Lesson 1.4 · Unit 1 · G-CO.9

The four relationships every later proof will cite

Four angle pairs appear in nearly every proof in the course. Two of them are defined by measure and two by position, and keeping that distinction straight prevents the most common error in the topic: assuming that angles which sum correctly must also be next to each other.

The method
  1. Complementary angles have measures summing to \( 90^\circ \); supplementary angles sum to \( 180^\circ \). Both are defined by measure alone, with no requirement that the angles touch.
  2. Adjacent angles share a vertex and a side and do not overlap. This is defined by position alone and says nothing about measure.
  3. A linear pair is two adjacent angles whose non-shared sides form a line. It requires both conditions.
  4. Vertical angles are the two non-adjacent angles formed by two intersecting lines, opposite each other at the intersection.
  5. A linear pair is supplementary, which is a theorem, not the definition. The definition is about position; supplementarity is what follows.
  6. Vertical angles are congruent, also a theorem, and it is proved from the linear pair result rather than assumed.
  7. Supplementary does not imply linear pair. Two angles in different parts of a diagram can sum to \( 180^\circ \) without touching.
  8. Set up an equation from the relationship, solve, and check both that the sum is right and that each angle is positive.

Where students lose marks: treating "supplementary" and "linear pair" as interchangeable. Every linear pair is supplementary; most supplementary pairs are not linear pairs. A proof that needs the angles adjacent must cite the linear pair, and one that only needs the sum may cite supplementarity.

Worked example

The problem. Two lines intersect, forming four angles numbered 1, 2, 3 and 4 in order around the point, so that \( \angle 1 \) and \( \angle 3 \) are vertical, as are \( \angle 2 \) and \( \angle 4 \). (a) Prove that vertical angles are congruent. (b) If \( m\angle 1 = (3x + 10)^\circ \) and \( m\angle 3 = (5x - 20)^\circ \), find every angle in the figure.

Step one: set up the proof in (a) with what is given. Given: two intersecting lines forming \( \angle 1 \), \( \angle 2 \), \( \angle 3 \), \( \angle 4 \) in order. Prove: \( \angle 1 \cong \angle 3 \).

Step two: identify the two linear pairs. Going around the point, \( \angle 1 \) and \( \angle 2 \) are adjacent and their non-shared sides lie on one of the two lines, so they form a linear pair. The same holds for \( \angle 2 \) and \( \angle 3 \).

Step three: apply the linear pair theorem to both. \( m\angle 1 + m\angle 2 = 180 \) and \( m\angle 2 + m\angle 3 = 180 \). Each line is licensed by the linear pair theorem, cited by name.

Step four: combine them. Both left sides equal 180, so they equal each other by the transitive property: \( m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3 \).

Step five: finish the proof. Subtracting \( m\angle 2 \) from both sides, by the subtraction property of equality, gives \( m\angle 1 = m\angle 3 \), and therefore \( \angle 1 \cong \angle 3 \) by the definition of congruent angles. Notice what the proof did not do: it never measured anything and never appealed to the picture looking symmetric. The result came from two applications of one theorem and some algebra.

Step six: use the result in (b). Since \( \angle 1 \) and \( \angle 3 \) are vertical, they are congruent by the theorem just proved, so their measures are equal: \( 3x + 10 = 5x - 20 \), giving \( 30 = 2x \) and \( x = 15 \).

Step seven: find \( \angle 1 \) and \( \angle 3 \). \( m\angle 1 = 3(15) + 10 = 55^\circ \), and checking with the other expression, \( m\angle 3 = 5(15) - 20 = 55^\circ \). They agree, confirming \( x \).

Step eight: find the other two and check the whole figure. \( \angle 1 \) and \( \angle 2 \) form a linear pair, so they are supplementary: \( m\angle 2 = 180 - 55 = 125^\circ \). And \( \angle 4 \) is vertical to \( \angle 2 \), so \( m\angle 4 = 125^\circ \) as well. Check: the four angles around the point sum to \( 55 + 125 + 55 + 125 = 360^\circ \), as angles filling a complete turn must. Two are acute and two obtuse, which is what two intersecting non-perpendicular lines always produce.

Practice · 10 problems

Problems 1 to 5 are routine. Problems 6 to 10 are multi-step or ask for reasoning.

