Lines, angles and triangles

What the exam asks

Geometry and Trigonometry makes up about 15% of the scored Math questions (5–7 of 40), and this topic usually supplies one or two of them. Expect four kinds of task:

• Angle chasing: intersecting lines, parallel lines cut by a transversal and triangle angle sums, often with algebraic labels such as $(3x + 12)^\circ$.
• Triangle and polygon properties: isosceles and equilateral triangles, the exterior angle theorem, and interior angles of polygons.
• Similar triangles: a segment parallel to a side, a “bow-tie”, the altitude to a hypotenuse, and shadow or ramp problems in context, plus scale factors for perimeter and area.
• Congruence and similarity reasoning: which extra fact proves two triangles congruent or similar, or which statement must be true.

The reference sheet gives the $180^\circ$ triangle angle sum. Figures are drawn to scale unless the question says “Note: Figure not drawn to scale,” so an angle that looks obtuse cannot measure $40^\circ$.

Core ideas

Angle relationships

When two parallel lines are cut by a transversal, only two measures appear: an acute angle $a$ and its supplement $180 - a$. Every acute angle in the picture equals $a$, and every obtuse angle equals $180 - a$.

For a path that zigzags between two parallel lines, draw a third parallel line through the corner. The corner angle equals the sum of the angles that the two segments make with the parallel lines.

Triangle facts

• The angles sum to $180^\circ$.
• Exterior angle theorem: an exterior angle equals the sum of the two remote interior angles.
• Isosceles: equal sides are opposite equal angles. The vertex angle, where the equal sides meet, is the odd one out.
• Equilateral: every angle is $60^\circ$.
• The longest side is opposite the largest angle, and each side is shorter than the sum of the other two (the triangle inequality).
• Chains of equal segments such as $AB = BD = DC$ hide two isosceles triangles. Find each one’s base angles, then link them with the exterior angle theorem.

Polygon angles

For an $n$-sided polygon, the interior angles sum to $(n - 2) \cdot 180^\circ$, and the exterior angles (one at each vertex) sum to $360^\circ$. In a regular polygon each exterior angle is and each interior angle is .

Congruence and similarity

Congruent triangles have the same shape and size. The valid tests are SSS, SAS, ASA, AAS, and HL for right triangles. SSA and AAA do not prove congruence.

Similar triangles have the same shape: equal corresponding angles and proportional corresponding sides. The valid tests are AA, SAS similarity (two pairs of sides in the same ratio with equal included angles) and SSS similarity.

If the scale factor for lengths is $k$, then perimeters scale by $k$, areas by $k^2$ and volumes by $k^3$.

These are the five similar-triangle set-ups the SAT uses most:

1. Parallel segment inside a triangle: $\overline{DE} \parallel \overline{BC}$ creates a small triangle similar to the whole one.
2. Bow-tie: two segments cross between parallel lines; vertical angles plus alternate interior angles give AA.
3. Altitude to the hypotenuse: it creates three similar right triangles, and the altitude squared equals the product of the two pieces of the hypotenuse.
4. Shadows and ramps: two right triangles share an angle of elevation.
5. Flipped triangles: a shared angle plus one other pair of equal angles, with no parallel lines. The correspondence is not the obvious one.

Worked examples

Exam technique

• Chase angles on scratch paper, writing each measure on your sketch as you find it.
• Use the two-measure rule for parallel lines: decide whether the target angle is acute or obtuse, then choose $a$ or $180 - a$.
• Add a line. A zigzag between parallel lines needs an auxiliary parallel line, and a trig or area question about a non-right triangle needs an altitude.
• Write the correspondence first. For similar triangles, list the matching vertices from the equal angles (for example $A \leftrightarrow D$, $B \leftrightarrow E$), then build every ratio from a whole side to its matching whole side.

Common mistakes

Quick recap

• Vertical angles are equal; a linear pair sums to $180^\circ$; angles around a point sum to $360^\circ$.
• Parallel lines cut by a transversal produce only two angle measures, $a$ and $180 - a$.