Lesson 1.3 · Unit 1 · G-CO.1
Naming an angle unambiguously, and adding angle measures
Angles work exactly like segments: there is an addition postulate, a bisector, and the same distinction between the figure and its measure. The one thing genuinely new is naming, where a careless single letter can name three different angles at once.
- An angle is two rays with a common endpoint, called the vertex, and the rays are its sides.
- Name an angle three ways: by three points with the vertex in the middle, \( \angle ABC \); by the vertex alone, \( \angle B \); or by a number written inside it.
- The single-letter name is only allowed when one angle has that vertex. If three angles share vertex \( B \), then \( \angle B \) is ambiguous and the three-letter name is required.
- \( m\angle ABC \) is the measure, a number; \( \angle ABC \) is the figure. Congruent angles have equal measures.
- Classify by measure: acute is less than \( 90^\circ \), right is exactly \( 90^\circ \), obtuse is between \( 90^\circ \) and \( 180^\circ \), and straight is exactly \( 180^\circ \).
- The angle addition postulate: if \( D \) is in the interior of \( \angle ABC \), then \( m\angle ABD + m\angle DBC = m\angle ABC \).
- An angle bisector is a ray through the interior dividing the angle into two congruent angles.
- Check an angle answer against its classification. An angle computed as \( 115^\circ \) that was marked as acute in the diagram means an error upstream.
Where students lose marks: naming an angle with the vertex letter not in the middle. \( \angle ABC \) has vertex \( B \). Writing \( \angle BAC \) for the same angle names a different angle entirely, the one with vertex \( A \).
The problem. Ray \( \overrightarrow{BD} \) lies in the interior of \( \angle ABC \), with \( m\angle ABD = 4x \), \( m\angle DBC = 2x + 10 \) and \( m\angle ABC = 70^\circ \). (a) Find all three measures. (b) Is \( \overrightarrow{BD} \) the angle bisector? (c) Classify each of the three angles. (d) Explain why \( \angle B \) would be an unacceptable name here.
Step one: name the postulate before writing the equation. Since \( D \) is in the interior of \( \angle ABC \), the angle addition postulate applies: \[ m\angle ABD + m\angle DBC = m\angle ABC \] Without the interior condition the postulate says nothing, so that condition is quoted from the given information rather than read off a picture.
Step two: substitute and solve. \( 4x + (2x + 10) = 70 \), so \( 6x + 10 = 70 \), giving \( 6x = 60 \) and \( x = 10 \).
Step three: find the measures. \( m\angle ABD = 4(10) = 40^\circ \) and \( m\angle DBC = 2(10) + 10 = 30^\circ \), with \( m\angle ABC = 70^\circ \) as given.
Step four: check against the postulate. \( 40 + 30 = 70 \). Correct, and both parts are positive as angle measures must be.
Step five: answer (b) from the definition. An angle bisector divides an angle into two congruent angles. Here the two parts measure \( 40^\circ \) and \( 30^\circ \), which are not equal, so \( \overrightarrow{BD} \) is not the bisector. Had it been the bisector, each part would measure \( 35^\circ \).
Step six: answer (c). \( \angle ABD \) at \( 40^\circ \) is acute; \( \angle DBC \) at \( 30^\circ \) is acute; \( \angle ABC \) at \( 70^\circ \) is acute. All three are less than \( 90^\circ \), which is consistent: two acute angles can sum to an obtuse one, but here they do not.
Step seven: answer (d). Three angles in this figure have vertex \( B \): \( \angle ABD \), \( \angle DBC \) and \( \angle ABC \). Writing \( \angle B \) would not say which of the three is meant, so the name is ambiguous and is not accepted. The three-letter name is required whenever more than one angle shares a vertex.
Step eight: note the naming rule that follows. In \( \angle ABD \) the middle letter \( B \) is the vertex and the outer letters name a point on each side. The outer two may be swapped, so \( \angle ABD \) and \( \angle DBA \) are the same angle, but the middle letter may not move. \( \angle BAD \) would name an angle with vertex \( A \), which is a different figure.
Problems 1 to 5 are routine. Problems 6 to 10 are multi-step or ask for reasoning.
- Classify an angle measuring \( 143^\circ \).
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Obtuse
- What is the vertex of \( \angle PQR \)?
