Glossary

10.3 Triangles

3 min read•june 18, 2024

Triangles come in various shapes and sizes, each with unique properties. From acute to obtuse, equilateral to scalene, understanding these differences helps us classify and analyze triangles effectively. Knowing the rules for congruence and allows us to compare triangles and solve real-world problems.

Triangle concepts extend beyond basic geometry, influencing fields like architecture and engineering. The , , and area calculations are powerful tools derived from triangle properties. These principles form the foundation for more advanced mathematical and practical applications.

Triangle Classification and Properties

Types of triangles

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  • Classified by angles measures whether the angles are acute (less than 90°), right (exactly 90°), or obtuse (greater than 90°)
    • has all three angles measuring less than 90° (equilateral, isosceles, scalene)
    • has one angle measuring exactly 90° (isosceles right, scalene right)
      • The side opposite the right angle is called the
    • has one angle measuring greater than 90° (isosceles obtuse, scalene obtuse)
  • Classified by side lengths determines if the sides are equal or unequal
    • has all three sides of equal length
      • Consequently, all three angles measure 60°
    • has two sides of equal length () and one side of different length ()
      • The two angles opposite the equal sides () are also equal
    • has no sides of equal length
      • Consequently, no angles are equal

Triangle congruence theorems

  • have exactly the same size and shape, with equal corresponding sides and angles
  • (Side-Side-Side) states if all three sides of one triangle are equal in length to the corresponding sides of another triangle, the triangles are congruent
    • Example: If AB‾≅DE‾\overline{AB} \cong \overline{DE}, BC‾≅EF‾\overline{BC} \cong \overline{EF}, and AC‾≅DF‾\overline{AC} \cong \overline{DF}, then △ABC≅△DEF\triangle ABC \cong \triangle DEF
  • (Side-Angle-Side) states if two sides and the included angle of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent
    • The included angle is the angle between the two given sides
    • Example: If AB‾≅DE‾\overline{AB} \cong \overline{DE}, ∠B≅∠E\angle B \cong \angle E, and BC‾≅EF‾\overline{BC} \cong \overline{EF}, then △ABC≅△DEF\triangle ABC \cong \triangle DEF
  • (Angle-Side-Angle) states if two angles and the included side of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent
    • The included side is the side between the two given angles
    • Example: If ∠A≅∠D\angle A \cong \angle D, AB‾≅DE‾\overline{AB} \cong \overline{DE}, and ∠B≅∠E\angle B \cong \angle E, then △ABC≅△DEF\triangle ABC \cong \triangle DEF
  • (Angle-Angle-Side) states if two angles and a non-included side of one triangle are equal to the corresponding parts of another triangle, the triangles are congruent
    • The non-included side is a side not between the two given angles
    • Example: If ∠A≅∠D\angle A \cong \angle D, ∠C≅∠F\angle C \cong \angle F, and AB‾≅DE‾\overline{AB} \cong \overline{DE}, then △ABC≅△DEF\triangle ABC \cong \triangle DEF

Similarity in triangles

  • have the same shape but not necessarily the same size, with equal corresponding angles and proportional corresponding sides
    • Corresponding angles are congruent (equal measure)
    • Corresponding sides are proportional (in the same ratio)
  • in similar triangles means the ratios of corresponding side lengths are equal
    • If △ABC∼△DEF\triangle ABC \sim \triangle DEF, then ABDE=BCEF=ACDF\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}
    • This proportionality allows for solving missing side lengths when given sufficient information
  • Similarity has real-world applications in fields like art (perspective), architecture (scale models), engineering (blueprints), and photography (resizing)
    • Example: Using shadow lengths to calculate the height of a tree or building
    • Example: Enlarging or reducing an image while maintaining its proportions

Additional Triangle Concepts

  • The relates the lengths of the sides in a right triangle: a² + b² = c², where c is the length of the hypotenuse
  • Trigonometry uses the ratios of sides in right triangles to define sine, cosine, and tangent functions
  • The can be calculated using the formula: A = ½bh, where b is the base and h is the height ()
  • A is a line segment that connects a vertex to the midpoint of the opposite side
  • The is the point where all three medians intersect, dividing each in a 2:1 ratio

Key Terms to Review (30)

