Proving angle pair theorems

When you reach for it

A proof needs two angles to be congruent, and the figure gives no measure for either. Three situations come up again and again, and each has a theorem that settles it in one line:

• both angles are right angles, wherever they sit;
• each angle has a partner it is supplementary or complementary to, and the two partners are congruent or are one and the same angle;
• the two angles sit opposite each other where two lines cross.

These are the proofs about angle pairs. Each fact is already known as a fact: vertical angles showed why opposite angles match, and congruent angles did the algebra of equal measures. Here each is proved with a reason on every line, as in giving a reason for every step. Once proved, its name may stand as the reason for a single line of any later proof, so each short proof here saves many lines elsewhere.

Right Angle Congruence Theorem. All right angles are congruent.

Congruent Supplements Theorem. If two angles are supplementary to the same angle, or to two congruent angles, then they are congruent.

Congruent Complements Theorem. If two angles are complementary to the same angle, or to two congruent angles, then they are congruent.

Vertical Angles Theorem. If two angles are vertical angles, then they are congruent.

The order they are proved in. Right angles first, straight from the definitions, as the warm-up. Then congruent supplements and complements, from the definitions and the properties of equality, with no vertical angles anywhere. Then vertical angles, whose shortest proof quotes the Congruent Supplements Theorem. The order matters: a proof may lean only on theorems proved before it, so the vertical angle proof is sound only because the supplements theorem was proved without vertical angles.

The procedure

Proving all right angles are congruent

A proof often needs two angles to be congruent, and the only thing it knows about either one is that it is a right angle. The two may be corners of different triangles, the angles two perpendicular lines make with one transversal, or the corners at the two ends of a segment. They are not vertical angles and not a linear pair; nothing in the figure ties them together except that each is square. That is when to cite the Right Angle Congruence Theorem. It turns “each of these is a right angle” into “these two are congruent” in one line, and congruence is the form that the triangle congruence criteria and the tests for parallel lines ask for.

The theorem is proved once, so that later proofs may quote it. Its set-up, as in writing the Given and the Prove, is:

Given. \angle PQR and \angle XYZ are right angles.

Prove. \angle PQR \cong \angle XYZ.

Each angle gets a square, because being right is a measure fact. The two are drawn apart and turned different ways, since the statement says nothing about where they sit, and no arc marks them as congruent, since that is the Prove.

Plan. A right angle is defined by its measure, and congruent angles are defined by equal measures, so pass from the angles to their measures and back again.

Nothing in the proof used where the angles are or which way they face, so it holds for any two right angles at all.

Citing it. From now on the theorem is a reason in its own right. It may be written on a line \angle 1 \cong \angle 2 only when earlier lines already show that \angle 1 is a right angle and that \angle 2 is a right angle, each with its own reason: given, a square in the Given’s figure, the definition of perpendicular, or an earlier theorem. The payoff is length. Every use would otherwise repeat lines 2 to 6 above; with the theorem it is one.

Proving supplements of congruent angles are congruent

Here a proof needs two angles to be congruent, and nothing about them is known directly: no measure, no mark, no shared side. But each one has a partner it is supplementary to, and the two partners are known to be congruent, or are one and the same angle. Then the two angles are congruent, and a single line of the proof can say so. The same holds with complementary partners. Both facts, and the algebra behind them, are in complementary and supplementary angles and congruent angles; here they are proved, and then put to work as reasons themselves.

Proving the Congruent Supplements Theorem. Prove the congruent-angles form, since it contains the other.

Given. \angle 1 and \angle 2 are supplementary; \angle 3 and \angle 4 are supplementary; \angle 2 \cong \angle 4.

Prove. \angle 1 \cong \angle 3.

The angles are drawn apart and turned different ways, since the Given says nothing about where they sit. “Supplementary” is a fact about a sum and has no mark; \angle 1 and \angle 3 are the Prove, so they carry nothing.

