Lesson 8: Angle Relationships, Similarity, and Applications

Property

For any $\triangle ABC$, the sum of the measures of the angles is $180^\circ$.

Examples

• The measures of two angles of a triangle are $60^\circ$ and $85^\circ$. The third angle, $x$, is found by solving $60^\circ + 85^\circ + x = 180^\circ$, which gives $145^\circ + x = 180^\circ$, so $x = 35^\circ$.

Property

The sum of the measures of interior angles in a triangle is $180^\circ$.
Once we set an expression equal to something, we can find a value for $x$ to make the equation true.
This is the very beginning of thinking about “$x$” as a variable rather than an unknown.

Examples

• A triangle has angles measuring $x$, $x+20$, and $x+40$ degrees. The sum is $x + (x+20) + (x+40) = 180$. Combining terms gives $3x + 60 = 180$, so $3x = 120$ and $x = 40$. The angles are $40^\circ, 60^\circ, 80^\circ$.
• A right triangle has one angle measuring $55^\circ$. The other two angles are $90^\circ$ and $x$. Since the sum is $180^\circ$, we have $55 + 90 + x = 180$, which means $145 + x = 180$. The missing angle is $35^\circ$.
• The angles of a triangle are in the ratio $1:2:3$. Let the angles be $x$, $2x$, and $3x$. Their sum is $x + 2x + 3x = 180$. This is $6x = 180$, so $x=30$. The angles are $30^\circ$, $60^\circ$, and $90^\circ$.

Explanation

Every triangle's three angles always add up to $180^\circ$. This fact lets us build an equation. If the angles are written with a variable $x$, we just add them all up and set the sum equal to 180 to solve.

Property

When a transversal intersects two parallel lines, corresponding angles are congruent. Corresponding angles occupy the same relative position at each intersection point.

Examples