∆: In mathematics, the symbol ∆ (Delta) is commonly used to represent a change or difference in a certain quantity. In the context of triangles, it often signifies the change in length or the variation in angle measurements, which are crucial when studying properties like congruence, similarity, and the relationships between angles and sides.
∠: The symbol ∠ represents an angle, which is formed by two rays that share a common endpoint known as the vertex. Angles are crucial in understanding the properties and relationships within triangles, including their types, measures, and how they interact with one another in geometric figures. They play a fundamental role in various concepts such as congruence, similarity, and the calculation of triangle area.
≅: The symbol '≅' is used to denote that two geometric figures, particularly triangles, are congruent. Congruent figures have the same shape and size, meaning that their corresponding sides are equal in length and their corresponding angles are equal in measure. This concept is fundamental in geometry, particularly when proving the properties and relationships of triangles.
AAS: AAS, or Angle-Angle-Side, is a criterion used in triangle congruence that states if two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the two triangles are congruent. This rule emphasizes the importance of angle measures in determining the similarity and congruence of triangles, showing that knowing just two angles is enough to establish congruence when combined with the length of one side.
Acute triangle: An acute triangle is a type of triangle where all three interior angles measure less than 90 degrees. This characteristic means that the angles are sharp, giving the triangle a distinct appearance compared to other types like right or obtuse triangles. The property of being acute also plays a vital role in determining various aspects such as area, perimeter, and relationships between the sides based on their angle measures.
Altitude: Altitude is the perpendicular distance from a vertex of a triangle to the line containing the opposite side. This measurement is crucial in understanding the properties and relationships within triangles, as it plays a key role in determining area and can also help establish congruence between different triangles through various geometric principles.
Area of a Triangle: The area of a triangle is the measure of the space enclosed within its three sides. This area can be calculated using various methods, with the most common formula being $$A = \frac{1}{2} \times b \times h$$, where 'b' represents the length of the base and 'h' represents the height perpendicular to that base. Understanding the area of a triangle is essential for solving problems in geometry and can also connect to real-world applications such as land measurement and architectural design.
ASA: ASA stands for Angle-Side-Angle, a method used in triangle congruence. This rule states that if two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the two triangles are congruent. The significance of ASA lies in its ability to establish the equality of two triangles without needing to measure all three sides.
Base: In mathematics, a base is a fundamental component that serves as the reference point for various operations, such as exponentiation or measurement in geometry. It indicates the number that is being multiplied by itself in exponentiation, the unit of measure in area and volume calculations, and the foundational side in geometric shapes like triangles. Understanding the concept of base is essential for grasping how numbers and shapes relate to one another in mathematical expressions and equations.
Base Angles: Base angles are the angles that are formed at the base of an isosceles triangle. In an isosceles triangle, the two sides that are equal in length are referred to as the legs, and the angles opposite these sides are the base angles. These angles have special properties, especially concerning congruence and relationships with the vertex angle.
Centroid: The centroid is the point where all three medians of a triangle intersect, effectively serving as the triangle's center of mass. This point is significant in understanding the balance and symmetry of triangles, as it divides each median into two segments with a 2:1 ratio, making it a crucial concept in both geometry and physics.
Congruent Triangles: Congruent triangles are triangles that are identical in shape and size, meaning that their corresponding sides and angles are equal. This concept is fundamental in geometry because it allows for the establishment of relationships between different triangles, proving that they can be manipulated without changing their fundamental properties. The congruence of triangles has significant applications in various proofs and theorems throughout geometry.
Equilateral Triangle: An equilateral triangle is a special type of triangle where all three sides are of equal length and all three angles are congruent, each measuring 60 degrees. This property makes equilateral triangles a significant subject of study, as they exhibit high levels of symmetry and are often used in various mathematical concepts, including geometric constructions and trigonometry.
Hypotenuse: The hypotenuse is the longest side of a right triangle, opposite the right angle. It plays a crucial role in various mathematical concepts, especially in trigonometry and geometry, where it serves as a reference for calculating the lengths of the other sides using the Pythagorean theorem and trigonometric ratios.
Isosceles triangle: An isosceles triangle is a type of triangle that has at least two sides of equal length. This unique property leads to specific characteristics, such as two equal angles opposite the equal sides, which are known as the base angles. Understanding the properties of isosceles triangles is essential when studying the relationships within triangles and how they fit into the larger category of polygons.