Plan. Turn every fact into an equation about measures. Both sums equal 180^\circ, so they equal each other; then take the equal parts m\angle 2 and m\angle 4 away from both sides.

Line 6 puts m\angle 3 + m\angle 4 in place of the 180^\circ it equals; substitution lets the right side of an equation replace its left, so line 5 first turns line 4 around. The same-angle form needs no proof of its own. If \angle 4 is \angle 2 itself, line 7 becomes \angle 2 \cong \angle 2, by the reflexive property of congruence, and every later line stands as written. Line 10 is Euclid’s third Common Notion: if equals are taken from equals, what remains is equal.

Proving the Congruent Complements Theorem. This time, the same-angle form.

Given. \angle 1 and \angle 2 are complementary; \angle 3 and \angle 2 are complementary.

Prove. \angle 1 \cong \angle 3.

Plan. As before, with 90^\circ in place of 180^\circ. With one shared angle there is no link to trade in.

The congruent-angles form needs the three extra lines, 7 to 9, of the first proof; Exercise 32 asks for it. Neither proof uses 180 or 90 except to say that two sums are equal, which is why, the link aside, the proofs match line for line.

Citing the theorems. Each theorem is a conditional with a three-part if. Before the line that cites it, earlier lines must show:

• that the first two angles are supplementary (or complementary);
• that the second two are, of the same kind;
• the link: one angle from each pair is congruent to the other, or they are the same angle.

The new line is a congruence between the two angles that are not the link. Pairs rarely arrive labeled “supplementary”; an earlier line usually makes them so, from one of these sources:

In the reason column write “Congruent Supplements Theorem” or “Congruent Complements Theorem”, and name the lines that supply the pairs and the link.

Proving vertical angles congruent

Two lines cross, and a proof needs the angles opposite each other to be congruent. Vertical angles showed why they always are: each shares a straight line with the same neighbor. Here that reason is written out as a proof, and once it is proved, the words Vertical Angles Theorem may stand as the reason for a single line in any later proof. That later use is the point: wherever two lines, two segments or a transversal cross, the theorem moves an angle to the other side of the crossing.

The set-up is:

Given. \overleftrightarrow{AB} and \overleftrightarrow{CD} meet at E, with E between A and B and between C and D.

Prove. \angle AEC \cong \angle BED.

The betweenness clause is what “vertical” means written out: \overrightarrow{EA} and \overrightarrow{EB} point in opposite directions along one line, and so do \overrightarrow{EC} and \overrightarrow{ED}. No arc marks the two angles, since their congruence is the Prove.

Plan. \angle AEC and \angle BED each form a linear pair with the shaded \angle AED. Two angles supplementary to one angle are congruent. That last step can be quoted as the Congruent Supplements Theorem or done by hand with measures, which gives two proofs.

Proof 1, from the Congruent Supplements Theorem.

Line 9 uses the theorem’s same-angle form: if two angles are supplementary to the same angle, then they are congruent. Here the same angle is \angle AED.

Proof 2, from angle measures. Lines 1 to 6 are the same. Then:

What each proof depends on. Both need lines 3 to 6, so that both lines really are straight through E, and both need the Linear Pair Theorem, which some courses take as a postulate. Proof 1 is shorter because lines 9 to 12 of Proof 2 are packed into one theorem: the Congruent Supplements Theorem is that subtraction, done once for any angles. So Proof 1 is sound only where that theorem is proved without vertical angles, as it is above. Proof 2 rests only on definitions, the Linear Pair Theorem and the properties of equality, so it can come first in a course.

Proof 2 is also Euclid’s, in Book I, Proposition 15, with the same letters and the same shared neighbor \angle AED. Where lines 7 and 8 say 180^\circ, Euclid says “two right angles” (his Proposition 13); lines 9 and 10 are his rule that things equal to the same thing are equal, and line 11 his rule that equals taken from equals leave equals.

