SAT Lines, angles & triangles
Parallel lines, angle sums, and similar triangles.
How to score it
- Mark every angle you can the moment you see parallel lines — corresponding and alternate angles are equal.
- Triangle angles sum to 180°; isosceles triangles have two equal base angles.
- Similar triangles: set up a ratio of corresponding sides.
Common traps
- Assumes a figure is to scale when it isn't.
- Pairs the wrong corresponding sides in a similarity ratio.
- Misses an isosceles or right-angle property.
The 4 question types, with real examples
Parallel lines cut by a transversal
“What is the measure of angle x?”
Two parallel lines are cut by a transversal. On one parallel line, an angle measures degrees. On the other parallel line, the corresponding angle on the opposite side of the transversal measures degrees. What is the value of x?
- A✓
- B
- C
- D
Alternate interior angles formed when a transversal cuts parallel lines are equal. Setting and solving: , so .
Similar triangles & proportionality
“[Similar triangles given] What is the length of … ?”
Triangle GHI has angles measuring 44°, 71°, and 65°. Triangle JKL is similar to triangle GHI, where vertices G, H, and I correspond to vertices J, K, and L, respectively. What is the measure of angle K?
- A
- B
- C
- D✓
Corresponding angles in similar triangles are congruent. Since vertex H in triangle GHI corresponds to vertex K in triangle JKL, angle K has the same measure as angle H. Therefore, angle K measures 71°.
Triangle angle sum (isosceles/equilateral/right)
“In triangle [ABC], the measure of angle [A] is [m] degrees. What is the value of [angle expression]?”
In triangle GHI, angle G measures 29° and angle H measures 73°. What is the measure of angle I, in degrees?
- A
- B
- C✓
- D
The sum of the interior angles of a triangle is 180°. The sum of angles G and H is 29° + 73° = 102°. Therefore, angle I measures 180° - 102° = 78°.
Quadrilateral / polygon angle reasoning
“[FIGURE: quadrilateral with one diagonal]. [question]”
In quadrilateral PQRS, diagonal PR is drawn. If angle QPR = 28°, angle SPR = 42°, angle PRS = 48°, and angle PRQ = 65°, what is the measure, in degrees, of angle PQR?
- A✓
- B
- C
- D
In triangle PQR, the sum of angles equals 180°. We have angle QPR = 28° and angle PRQ = 65°, so angle PQR = 180° - 28° - 65° = 87°.
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