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Year 7 Maths Angles

Complementary and Supplementary Angles

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Theory

Two angles are complementary if they add to \(90^\circ\). Two angles are supplementary if they add to \(180^\circ\). This Year 7 maths page shows how to find complements and supplements, how to find unknown angles with reasons, and how to solve angle equations such as \(5x + 4x = 180\).

The complement of an angle is what you add to make \(90^\circ\). The complement of \(35^\circ\) is \(90^\circ - 35^\circ = 55^\circ\).

The supplement of an angle is what you add to make \(180^\circ\). The supplement of \(55^\circ\) is \(180^\circ - 55^\circ = 125^\circ\).

Complementary angles often come from a right angle split by an arm. Supplementary angles often come from a straight line split by an arm. These are adjacent angles.

To find an unknown angle: write the reason in words, then the equation. Solve it and check.

Complementary angles in a right angle Right angle ABD at B split by arm BC into 55 degrees (top) and 35 degrees (bottom). 55° 35° B A C D
\(\angle ABC\) and \(\angle CBD\) are complementary, since \(\angle ABD\) is a right angle: \(55 + 35 = 90\).
Supplementary angles on a straight line Straight line FHJ with arm HG. Angle FHG is 55 degrees and angle GHJ is 125 degrees. 55° 125° H F J G
\(\angle FHG\) and \(\angle GHJ\) are supplementary, since \(\angle FHJ\) is a straight angle: \(55 + 125 = 180\).

Reasons to write:

\[ \text{Angles in a right angle add to } 90^\circ \]
Angles in a right angle add to 90
\[ \text{Angles on a straight line add to } 180^\circ \]
Angles on a straight line add to 180
\[ \text{complement} = 90^\circ - \text{angle} \]
complement =90− angle
\[ \text{supplement} = 180^\circ - \text{angle} \]
supplement =180− angle
C for Corner, S for Straight. Complementary makes a corner (\(90^\circ\)); supplementary makes a straight line (\(180^\circ\)).
Only acute angles have a complement. \(110^\circ\) has no complement, because it is already more than \(90^\circ\).

How to find an unknown angle

  1. Spot the right angle or the straight line.
  2. Write the reason in words.
  3. Write and solve the equation.
  4. Check that the angles add to \(90^\circ\) or \(180^\circ\).
Example 1 — Pairs of angles
Are the angles complementary, supplementary or neither?
(i) \(47^\circ\) and \(43^\circ\)
(ii) \(125^\circ\) and \(55^\circ\)
(iii) \(80^\circ\) and \(20^\circ\)
Solution
\(47 + 43\)\(=\)\(90\)
\(125 + 55\)\(=\)\(180\)
\(80 + 20\)\(=\)\(100\)
47+43=90,125+55=180,80+20=100

(i) complementary
(ii) supplementary
(iii) neither

Example 2 — Unknown angles with reasons
Find the value of each pronumeral.
Unknown angles a and b Left: a right angle split into 27 degrees (bottom) and a degrees (top). Right: a straight line split into b degrees (left) and 114 degrees (right). a° 27° b° 114° (i) (ii)
Solution

(i) Angles in a right angle add to \(90^\circ\).
(ii) Angles on a straight line add to \(180^\circ\).

\(a\)\(=\)\(90 - 27 = 63\)
\(b\)\(=\)\(180 - 114 = 66\)
a =90−27=63, b =180−114=66

(i) \(a = 63\)
(ii) \(b = 66\)

Example 3 — Two lines crossing
Find the size of
(i) \(\angle DBA\)
(ii) \(\angle ABE\)
(iii) \(\angle EBC\)
Two lines crossing at B Line AC is horizontal and line DE crosses it at B. Angle DBC is 70 degrees. 70° B A C D E
Solution

Each pair of adjacent angles on line \(AC\) or line \(DE\) makes a straight angle. Angles on a straight line add to \(180^\circ\).