The other pair. \angle AED \cong \angle BEC by the same lines with C and D swapped; the shared neighbor is then \angle AEC.

Citing it. Before the theorem is the reason for \angle 1 \cong \angle 2, the page must show the two angles are vertical: their sides are two pairs of opposite rays. Lines or segments given as crossing supply that, and so do lines drawn crossing, a position fact (what a diagram may assume). The theorem gives a congruence; to get numbers, follow it with the definition of congruent angles.

Worked examples

Boundaries

Every fact a theorem needs must be established, not seen. A corner that looks square is not a reason: each angle needs its own line showing it is right before the Right Angle Congruence Theorem can be cited. Two angles that look as if they fill a straight line form a linear pair only when their outer sides are opposite rays; unless the Given or the figure puts the three points on one line, the Linear Pair Theorem cannot be cited. And vertical angles need both pairs of sides to be opposite rays, by the Given, by lines drawn crossing, or by an earlier line (Example 14).

The Right Angle Congruence Theorem works because right angles have one fixed size. The same steps prove that all straight angles are congruent. They prove nothing about acute or obtuse angles, which come in every size.

For the Congruent Supplements and Complements Theorems, both pairs must be of the same kind. A complementary pair and a supplementary pair give nothing. If \angle 1 and \angle 2 are complementary and \angle 3 and \angle 2 are supplementary, then \angle 3 is 90^\circ larger than \angle 1.

The conclusion joins the two angles that are not linked (Example 9).

Complements exist only for acute angles, and supplements only for angles under 180^\circ. So the Congruent Complements Theorem can only ever conclude that two acute angles are congruent.

The right angle and supplement theorems compare angles, not figures, and need nothing about position. Two right angles are congruent however long their arms are drawn and however the figures around them differ. Two supplements of congruent angles need not be near each other (Example 8) or turned the same way (Figure 2). Each theorem gives equal measures only.

The converses of the Right Angle Congruence and Vertical Angles Theorems are false. Two congruent angles of 40^\circ are not right. And when \overleftrightarrow{BD} \perp \overleftrightarrow{AC} at B, \angle ABD and \angle DBC are congruent and share a vertex, but they are a linear pair, not vertical angles. The Congruent Supplements Theorem is different: trade its link for its conclusion and it still holds, since the same theorem applies with the roles swapped (Exercise 46).

The order of theorems matters. Proof 1 of the Vertical Angles Theorem is only as good as the proof of the Congruent Supplements Theorem; if that proof cited vertical angles, each theorem would rest on the other and neither would be proved. Here it does not, so Proof 1 stands. Likewise the Alternate Interior Angles Theorem is proved from corresponding angles using the Vertical Angles Theorem, so the Vertical Angles Theorem may not be proved from it.

A third route to vertical angles, a half turn. Turning the plane 180^\circ about E carries each line through E onto itself with its rays swapped, laying \angle AEC on \angle BED. That proof assumes a half turn keeps angle measures. Some axiomatic treatments go the other way, using vertical angles to prove a half turn is a rigid motion. Either order works, not both.

In Euclid, all right angles being equal is not a theorem. Book I of the Elements defines a right angle without numbers: when a line standing on a line makes the neighboring angles equal, each is right (Definition 10). Nothing there compares a right angle here with one somewhere else, so Euclid assumed it: “That all right angles equal one another” is Postulate 4. School courses give every angle a measure instead, and define a right angle as one of 90^\circ, which makes the fact provable in a few lines. The assumption has not vanished. It has moved into the protractor postulate, which gives every angle exactly one measure and every straight angle 180^\circ, wherever it is drawn.

Where this goes next

From here the three theorems are reasons in almost every angle proof, cited without a second thought.

