Subject: Geometry

Triangles

A triangle is a 3-sided polygon, which consists of lines segments connected at their endpoints. It is named using its vertices, with a symbol ∆. The sum of all the measures of the angles within a triangle is equal to 180o.


ABC

A point is in the interior of a triangle if it is within the interior of the angles of the triangle; exterior if it's outside or not within the triangle. Included angle is the angle formed between two sides of a triangle. In the image above, \angle{B} is the included angle between the sides \overline{AB} and \overline{BC}

Classification:

Triangles can be classified according to the measure of its angles and its sides.

CLASSIFICATION ACCORDING TO ANGLE:

Acute triangle- a triangle is said to be acute if all the measure of its angles are acute (<90o).

Right triangle- a triangle is classified as a right triangle if one of the measure of its angles is equal to 90o. The longest side of a right triangle is called hypotenuse; its other sides are called legs.

Obtuse triangle- A triangle is classified as obtuse if one its angles measures more than 90o.


a. acute triangle, b. Right triangle, c. Obtuse triangle

ACCORDING TO SIDE:

Isosceles triangle- a triangle is classified as an isosceles triangle if it has two congruent sides. Its congruent sides are called legs, and the third side is the base. The angles opposite to its congruent sides are called its base angles, and the third angle is the vertex angle.

Scalene triangle- a triangle is scalene if it has no two congruent sides.

Equilateral triangle- triangle is equilateral if all of its sides are equal. Also called equiangular since all of its angles are equal too.


a. Isosceles triangle, b. Scalene triangle, c. Equilateral(equiangular) triangle

Triangle classifications can be mixed too, combining the characteristics in the angle classification and side classification.

Triangle Correspondence and Congruence

Triangle correspondence in geometry, means to pair up or to match up. It is symbolized with \leftrightarrow (read as corresponds to) in between the paired up vertices.

Suppose we pair up the two triangles in the image below.

In the image above we can match \angle{A} with \angle{1} , \angle{B} with \angle{2} and \angle{C} with \angle{3} . In other words \angle{A} corresponds to \angle{1} , \angle{B} corresponds to \angle{2} , and \angle{C} corresponds to \angle{3} .

Other corresponding parts are as follows:

∆ABC \leftrightarrow ∆321

∆ABC \leftrightarrow ∆123

∆ABC \leftrightarrow ∆132

∆ABC \leftrightarrow ∆312

∆ABC \leftrightarrow ∆213

∆ABC \leftrightarrow ∆231

The correspondence mentioned above also identifies pairs of corresponding angles and pairs of corresponding sides.

Suppose, ∆ABC \leftrightarrow ∆123, this implies that:

\angle{A} \leftrightarrow \angle{1}

\angle{B} \leftrightarrow \angle{2}

\angle{C} \leftrightarrow \angle{3}

\overline{AB} \leftrightarrow \overline{12}

\overline{BC} \leftrightarrow \overline{23}

\overline{AC} \leftrightarrow \overline{13}

The correspondence listed above is an example of a one-to-one correspondence.

The following correspondence below is an example of of equivalent correspondence:

∆ABC \leftrightarrow ∆123

∆BAC \leftrightarrow ∆213

∆CBA \leftrightarrow ∆321

Notice that, in all the equivalent correspondence mentioned above, one endpoint is equivalent to one and only endpoint in another triangle, such that:

\angle{A} corresponds only to \angle{1}

\angle{B} corresponds only to \angle{2}

\angle{C} corresponds only to \angle{3}

Triangles are said to be only congruent if and only if there is an existing correspondence between its vertices, so that its corresponding angles and corresponding sides are congruent. From the following example of correspondence listed above, it can be implied that if:

\angle{A} \leftrightarrow \angle{1} , then \angle{A}\cong\angle{1}

\angle{B} \leftrightarrow \angle{2} , then \angle{B}\cong\angle{2}

\angle{C} \leftrightarrow \angle{3} , then \angle{C}\cong\angle{3}

\overline{AB} \leftrightarrow \overline{12} , then \overline{AB}\cong\overline{12}

\overline{BC} \leftrightarrow \overline{23} , then \overline{BC}\cong\overline{23}

\overline{AC} \leftrightarrow \overline{13} , then \overline{AC}\cong\overline{13}

Postulates and Theorem for Triangle Congruence

SSS postulate- If three sides of a triangle are congruent to the corresponding sides of another triangle, then the two triangles are congruent.

