Lines, angles, and triangles

Lines, angles, and triangles form the geometric foundation of the SAT, covering parallel line properties, angle relationships, and geometric properties of triangles.

Lines and Angles

When two parallel lines are intersected by a transversal line, specific angle relationships are created:

• Adjacent Angles on a Line: Angles that lie on a straight line add up to $180^\circ$.

  Example: If $\angle A$ and $\angle B$ form a line and $\angle A = 70^\circ$, then $\angle B = 180^\circ - 70^\circ = 110^\circ$.

• Vertical Angles: Opposite angles formed by two intersecting lines are always equal.

  Example: If two lines intersect to form opposite angles $x^\circ$ and $45^\circ$, then $x = 45^\circ$.

• Parallel Lines Cut by a Transversal:

  • Corresponding Angles: Equal to each other (e.g., top-left matches top-left).

  • Alternate Interior Angles: Equal to each other (e.g., inside angles on opposite sides of the transversal).

  • Same-Side Interior Angles: Supplementary (add up to $180^\circ$).

Triangle Fundamentals and Special Triangles

The interior angles of any triangle always add up to $180^\circ$.

• Angle-Side Relationships:

  • Isosceles Triangle: Has $2$ equal sides and $2$ equal angles opposite those sides.

  • Equilateral Triangle: Has $3$ equal sides and $3$ equal angles ($60^\circ$ each).

• Exterior Angle Theorem: The measure of an exterior angle of a triangle equals the sum of its two remote interior angles.

  Example: For $\triangle ABC$, an exterior angle at vertex $C$ equals $\angle A + \angle B$.

• Similar Triangles: Triangles with equal corresponding angles have proportional side lengths.

  Example: If $\triangle ABC \sim \triangle DEF$ with a scale factor of $2$, then every side in $\triangle DEF$ is twice as long as the corresponding side in $\triangle ABC$.

Right Triangles and Trigonometry

Right triangles ($90^\circ$ angle) follow the Pythagorean theorem: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.

Special Right Triangles

1. Use special right triangle ratios to skip using the Pythagorean theorem.

   Example: A $30^\circ-60^\circ-90^\circ$ triangle has a shorter leg ($x$) of $5$. What are the other sides?

   Side	Ratio Formula	Value
   Short Leg ($30^\circ$)	$x$	$5$
   Long Leg ($60^\circ$)	$x\sqrt{3}$	$5\sqrt{3}$
   Hypotenuse ($90^\circ$)	$2x$	$10$

2. Apply the Exterior Angle Theorem for quick mental math on triangle problems.

   Example: A triangle has interior angles of $50^\circ$ and $60^\circ$. Find the exterior angle at the third vertex.

   Method	Steps	Calculation
   Standard Method	Find 3rd angle, then subtract from $180^\circ$	$180 - (50 + 60) = 70 \implies 180 - 70 = 110^\circ$
   Exterior Angle Rule	Sum the two remote interior angles directly	$50^\circ + 60^\circ = 110^\circ$

3. Set up proportions between corresponding sides of similar triangles.

   Example: $\triangle ABC \sim \triangle DEF$. Side $AB = 6$, $BC = 8$, and corresponding side $DE = 9$. Find $EF$.

   Step	Proportional Setup	Calculation
   Set Ratio	$\frac{AB}{DE} = \frac{BC}{EF}$	$\frac{6}{9} = \frac{8}{EF}$
   Cross-Multiply	$6 \cdot EF = 72$	$EF = \frac{72}{6} = 12$