Triangles

Trianglesare多边形(shapes) with three sides and threeangles, which can be formed by connecting any three points in a plane. They are one of the first shapes studied ingeometry.

Triangles are particularly important because arbitrary polygons (with 4, 5, 6, or \(n\) sides) can bedecomposed into triangles. Thus, understanding the basic properties of triangles allows for deeper study of these larger polygons as well. Interestingly, the triangle is the only rigid polygon formed out of straight line segments, meaning that if the three lengths of the sides are given, the measurements correspond to a unique triangle. Because of this, it is often possible, given some information about a triangle (e.g. some side lengths and some angles), to determine additional facts about the triangle.

An equilateral triangle, a right triangle, and a scalene triangle

如果\ (ABC \)是一个三角形,然后\ (ABC \角+ \盎le BCA + \angle CAB = 180^\circ.\) In other words, the sum of the \(3\) internalanglesin any triangle is \(180^\circ.\)

Draw line \(DE\) that is parallel to \(AB\) and passes through \(C\) as in the above figure.
Since \( DE \parallel AB \), applying the principle ofalternate interior anglesshows that \( \angle DCA = \angle CAB \) and \( \angle BCE = \angle CBA.\)
Since angles in a line sum up to 180 degrees, \( \angle DCA + \angle ACB + \angle BCE = 180^ \circ \).
Thus, we conclude that \( \angle CAB + \angle ACB + \angle CBA = \angle DCA + \angle ACB + \angle BCE = 180^ \circ \). \(_\square\)

Main article:Triangle Inequality

Triangles have the property that the sum of any two sides of the triangle is always strictly greater than the third side. This property, known as thetriangle inequality, is explored in the wiki linked above.

Main article:Classification of Triangles

Triangles can be classified into different categories based on their sides and angles. For example, a triangle with one angle of measure \(90^\circ\) is known as a right triangle, while a triangle with sides of all equal length is known as an equilateral triangle. These classifications and many others are explored in the wiki linked above.

Main article:Area of a Triangle

When determining the area of a triangle, note that a triangle can be thought of as half of a平行四边形. The following picture should make this point clear:

Because thearea of a parallelogramis equal to the product of its base and height, the area of a triangle is simply half of that area.

The area of a triangle is \(A = \frac{bh}{2}\), where \(b\) is the length of the base and \(h\) is the height.

What is the area of a triangle with base 10 and height 6?

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The area of this triangle is \(\frac{10 \times 6}{2} = 30.\) \(_\square\)

For more advanced methods of finding the area of a triangle, such asHeron's formula, see the wiki linked at the top of this section.

The measure of anexterior angleof a triangle is the sum of its two remote interior angles.Remote interior anglesare the interior angles of a triangle that are opposite to the exterior angle under consideration.

Forregular polygons, the formula to find the exterior angle of a polygon is \(\frac{360^\circ}{n},\) where \(n\) is the number of sides.

Prove that \(\angle ABC+\angle CAB=\angle DCA. \)

ExteriorAngle

ANSWER

From the property oftriangles-angle sum, the sum of measures of all angles in a triangle is \(180^\circ\). So, in \(\triangle ABC\), it follows that \( \angle ABC + \angle BCA + \angle CAB = 180^\circ \).

Since \( \angle BCA \) and \( \angle DCA \) form a straight line, \( \angle BCA + \angle DCA = 180^\circ, \) It follows that both sets of angles are equal to \( 180^\circ: \)

\[ \angle ABC + \angle BCA + \angle CAB = \angle BCA + \angle DCA.\]

Therefore,

\[ \angle ABC + \angle CAB = \angle DCA. \ _\square \]

Interested in learning more about triangles? Check out these pages:

• Congruent and Similar Triangles
• Pythagorean Theorem
• Triangles - Calculating Area