\(\angle DBA\)\(=\)\(180 - 70 = 110\)
\(\angle ABE\)\(=\)\(180 - 110 = 70\)
\(\angle EBC\)\(=\)\(180 - 70 = 110\)
DBA =180−70=110, ABE =180−110=70, EBC =180−70=110

(i) \(110^\circ\) (supplementary to \(\angle DBC\))
(ii) \(70^\circ\) (supplementary to \(\angle DBA\))
(iii) \(110^\circ\) (supplementary to \(\angle ABE\))

Example 4 — A right angle on a line
\(BD\) is perpendicular to line \(AC\). Find the size of
(i) \(\angle DBC\)
(ii) \(\angle DBE\)
Right angle at B with a 25 degree angle Line AC is horizontal; BD is vertical, making a right angle at B. Arm BE makes 25 degrees with BC. 25° B A C D E
Solution
\(\angle DBC\)\(=\)\(180 - 90 = 90\)
\(\angle DBE\)\(=\)\(90 - 25 = 65\)
DBC =180−90=90, DBE =90−25=65

(i) \(90^\circ\) (supplementary to \(\angle ABD\))
(ii) \(65^\circ\) (complementary to \(\angle EBC\))

Example 5 — Pronumerals in both angles
\(PQR\) is a straight line, split by arm \(QS\) into \(5x^\circ\) and \(4x^\circ\). Find \(x\), then the size of each angle.
Solution

Angles on a straight line add to \(180^\circ\). Collect like terms.

\(5x + 4x\)\(=\)\(180\)
\(9x\)\(=\)\(180\)
\(x\)\(=\)\(20\)
5 x +4 x =180,9 x =180, x =20

\(x = 20\). The angles are \(5 \times 20 = 100^\circ\) and \(4 \times 20 = 80^\circ\). Check: \(100 + 80 = 180\).

Example 6 — Three parts and a difference
(i) A right angle is split into \(21^\circ\), \(x^\circ\) and \(33^\circ\). Find \(x\).
(ii) A straight angle is split into \(x^\circ\) and \((x + 50)^\circ\). Find \(x\).
(iii) An angle is \(16^\circ\) more than its complement. Find the angle.
Solution
\(21 + x + 33\)\(=\)\(90\)
\(x\)\(=\)\(36\)
\(2x + 50\)\(=\)\(180\)
\(x\)\(=\)\(65\)
\(2c + 16\)\(=\)\(90\)
\(c\)\(=\)\(37\)
21+ x +33=90, x =362 x +50=180, x =652 c +16=90, c =37

(i) \(x = 36\)
(ii) \(x = 65\)
(iii) Let the complement be \(c^\circ\), so the angle is \((c + 16)^\circ\). The angle is \(37 + 16 = 53^\circ\).

Example 7 — Two steps with reasons
\(XOY\) is a straight line and \(\angle ZOY = 90^\circ\). Arm \(OW\) splits \(\angle XOZ\) into \(\angle XOW = 3m^\circ\) and \(\angle WOZ = 2m^\circ\). Find \(m\), then \(\angle WOY\).
Solution

Angles on a straight line add to \(180^\circ\), so \(\angle XOZ = 180 - 90 = 90^\circ\). Angles in a right angle add to \(90^\circ\):

\(3m + 2m\)\(=\)\(90\)
\(m\)\(=\)\(18\)
\(\angle WOY\)\(=\)\(180 - 3 \times 18 = 126\)
3 m +2 m =90, m =18, WOY =180−54=126

\(m = 18\) and \(\angle WOY = 126^\circ\). Check: \(\angle WOZ + \angle ZOY = 36 + 90 = 126\).

Common pitfalls

Mixing up 90 and 180. The complement of \(40^\circ\) is \(50^\circ\), not \(140^\circ\).
Looking for a complement of an obtuse angle. \(110^\circ\) has no complement.
Stopping at x. If the angles are \(2x^\circ\) and \(3x^\circ\), finding \(x\) is not the end when the question asks for the angles.
Leaving out the reason. Write the reason in words next to each equation.

Frequently asked questions

What are complementary angles?

Two angles that add to \(90^\circ\).

What are supplementary angles?

Two angles that add to \(180^\circ\).

What is the complement of 35°?

\(90^\circ - 35^\circ = 55^\circ\).

What is the supplement of 55°?

\(180^\circ - 55^\circ = 125^\circ\).

How do you find an unknown angle on a straight line?

Angles on a straight line add to \(180^\circ\). Subtract the known angles from \(180^\circ\).

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