SAT-66: Lines and Angles — Vertical, Supplementary, and Complementary
Use vertical, supplementary (180°), and complementary (90°) angle relationships to solve for unknowns.
SAT-66: Lines and Angles — Vertical, Supplementary, and Complementary
Description: Geometry on the SAT starts with angle relationships at lines and points. Three relationships solve a huge share of problems: vertical, supplementary, and complementary angles. Master these and you can set up equations for almost any figure.
Supplementary angles (straight line = 180°)
Two angles that form a straight line add to 180°. They are called supplementary. (Oʻzbekcha: toʻgʻri chiziq hosil qiluvchi ikki burchak yigʻindisi 180° ga teng.)
Complementary angles (right angle = 90°)
Two angles that together form a right angle add to 90°. They are complementary. (Oʻzbekcha: birgalikda toʻgʻri burchak hosil qiluvchi burchaklar 90° ga teng.)
Vertical angles (the X)
When two lines cross, the angles opposite each other are vertical angles, and they are always equal. The two angles next to each other on a line are supplementary. (Oʻzbekcha: ikki chiziq kesishganda qarama-qarshi burchaklar (vertikal) oʻzaro teng boʻladi.)
Worked Example 1 — supplementary
Two angles on a straight line are (2x) and (x + 30). Find x.
- They add to 180: 2x + (x + 30) = 180 → 3x + 30 = 180 → 3x = 150 → x = 50.
Worked Example 2 — complementary
Two complementary angles are (3x) and (x + 10). Find each angle.
- They add to 90: 3x + x + 10 = 90 → 4x = 80 → x = 20.
- Angles: 3(20) = 60° and 20 + 10 = 30° (and 60 + 30 = 90 ✓).
(Oʻzbekcha: ikkala burchak yigʻindisi 90° boʻlishi kerak.)
Worked Example 3 — vertical + supplementary together
Two lines cross. One angle is 110°. Find its vertical angle and an adjacent angle.
- Vertical angle (opposite) = 110° (equal).
- Adjacent angle (on the line) = 180 − 110 = 70° (supplementary).
Strategy: label what you know, then write "these add to 180" or "these are equal." Most SAT angle problems are just one linear equation. (Oʻzbekcha: bilganingizni belgilang, soʻng tenglama tuzing — koʻpincha bitta chiziqli tenglama yetarli.)
Worked Example 4 — angles around a point
Three angles meet at a single point and go all the way around: (2x)°, (3x)°, and (4x)°. Find x.
- Angles around a point add to 360°: 2x + 3x + 4x = 360 → 9x = 360 → x = 40.
This is the fourth key fact: a full turn around a point is 360°, while a straight line is only 180°. (Oʻzbekcha: nuqta atrofidagi burchaklar yigʻindisi 360°, toʻgʻri chiziqdagi esa 180°.)
Don't trust the picture — trust the numbers
SAT figures are usually drawn close to scale, but they are not guaranteed to be exact unless the problem says "figure drawn to scale." An angle that looks like 90° might be 88°. So never measure with your eye and assume; use the stated values and the angle rules to write an equation. The picture is a guide for setting up, not a source of measurements. (Oʻzbekcha: rasmga ishonib oʻlchamang — berilgan sonlar va qoidalar asosida tenglama tuzing.)
Practice 1
An angle is 4 times its supplement. Find the angle.
Show answer
Let the supplement be x; the angle is 4x. Then x + 4x = 180 → 5x = 180 → x = 36. The angle = 4(36) = 144°.
Practice 2
Two lines intersect; one of the four angles is (5x − 5)° and the angle vertical to it is 70°. Find x.
Show answer
Vertical angles are equal: 5x − 5 = 70 → 5x = 75 → x = 15.
Key words — Kalit soʻzlar
- Angle — burchak
- Vertical angles — vertikal burchaklar
- Supplementary — qoʻshni (yigʻindisi 180°)
- Complementary — toʻldiruvchi (yigʻindisi 90°)
- Straight line — toʻgʻri chiziq
- Right angle — toʻgʻri burchak
- Adjacent — yondosh
- Intersect — kesishish
- Degree (°) — gradus
Summary
- Supplementary angles add to 180° (straight line); complementary add to 90°.
- Vertical (opposite) angles are equal.
- Translate the relationship into one equation and solve.