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Unit 8: Geometry and Trigonometry

Topic 1

Lines and Angles — Parallel, Perpendicular, and Transversals

Lines and angles are foundational to all geometry problems on the SAT. A line extends infinitely in both directions, while a ray has one endpoint and extends in one direction. When two lines meet, they form angles — measured in degrees.
A transversal is a line that cuts across two or more other lines. If those lines are parallel, several angle relationships form:
Angle PairRelationshipCorresponding AnglesEqual (congruent)Alternate Interior AnglesEqual (congruent)Alternate Exterior AnglesEqual (congruent)Consecutive Interior AnglesSupplementary (sum = 180°)\begin{array}{c|l} \textbf{Angle Pair} & \textbf{Relationship} \\ \hline \text{Corresponding Angles} & \text{Equal (congruent)} \\ \text{Alternate Interior Angles} & \text{Equal (congruent)} \\ \text{Alternate Exterior Angles} & \text{Equal (congruent)} \\ \text{Consecutive Interior Angles} & \text{Supplementary (sum = 180°)} \end{array}
The figure below numbers the eight angles formed: 1-4 at the top intersection, 5-8 at the bottom, each in top-left/top-right/bottom-left/bottom-right order.
Using that numbering: corresponding angles are 1 & 5, 2 & 6, 3 & 7, and 4 & 8. Alternate interior angles are 3 & 6 and 4 & 5. Consecutive interior angles are 3 & 5 and 4 & 6. Alternate exterior angles are 1 & 8 and 2 & 7.
Perpendicular lines intersect to form right angles (90°). If two lines are perpendicular, their slopes multiply to −1-1:
m1⋅m2=−1.m_1 \cdot m_2 = -1.
Parallel lines never meet and have the same slope:
m1=m2.m_1 = m_2.
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Core Skills

  • Identify angle pairs formed by a transversal.
  • Use properties of parallel and perpendicular lines to find unknown angles.
  • Apply the sum of angles on a line (180°) and at a point (360°).
  • Relate slopes of parallel and perpendicular lines in coordinate geometry.
  • Use logical reasoning to find missing measures in angle diagrams.

Example 1: Parallel Lines and Transversal

Lines l1l_1 and l2l_2 are parallel, and transversal tt crosses them. If one acute angle measures 65∘,65^\circ, find its alternate interior angle. Alternate interior angles are congruent:
65∘.\boxed{65^\circ.}

Example 2: Supplementary Angles

When two parallel lines are cut by a transversal, a pair of consecutive interior angles are supplementary. If one angle is 110∘,110^\circ,
180−110=70.180 - 110 = 70.
70∘.\boxed{70^\circ.}

Example 3: Perpendicular Lines

Two lines intersect perpendicularly. One angle measures x+15∘,x + 15^\circ, and the adjacent angle measures 3x−45∘.3x - 45^\circ. Set their sum equal to 90∘:90^\circ:
(x+15)+(3x−45)=90⇒4x−30=90⇒4x=120⇒x=30.(x + 15) + (3x - 45) = 90 \Rightarrow 4x - 30 = 90 \Rightarrow 4x = 120 \Rightarrow x = 30.
x=30\boxed{x = 30}

Example 4: Slopes of Parallel and Perpendicular Lines

If a line has equation y=2x+3,y = 2x + 3,
  • A parallel line has slope m=2.m = 2.
  • A perpendicular line has slope m=−12.m = -\frac{1}{2}.
Parallel: y=2x+b,Perpendicular: y=−12x+b.\boxed{\text{Parallel: } y = 2x + b, \quad \text{Perpendicular: } y = -\tfrac{1}{2}x + b.}

Example 5: Angle Relationships at a Point

At a point, four angles form. Two adjacent ones are xx and 2x+30.2x + 30. Since angles around a point sum to 360°:
2(x+2x+30)=360⇒6x+60=360⇒6x=300⇒x=50.2(x + 2x + 30) = 360 \Rightarrow 6x + 60 = 360 \Rightarrow 6x = 300 \Rightarrow x = 50.
x=50∘\boxed{x = 50^\circ}

Example 6: SAT-Style Interpretation

In the coordinate plane, line kk has slope 3.3. A line perpendicular to kk passes through (1,4)(1, 4). Find its equation. Slope of perpendicular line = −13-\frac{1}{3}. Point-slope form:
y−4=−13(x−1)⇒y=−13x+13+4=−13x+133.y - 4 = -\tfrac{1}{3}(x - 1) \Rightarrow y = -\tfrac{1}{3}x + \tfrac{1}{3} + 4 = -\tfrac{1}{3}x + \tfrac{13}{3}.
y=−13x+133\boxed{y = -\tfrac{1}{3}x + \tfrac{13}{3}}

Key Takeaways

  • Parallel lines have equal slopes; perpendicular lines’ slopes multiply to −1.-1.
  • Corresponding, alternate interior, and alternate exterior angles are congruent for parallel lines.
  • Consecutive interior angles are supplementary.
  • Angles on a straight line sum to 180°; angles around a point sum to 360°.
  • Always mark known relationships on diagrams before solving.