General properties of triangles
Definition and vocabulary
Triangle
A triangle is a closed plane figure with three sides.
- The three vertices of the triangle are labeled A,B, and C.
- The triangle is denoted by \triangle ABC.
- The side connecting the vertices A and B is denoted \overline{AB}.
- The side connecting the vertices B and C is denoted \overline{BC}.
- The side connecting the vertices A and C is denoted \overline{AC}.
- The angle formed by the sides \overline{AB} and \overline{BC} is denoted \angle ABC.
- The angle formed by the sides \overline{AB} and \overline{AC} is denoted \angle BAC.
- The angle formed by the sides \overline{BC} and \overline{AC} is denoted \angle BCA.
Sum of angles of a triangle
The sum of the three angles of any triangle is 180^\circ.
In the following graphic of a triangle, the measurements of two angles is given:
Solve the following equation to find the missing angle x:
70+50+x=180\\x=180-120\\x=60
The missing angle is 60^\circ.
Basic constructions with triangles
Perpendicular bisectors
Circumcircle of a triangle
There is only one circle which contains all three of the triangles vertices. This circle is the circumcircle of the triangle.
Circumcenter
Given a triangle, the center of the circumcircle is called the circumcenter of the triangle.
Perpendicular bisector
The perpendicular bisector of a line segment is the line perpendicular to the line segment at its midpoint.
The three perpendicular bisectors of the sides of a triangle intersect at the circumcenter of the triangle.
Angle bisectors
Incircle
There is a unique circle which is tangent to all three sides of a triangle. This circle is called the incircle of the triangle.
Incenter
Given a triangle the center of its incircle is called the incenter of the triangle.
Angle bisector
Given an angle, the angle bisector is the line which separates the angle into two congruent angles.
Angle bisectors of a triangle intersect at incenter
The three angle bisectors of a triangle intersect at the incenter of the triangle.
Medians
Median of a triangle
A median of a triangle is a line segment connecting a vertex of the triangle with the middle point of the side opposite to the angle.
Centroid of a triangle
The three medians of a triangle intersect at a single point. That point is referred to as the centroid of the triangle.
Altitudes
Altitude
An altitude of a triangle is a line which passes through a vertex of the triangle and is perpendicular to the side opposite of the triangle. Observe that a triangle has three altitudes.
Orthocenter of a triangle
The three altitudes of a triangle intersect at a common point. This point of intersection is called the orthocenter of the triangle.
The orthocenter of a triangle does not necessarily have to be in the interior of the triangle.
Midsegments
Midsegment
A midsegment of a triangle is a line segment which connects the midpoints of two sides of a triangle. Observe that a triangle has three midsegments.
Consider a triangle and one of its midsegments as in the following graphic:
The triangle with vertices A,B,C is similar to the triangle A,b,c and the following hold:
- Vertex A corresponds to vertex A.
- Vertex B corresponds to vertex b.
- Vertex C corresponds to vertex c.
- The ratio of the sides of the smaller triangle to the larger triangle is \dfrac{1}{2}.
- In particular, the length of the median connecting the points b and c is exactly \dfrac{1}{2} the length of the side of the triangle connecting vertices B and C.
Three midsegments of a triangle form a similar triangle. Recall that two triangles are similar if they have equal corresponding angles and proportionate sides.
Suppose that we are given the following triangle and its three midsegments:
The triangle with vertices a,b,c is similar to the triangle with vertices A,B,C and the following hold:
- Vertex A corresponds to vertex a.
- Vertex B corresponds to vertex b.
- Vertex C corresponds to vertex c.
- The ratio of the smaller triangle to the larger triangle is \dfrac{1}{2}.
Relationships with angles and sides
The angles and sides of a triangle are related to one another through the Law of Cosines.
Law of Cosines
Suppose we are given a triangle as in the following graphic:
We have the following equations:
- a^2=b^2+c^2-2bc\cos\left(\alpha\right)
- b^2=a^2+c^2-2ac\cos\left(\beta\right)
- c^2=a^2+b^2-2ab\cos\left(\gamma\right)
Consider the following triangle:
To solve for the missing side length, use the Law of Cosines:
x^2=4^2+5^2-2\left(4\right)\left(5\right)\cos\left(60^\circ\right)\\x^2=16+25-40\left(\dfrac{1}{2}\right)\\x^2=41-20\\x^2=21\\x=\sqrt{21}
Area and perimeter of triangles
Area of a triangle
Suppose we are given a triangle as in the following graphic:
The value of h is the altitude of the triangle. The area of the triangle is:
\dfrac{1}{2}bh
Consider the following triangle:
The area of the triangle is:
\dfrac{1}{2}8\left(7\right)=4\left(7\right)=28
Perimeter
The perimeter of a triangle is the sum of the lengths of the side of the triangle.
Consider the following triangle:
The perimeter of the triangle is:
30+25+33=88
Special triangles
Right triangles and the Pythagorean theorem
Right triangle
A triangle which has an angle measuring 90^\circ is called a right triangle.