  1. Find the complement of \( 37^\circ \).
    Show the full solution

    \( 90 - 37 \). \( 53^\circ \)

  2. Find the supplement of \( 112^\circ \).
    Show the full solution

    \( 180 - 112 \). \( 68^\circ \)

  3. Two angles form a linear pair and one measures \( 43^\circ \). Find the other.
    Show the full solution

    A linear pair is supplementary. \( 137^\circ \)

  4. Two vertical angles measure \( (4x)^\circ \) and \( (2x + 30)^\circ \). Find \( x \).
    Show the full solution

    Vertical angles are congruent: \( 4x = 2x + 30 \). \( x = 15 \)

  5. Does an obtuse angle have a complement?
    Show the full solution

    Its complement would have to be negative. No

  6. An angle is \( 20^\circ \) more than its complement. Find both.
    Show the full solution

    Let the smaller angle be \( x \), so the larger is \( x + 20 \). Complementary means they sum to 90: \( x + (x + 20) = 90 \), so \( 2x = 70 \) and \( x = 35 \). The angles are \( 35^\circ \) and \( 55^\circ \). Check: they sum to 90, and 55 is 20 more than 35. Both conditions hold. \( 35^\circ \) and \( 55^\circ \)

  7. An angle is three times its supplement. Find both.
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    Let the smaller be \( x \), so the larger is \( 3x \). Supplementary means they sum to 180: \( x + 3x = 180 \), so \( 4x = 180 \) and \( x = 45 \). The angles are \( 45^\circ \) and \( 135^\circ \). Check: \( 45 + 135 = 180 \), and \( 135 = 3 \times 45 \). Both conditions hold. \( 45^\circ \) and \( 135^\circ \)

  8. Give an example of two supplementary angles that do not form a linear pair.
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    Supplementary is defined by measure alone, so the angles need not touch at all. Take an angle of \( 60^\circ \) in one corner of a page and an angle of \( 120^\circ \) in the opposite corner. Their measures sum to \( 180^\circ \), so they are supplementary, and they share no vertex, so they cannot form a linear pair. A more useful example from later in the course: in a parallelogram, two consecutive angles are supplementary but the pair is not a linear pair, since their non-shared sides are two sides of the figure rather than one straight line. The general principle: a linear pair requires adjacency and a straight line, while supplementary requires only that the measures sum to 180. Linear pair is the stronger condition, and a proof that needs adjacency must establish it separately. Any two angles summing to \( 180^\circ \) that do not share a vertex and a side, such as \( 60^\circ \) and \( 120^\circ \) drawn separately

  9. Two lines intersect. One angle measures \( 68^\circ \). Find all four angles.
    Show the full solution

    Call the given angle \( \angle 1 \), with \( \angle 2 \) and \( \angle 4 \) adjacent to it and \( \angle 3 \) opposite. \( \angle 1 \) and \( \angle 2 \) form a linear pair, so they are supplementary: \( m\angle 2 = 180 - 68 = 112^\circ \). \( \angle 3 \) is vertical to \( \angle 1 \), so it is congruent: \( m\angle 3 = 68^\circ \). \( \angle 4 \) is vertical to \( \angle 2 \), so \( m\angle 4 = 112^\circ \). Check: \( 68 + 112 + 68 + 112 = 360^\circ \), a complete turn. \( 68^\circ \), \( 112^\circ \), \( 68^\circ \), \( 112^\circ \)

  10. Two angles are both supplementary and congruent. Find their measures, and prove the result in general.
    Show the full solution

    Let the measures be \( a \) and \( b \). Supplementary gives \( a + b = 180 \). Congruent gives \( a = b \) by the definition of congruent angles. Substituting: \( a + a = 180 \), so \( 2a = 180 \) and \( a = 90 \). Then \( b = 90 \) as well. So both angles are right angles. This is a small theorem worth remembering: if two angles are both congruent and supplementary, each is a right angle. It is the reason that two intersecting lines forming one right angle form four of them, which lesson 1.5 uses to define perpendicularity from a single right angle. The proof used both given facts and nothing from a diagram, which is what makes it general rather than a statement about one picture. Each measures \( 90^\circ \); two congruent supplementary angles are always right angles

← 1.3 Angles and angle measure1.5 Perpendicularity, bisectors and proof-ready definitions →

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