Two lines perpendicular to one transversal make right angles at two different points, and the Right Angle Congruence Theorem makes those corresponding angles congruent: that is the first case of proving lines parallel, and the perpendicular transversal theorem runs it the other way. Once the triangle congruence criteria arrive, it supplies the angle they need between two right triangles, and it is the pair of congruent angles that HL congruence takes for granted. That is how equal perpendiculars at the two ends of a segment lead to the diagonals of a rectangle being congruent, one of the properties of rectangles.

Example 8 is the kind of trade that proofs about parallel lines and proving lines parallel make again and again: one angle of a pair swapped, through a linear pair, for its neighbor at the same crossing, using nothing about parallels. The Alternate Interior Angles Theorem comes from corresponding angles by one vertical-angle step, as in Example 12. In triangle congruence proofs an angle pair that is not given directly often arrives through its complements or supplements, as in Exercise 33, or as the vertical pair between two given sides, as in Example 13, which is also the core of showing that a quadrilateral whose diagonals bisect each other is a parallelogram. Euclid’s very next proposition after vertical angles, on the exterior angle of a triangle, leans on it, and so does the angle between two chords. Set out as boxes and arrows, any of these proofs is a flowchart proof; how the two-column table itself is laid out, and why each line needs its reason, is justifying each step.

Exercises

Reading the proofs

1. Suppose line 2 of the vertical angle proofs, E between C and D, is dropped from the Given. Which is the first line after it that has no support, and which later lines of Proof 2 fall with it?
2. Proof 1 of the Vertical Angles Theorem used \angle AED as the shared neighbor. Which other angle in Figure 4 could it use for the same Prove, and what do lines 5 and 6 become?
3. Show how line 10 of Proof 2 could be justified by the transitive property of equality instead of substitution.

Which angles are congruent?

Say which two angles are congruent and which theorem says so, or explain why neither theorem applies.

4. \angle P and \angle Q are supplementary; \angle R and \angle S are supplementary; \angle Q \cong \angle S.
5. \angle 1 and \angle 2 are complementary; \angle 3 and \angle 1 are complementary.
6. \angle A and \angle B are supplementary; \angle C and \angle D are supplementary; \angle A \cong \angle D.
7. \angle X and \angle Y are complementary; \angle X and \angle Z are supplementary.

Right angles and their measures

8. \angle J and \angle K are right angles, and m\angle J = (4x + 18)^\circ. Find x.
9. \angle 1 is a right angle and \angle 1 \cong \angle 2, with m\angle 2 = (5y - 15)^\circ. Find y.
10. \angle P and \angle Q are right angles, with m\angle P = (a^2 + 9)^\circ and m\angle Q = (6a + 36)^\circ. Find every value of a that works.