Legs: In the context of triangles, 'legs' refers to the two sides that form the right angle in a right triangle. These sides are crucial for understanding various properties of triangles, especially when calculating angles and lengths using the Pythagorean theorem. The legs play a vital role in determining the triangle's area, as well as in solving for unknown lengths when applying trigonometric ratios.
Median: The median is the middle value in a data set when the numbers are arranged in ascending or descending order. If there is an even number of observations, the median is the average of the two middle numbers.
Median: The median is the middle value in a data set when the numbers are arranged in ascending or descending order. It effectively divides the data into two equal halves, making it a useful measure of central tendency, especially when dealing with skewed distributions. The median helps to represent the typical value in a dataset and can be more informative than the mean when there are outliers or extreme values present.
Obtuse Triangle: An obtuse triangle is a type of triangle that contains one angle measuring greater than 90 degrees. This characteristic differentiates it from other triangles, such as acute triangles, which have all angles less than 90 degrees, and right triangles, which contain one angle exactly equal to 90 degrees. The presence of an obtuse angle fundamentally affects the triangle's properties and relationships among its sides.
Proportionality: Proportionality is the mathematical relationship where two quantities maintain a constant ratio to each other. This concept is crucial for understanding various properties of shapes, sizes, and distributions, as it helps compare different elements by their relative sizes or amounts. It can also play a significant role in allocation processes and fair representation in diverse contexts.
Pythagorean Theorem: The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse and \(a\) and \(b\) are the other two sides.
Pythagorean theorem: The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This relationship connects geometry and algebra, forming the basis for understanding distances and relationships within triangular structures and applications in real-world scenarios.
Right triangle: A right triangle is a type of triangle that has one angle measuring exactly 90 degrees, which is known as the right angle. This unique feature allows right triangles to follow specific properties and relationships that are essential in geometry and trigonometry, particularly involving the lengths of their sides and the angles formed between them. The presence of the right angle makes right triangles fundamental for various applications, including calculations related to distance, height, and angles.
SAS: SAS stands for Side-Angle-Side, which is a congruence criterion used in geometry to establish that two triangles are congruent. This means if two sides and the angle between them in one triangle are respectively equal to two sides and the angle between them in another triangle, then the two triangles are congruent. This principle is crucial in solving various problems related to triangles and their properties.
Scalene Triangle: A scalene triangle is a type of triangle where all three sides have different lengths and all three angles have different measures. This distinct characteristic makes scalene triangles unique among other types of triangles, such as equilateral and isosceles triangles, which have equal sides or angles. Understanding the properties of scalene triangles is essential for solving various geometric problems and applying concepts related to angles and side lengths.
Scaling factor: A scaling factor is a number by which all dimensions of an object are multiplied to achieve a proportional enlargement or reduction. It maintains the geometric similarity between the original and scaled objects.
Similar Triangles: Similar triangles are triangles that have the same shape but may differ in size. This means that their corresponding angles are equal and the lengths of their corresponding sides are proportional. Understanding similar triangles is essential in geometry, especially when working with scale factors, properties of proportionality, and applications in real-world scenarios like map reading and architecture.
Similarity: Similarity, denoted by the symbol ∼, refers to a fundamental geometric concept where two figures are the same shape but may differ in size. When two triangles are similar, their corresponding angles are equal and the lengths of their corresponding sides are proportional. This concept is crucial for understanding properties of triangles and can be applied in various mathematical situations involving scaling and transformations.
SSS: SSS stands for Side-Side-Side, a criterion used in geometry to establish the congruence of triangles. If the three sides of one triangle are equal in length to the three sides of another triangle, then the two triangles are congruent. This concept is pivotal in proving that two triangles are identical in shape and size, emphasizing the importance of side lengths in triangle congruence.
Trigonometry: Trigonometry is the branch of mathematics that studies the relationships between the angles and sides of triangles, particularly right-angled triangles. It involves concepts such as sine, cosine, and tangent, which are crucial for understanding the properties of angles and lengths in various geometric shapes. This field has wide applications in science, engineering, and everyday life, making it essential for solving problems related to triangles and angles.
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