The side of a right triangle which is opposite of the right angle is called the hypotenuse of the triangle.
Pythagorean Theorem
Suppose we are given a right triangle as in the following graphic:
Then:
c^2=a^2+b^2
Consider the following right triangle:
We can use the Pythagorean Theorem to solve for the missing side length of the right triangle:
x^2=4^2+3^2\\x^2=25\\x=5
The sum of the angles in a triangle is 180^\circ. Therefore, if you know one of the non- 90^\circ angles of a right triangle, then you can solve for the missing angle.
Consider the right triangle in the following graphic:
We can solve for the missing angle:
90^\circ+70^\circ+\alpha=180^\circ\\\alpha=20^\circ
Isosceles triangles
Isosceles triangle
A triangle in which two sides have the same length is referred to as an isosceles triangle.
Suppose we are given an isosceles triangle. The two angles that are opposite the sides of the isosceles triangle of equal length are congruent.
If you know one angle of an isosceles triangle, then you can solve for the missing two angles.
Consider the isosceles triangle in the following graphic:
Because the triangle is isosceles:
\beta=75^\circ
To solve for \alpha, we use the fact that the sum of the angles of any triangle is 180^\circ :
\alpha+75^\circ+75^\circ=180^\circ\\\alpha=30^\circ
Equilateral triangles
Equilateral Triangle
An equilateral triangle is a triangle whose sides all have the same lengths.
The three angles of an equilateral triangle are all 60^\circ.
Any two equilateral triangles are similar.
Isosceles right triangles
Isosceles right triangle
A right triangle which also an isosceles triangle is an isosceles right triangle.
Angle measurements of an isosceles right triangle
The angle measurements of an isosceles right triangle are 90^\circ, 45^\circ, and 45^\circ.
Any two isosceles right triangles are similar.
The Pythagorean Theorem reveals information on the sides of an isosceles right triangle.
Suppose we are given the following isosceles right triangle:
Then:
c=\sqrt{2}a
Consider the following isosceles right triangle.
We can solve for the side length a :
3\sqrt{2}=\sqrt{2}a\\a=3
Congruent triangles
SSS, SAS, ASA, SSA and HL theorems
SSS (side-side-side)
If the lengths of the three sides of one triangle are equal to the lengths of the three sides of another triangle, then the two triangles are congruent.
SAS (side-angle-side)
If two sides and the induced angle of a triangle are congruent to two sides and the induced triangle of another triangle, then those two triangles are congruent.
ASA (angle-side-angle)
If two angles and the included side of a triangle are congruent to two angles and the included side of another triangle then those two triangles are congruent.
Congruency of isosceles and equilateral triangles
Congruency of isosceles triangles
If two sides of different length of an isosceles triangle are congruent to two sides of another isosceles triangle, then those two isosceles triangles are congruent.
Congruency of equilateral triangles
If one side of an equilateral triangle is congruent to one side of another equilateral triangle, then the two triangles are congruent.
Congruency of right triangles
HL (hypotenuse-leg)
If the hypotenuse and leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the two right triangles are congruent.
Proofs involving triangles
The above theorems on triangles allow us to derive new geometric properties.
Consider the following geometric object. We want to prove:
\angle BAC+\angle ABC=\angle BCD
The sum of the angles of a triangle is 180^\circ. Therefore:
\angle BAC +\angle ABC=180^\circ-\angle ACB
Furthermore, the line segment \overline{BC} separates the straight angle \angle ACD. Therefore:
\angle ACB=180^\circ -\angle BCD
Substituting the second equation into the first equation gives:
\angle BAC +\angle ABC =180^\circ-\left(180^\circ-\angle BCD\right)
And:
\angle BAC +\angle ABC =\angle BCD
Consider the following geometric graphic. We will prove that \triangle AEC is congruent to \triangle BED.
We are given the following pieces of information:
- \overline{AB}=\overline{CD}
- \angle EBC=\angle ECB
To prove the triangle \triangle AEC is congruent to \triangle BED, we will show that \overline{AC}=\overline{BD} and \overline{EB}=\overline{EC}. Using the given piece of information that \angle EBC=\angle ECB, we will be able to invoke SAS to conclude the two triangles are congruent.
The congruence of the sides \overline{AC} and \overline{BD} is proven as follows:
\overline{AC}=\overline{AD}-\overline{CD}\\=\overline{AD}-\overline{AB}\\=\overline{BD}
The congruence of the sides \overline{EB} and \overline{EC} is proven as follows. We know that \angle EBC=\angle ECB, therefore the \triangle EBC is an isosceles triangle and \overline{EB} is congruent to \overline{EC}.
We can now conclude \triangle AEC is congruent to \triangle BED as follows:
- \overline{AC} is congruent to \overline{BD}.
- \overline{EC} is congruent to \overline{ED}.
- \angle EAC is congruent to \angle EBD.
Therefore, \triangle AEC is congruent to \triangle BED by SAS.