In the image above,

\overline{AB}\cong\overline{12}

\overline{BC}\cong\overline{23}

\overline{AC}\cong\overline{13} ,

∆abc\cong∆123

SAS postulate- if the two sides and the included angle of one triangle are congruent to the other two sides and included angle of another triangle, then the two triangles are congruent.

in the image above:

\overline{ab}\cong\overline{xy}

\overline{bc}\cong\overline{yz}

\angle{b} \leftrightarrow \angle{y}

∴ ∆abc\cong∆xyz

ASA postulate- if the two angles and the included side of a triangle are congruent to corresponding two angles and included side of another triangle, then the two triangles are equal or congruent.

(included side is the side by which its endpoints are the vertices of the two given angles)

In the image aboe \overline{xz} is the included side between \angle{x} and \angle{Z} ; \overline{23} is the included side between \angle{2} and \angle{3} . That makes:

\angle{x}\cong\angle{2}

\angle{z}\cong\angle{3}

\overline{xz}\cong\overline{23}

∴ ∆xyz\cong ∆123

SAA or AAS Theorem- if the two angles and a non-included side of triangle corresponds to the two angles and a non-included side of another triangle, then the two triangles are congruent.

\angle{A}\cong\angle{X} and \angle{B}\cong\angle{Y} (two angles are congruent to both triangle)

\overline{BC}\cong\overline{YZ} (non-included side of both triangle are congruent)

thus ∆ ABC\cong∆XYZ

L-L Theorem (leg-leg Theorem- if two legs of a right triangle corresponds to 2 legs of another right triangle then the two triangles are congruent.

\angle{B} and \angle{2} are both right angles. thus they are congruent, since any two right angles are always congruent. In the image above it is given that \overline{BC}\cong\overline{23} , and \overline{AB}\cong\overline{12} . Thus, ∆ABC\cong∆123

L-AA Theorem (Leg-acute angle Thereom)- if a leg and one acute angle of a right triangle corresponds to a leg and acute angle of another right triangle, then the two triangles are congruent.

Given on the image above, \angle{b} and \angle{2} are both right angles thus the two are equal or congruent. \angle{a}\cong\angle{1} and are both acute angles, thus ∆abc\cong∆123

Isosceles triangle Theorem- if two sides of a triangle are congruent then the angles opposite those sides are also congruent.

Converse of Isosceles Triangle- If two angles of a triangle are congruent then the opposite sides of those congruent angles are also equal.

Other properties of a triangle

median- is a segment which extends from a vertex to the midpoint of the opposite side.


\overline{AB} in the image above is the median

Altitude- is a segment from a vertex that is perpendicular to the opposite line (side).

Notice in the picture that not all altitudes lie in the interior part of the triangle.

Angle bisector- is a segment that bisects an angle in the triangle.

  • if a point lies on the bisector of an angle, then this point is equidistant from the endpoints of the line where the bisector ended perpendicularly.
  • If a point is within the bisector of an angle, then that point is equidistant from the sides of the angle
  • if a point is equidistant from the sides of an angle, then that point is within the bisector of an angle.

Try to answer the following items:

Determine the length of the sides of the following triangles and classify the triangle.

1. If x=6

If x =6,then:

3x-6

3(6)-6

18-6= 8 first side

2x-4

2(6)-4

12-4= 8 second side

the third side is equal to 6

The triangle is an Isosceles Triangle

2. if x=9

If x =9,then:

2x+2

2(9)+2

18+2= 20 first side

x-5

9-5= 4 second side

x+9

9+9= 18 third side

The triangle is a scalene triangle

3. In the image below, suppose \triangle{ABC} corresponds to \triangle{XYZ} , Write all the possible corresponding parts of the two triangles.

\angle{A} \leftrightarrow \angle{X}

\angle{B} \leftrightarrow \angle{Y}

\angle{C} \leftrightarrow \angle{Z}

\overline{AB} \leftrightarrow \overline{XY}

\overline{AC} \leftrightarrow \overline{XZ}

\overline{BC} \leftrightarrow \overline{YZ}

4. On the images below, determine if the triangles are congruent, if so, state the theorem that proves its congruence.

1. SAS 2. SSS 3. ASA