.
Rectangle: parallelogram with perpendicular adjacent sides ( $m_{A B} \cdot m_{BC} = - 1$ ).
Rhombus: parallelogram with all four sides equal.
Square: both rectangle and rhombus conditions hold.
Isosceles trapezoid: exactly one pair of parallel sides, with non-parallel sides equal in length.
Kite: two pairs of consecutive equal sides, no pair of opposite sides equal.

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

Special quadrilateral Venn diagram

The diagram below shows how special quadrilaterals nest inside broader categories - for example, every square is both a rectangle and a rhombus, and every rectangle is a parallelogram.

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: Exactly two sides are equal; the two angles opposite those sides are also 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: Exactly one pair of opposite sides parallel.
Isosceles trapezoid: A trapezoid with non-parallel sides equal.
Kite: Two distinct pairs of consecutive (adjacent) sides that are equal in length, with no pair of opposite sides equal.
Angle: The union of two rays with a common endpoint; measured in degrees.

Identifying angles

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

Right angle: exactly $9 0^{\circ}$ ; marked in diagrams with a small square.
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.

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 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: $7 0^{\circ}$

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

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.

When triangles are given by coordinates, you typically classify them using distances (for side lengths) and 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 a triangle with sides $4$ , $5$ , and $6$

Assign $a = 4$ , $b = 5$ , $c = 6$ (longest side).

Compare $c^{2}$ with $a^{2} + b^{2}$ :

$c^{2} = 36, a^{2} + b^{2} = 16 + 25 = 41$

Since $c^{2} < a^{2} + b^{2}$ , the triangle is acute.

Answer: Acute triangle

Sidenote

Diagram notation

In triangle diagrams, congruent angles are marked with matching arcs and congruent sides are marked with matching tick marks.

Finding plausible third sides of triangles

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

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

These are 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: $3 < x < 13$

Common pitfalls

Triangle inequality boundaries are excluded. In the example above, $x = 3$ or $x = 13$ would make the three points fall on a straight line - a degenerate triangle with zero area, not a valid triangle. The bounds are strict: $3 < x < 13$ .

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	Exactly 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

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.

For any two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ :

$d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$

Equal slopes indicate parallel sides; slopes with a product of $- 1$ indicate perpendicular sides. A quick reference for common shapes:

Parallelogram: opposite sides parallel ( $m_{A B} = m_{C D}$ , $m_{BC} = m_{D A}$ ) and equal in length.
Rectangle: parallelogram with perpendicular adjacent sides ( $m_{A B} \cdot m_{BC} = - 1$ ).
Rhombus: parallelogram with all four sides equal.
Square: both rectangle and rhombus conditions hold.
Isosceles trapezoid: exactly one pair of parallel sides, with non-parallel sides equal in length.
Kite: two pairs of consecutive equal sides, no pair of opposite sides equal.

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

Special quadrilateral Venn diagram

The diagram below shows how special quadrilaterals nest inside broader categories - for example, every square is both a rectangle and a rhombus, and every rectangle is a parallelogram.