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The middle letter. \( Q \)
- \( \overrightarrow{BD} \) bisects \( \angle ABC \) and \( m\angle ABC = 84^\circ \).
Find \( m\angle ABD \).
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A bisector makes two congruent angles. \( 42^\circ \)
- \( m\angle ABD = 25^\circ \) and \( m\angle DBC = 40^\circ \), with \( D \) interior.
Find \( m\angle ABC \).
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Angle addition postulate. \( 65^\circ \)
- What is the difference between \( \angle ABC \) and \( m\angle ABC \)?
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The first is the figure; the second is its measure, a number
- \( \overrightarrow{BD} \) is in the interior of \( \angle ABC \) with
\( m\angle ABD = 3x + 5 \), \( m\angle DBC = x + 15 \) and \( m\angle ABC = 88^\circ \).
Find all three.
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By the angle addition postulate, \( (3x + 5) + (x + 15) = 88 \), so \( 4x + 20 = 88 \), giving \( 4x = 68 \) and \( x = 17 \). \( m\angle ABD = 3(17) + 5 = 56^\circ \) and \( m\angle DBC = 17 + 15 = 32^\circ \). Check: \( 56 + 32 = 88 \). Correct. \( 56^\circ \), \( 32^\circ \) and \( 88^\circ \)
- \( \overrightarrow{BD} \) bisects \( \angle ABC \), with \( m\angle ABD = 5x - 4 \) and
\( m\angle DBC = 3x + 10 \). Find \( m\angle ABC \).
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A bisector makes the two parts congruent, so their measures are equal: \( 5x - 4 = 3x + 10 \), giving \( 2x = 14 \) and \( x = 7 \). Each part measures \( 5(7) - 4 = 31^\circ \), and checking the other expression, \( 3(7) + 10 = 31^\circ \). They agree. By the angle addition postulate, \( m\angle ABC = 31 + 31 = 62^\circ \). The question asked for the whole angle, not for \( x \) or for one part, which is the step most often skipped. \( 62^\circ \)
- Two angles share vertex \( M \). A student refers to "\( \angle M \)". What should they
write instead, and why?
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The single-letter name is only unambiguous when exactly one angle has that vertex. With two angles at \( M \), the name \( \angle M \) could mean either, so a reader cannot tell which is being claimed about. They should use the three-letter name, with \( M \) in the middle and a point from each side on the outside: \( \angle PMQ \) and \( \angle QMR \), for instance. This matters more in geometry than it looks. A proof is an argument a reader must be able to check line by line, and a line that could mean two things cannot be checked. Ambiguous naming is marked wrong even when the intended meaning is guessable. A three-letter name with \( M \) in the middle, because two angles share that vertex
- \( m\angle ABD = 2x \), \( m\angle DBC = 3x \), and \( \angle ABC \) is a right angle
with \( D \) interior. Find both parts.
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A right angle measures \( 90^\circ \) by definition, so that is the whole. By the angle addition postulate, \( 2x + 3x = 90 \), so \( 5x = 90 \) and \( x = 18 \). \( m\angle ABD = 2(18) = 36^\circ \) and \( m\angle DBC = 3(18) = 54^\circ \). Check: \( 36 + 54 = 90 \). Correct, and both are acute, which is necessarily true when two positive angles sum to a right angle. \( 36^\circ \) and \( 54^\circ \)
- An angle measures \( (7x - 12)^\circ \) and its bisector creates two angles each
measuring \( (2x + 9)^\circ \). Find the angle, and check the answer three ways.
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The two halves must sum to the whole, by the angle addition postulate: \( (2x + 9) + (2x + 9) = 7x - 12 \), so \( 4x + 18 = 7x - 12 \), giving \( 30 = 3x \) and \( x = 10 \). The whole angle measures \( 7(10) - 12 = 58^\circ \), and each half measures \( 2(10) + 9 = 29^\circ \). Check one, addition: \( 29 + 29 = 58 \). Correct. Check two, halving: \( 58 \div 2 = 29 \), which is what a bisector must produce. Check three, classification: \( 58^\circ \) is acute and each half at \( 29^\circ \) is acute, which is consistent; bisecting an acute angle cannot produce anything but acute angles. \( 58^\circ \), with halves of \( 29^\circ \)