Triangles - Angles

SAT Questions that focus on Triangles - Angles require knowledge of the following topics.

Transversal Angles

The three lines in the figure below are on the same plane. Lines "a" and "b" are called concurrent. Line "t" is called transversal.
The pairs of angles in this figure have special names:

Alternate interior angles: (4 and 6); (3 and 5)
Alternate exterior angles: (2 and 8); (1 and 7)

Colateral interior angles (or Co-interior angles): (4 and 5); (3 and 6)
Colateral exterior angles (or Co-exterior angles): (1 and 8); (2 and 7)

Corresponding angles: (1 and 5); (2 and 6); (4 and 8); (3 and 7)

Parallel Lines

Two lines are parallel if, and only if, a transversal line forms congruent alternate interior angles. That is, in the figure below, "r" and "s" are parallel if, and only if, angles "b" and "e" are congruent ("r", "s" and "t" are on the same plane).

Demonstration.

Let's assume that "r" and "s" are not parallel. In this case, these two lines will intersect at point "P".

Angles "a" and "e" are supplementary: $a+e=180$.

The sum of the internal angles of a triangle is 180º. Thus, $a+b+p=180$

Subtracting the second equation from the first one:
$a+e-a-b-p=0$
$e=b+p$

Since $p\neq0$, "e" cannot be equal to "b", when the two lines are not parallel. Therefore, if the alternate interior angles are congruent, the lines are parallel.


Consider now the following figure in which the three lines are on the same plane, and lines "r" and "s" are parallel:

Since "r" and "s" are parallel the alternate interior angles are congruent:
"3" = "5" and "4" = "6".

Since vertically opposite angles are congruent ("2"="4", "3"="1", ...), when "r" and "s" are parallel, the alternate exterior angles are also congruent:
"2" = "8" and "1" = "7".

Colateral interior angles and colateral exterior angles are supplementaries.
"4"+"5"=180º
"3"+"6"=180º
"1"+"8"=180º
"2"+"7"=180º

Last, since "r" and "s" are parallel, the corresponding angles are also congruent:
"1"="5"
"2"="6"
"4"="8"
"3"="7"

Sum of the Interior Angles in Triangles

The sum of the measures of the three interior angles of any triangle is 180 °.

Consider triangle ABC in the figure below. The dotted line "r" is parallel to side BC:

Sides AC and AB are transversals to the two parallel lines BC and "r".
Thus angles "c" and "ce" are congruent (alternate interior).
And angles "b" and "be" are also congruent (alternate interior).

Since $be+a+ce=180$, and $c=ce$ and $b=be$:

$be+a+ce=180$
$b+a+c=180$

Exterior Angles in Triangles

The measure of an exterior angle in any triangle is equal to the sum of the measures of the two non-adjacent interior angles.

Demonstration:

In the figure below, the dotted line is an extension of side AB. Angle "e" is an exterior angle to vertex "A".

"a" and "e" are supplementary: $a+e=180$.

Since the sum of the interior angles of any triangle is 180º: $a+b+c=180$.

Subtracting the second equation from the first one:
$a+e-a-b-c=180-180$
$e-b-c=0$
$e=b+c$

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SAT Practice Tests - Triangles - Angles

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