Finding measures with reasons

11. In Figure 6, m\angle ABD = (6t - 4)^\circ and m\angle CBE = (4t + 10)^\circ. Find t, m\angle ABD and m\angle DBC.
12. In Figure 6, m\angle DBC is three times m\angle ABD. Find m\angle ABD, m\angle DBC and m\angle CBE.
13. In Figure 7, with the Given of Example 3, m\angle ABD = (4x + 10)^\circ and m\angle DBC = (6y - 12)^\circ. Find x and y.
14. In Figure 8, m\angle BAC = (3x + 5)^\circ and m\angle ACE = (x + 5)^\circ. Find x, m\angle BAC and m\angle ABD.
15. In Figure 8, m\angle ABD = 24^\circ. Find m\angle BAC, m\angle ACE and m\angle BHE.
16. \angle 1 and \angle 2 are supplementary, \angle 3 and \angle 4 are supplementary, and \angle 2 \cong \angle 4. If m\angle 1 = (5x - 22)^\circ and m\angle 3 = (3x + 14)^\circ, find x and m\angle 2.
17. \angle A and \angle B are complementary, \angle C and \angle D are complementary, and \angle A \cong \angle C. If m\angle B = (2k + 9)^\circ and m\angle D = (4k - 15)^\circ, find k and m\angle A.
18. In Figure 11, \angle ABD \cong \angle ACE, m\angle ABD = (6n + 10)^\circ and m\angle ACE = (4n + 40)^\circ. Find n and all three angles of \triangle ABC.
19. In the set-up of Example 6, m\angle DBC = (t^2)^\circ and m\angle SQR = (4t + 21)^\circ. Find every value of t that works, and m\angle ABD for each.
20. In Example 8, m\angle 1 = (7y - 5)^\circ and m\angle 3 = (4y + 40)^\circ. Find y and the measures of all three angles.
21. In Figure 4, m\angle AEC = (5x - 12)^\circ and m\angle BED = (3x + 18)^\circ. Find x and m\angle AED.
22. In Figure 14, with \overrightarrow{OB} bisecting \angle AOC, m\angle AOB = (3x + 7)^\circ and m\angle EOF = (5x - 19)^\circ. Find x and m\angle COD.
23. In Figure 15, \ell \parallel m, m\angle 1 = (6k - 20)^\circ and m\angle 2 = (4k + 14)^\circ. Find k and m\angle 3.
24. In Figure 16, m\angle A = 48^\circ and m\angle AEB = 71^\circ. Find m\angle DEC, m\angle AEC and m\angle ABE.
25. Two lines cross, and three of the four angles add to 250^\circ. Find all four angles.
26. Three lines meet at O as in Figure 14, but \overrightarrow{OB} need not bisect \angle AOC. m\angle DOE = 50^\circ, m\angle EOF = (2x)^\circ and m\angle AOC = (3x + 20)^\circ. Find x and m\angle AOC.

Short proofs

Write each as a two-column proof.

27. Given. \angle 1 and \angle 2 are right angles; \angle 2 \cong \angle 3. Prove. \angle 1 \cong \angle 3.
28. Given. \overline{BD} \perp \overline{AC} at D, with D between A and C. Prove. \angle ADB \cong \angle CDB.
29. Given. \angle ABC and \angle DEF are right angles; \overrightarrow{BP} bisects \angle ABC; \overrightarrow{EQ} bisects \angle DEF. Prove. \angle ABP \cong \angle DEQ.
30. Given. In Figure 8, \overline{BD} \perp \overline{AC} at D and \overline{CE} \perp \overline{AB} at E, and the two segments cross at H. Prove. \angle BHE \cong \angle BAC.
31. Prove that all straight angles are congruent.
32. Given. \angle 1 and \angle 2 are complementary; \angle 3 and \angle 4 are complementary; \angle 2 \cong \angle 4. Prove. \angle 1 \cong \angle 3.
33. Given. \triangle ABC with a right angle at C; \triangle DEF with a right angle at F; \angle A \cong \angle D. Prove. \angle B \cong \angle E.
34. A, O, B lie on a line in that order, and rays \overrightarrow{OC} and \overrightarrow{OD} rise from O on the same side of it. Given. \angle AOC \cong \angle BOD. Prove. \angle AOD \cong \angle BOC.
35. Given. \overleftrightarrow{AB} and \overleftrightarrow{CD} meet at E as in Figure 4; \angle AEC \cong \angle F. Prove. \angle BED \cong \angle F.
36. Given. \overleftrightarrow{AB} and \overleftrightarrow{CD} meet at E as in Figure 4; \angle AEC and \angle G are complementary. Prove. \angle BED and \angle G are complementary.
37. Given. In Figure 15, \ell \parallel m. Prove. \angle 1 \cong \angle 3.
38. Given. Three lines meet at O as in Figure 14, with no bisector assumed; \angle AOB \cong \angle COD. Prove. \angle DOE \cong \angle FOA.

Reasoning

39. Show by an example that “all acute angles are congruent” is false.
40. Is the converse of the Right Angle Congruence Theorem true: if two angles are congruent, are they right angles? Explain.
41. A proof’s Given is “\angle A and \angle B are complementary”, and its second line reads \angle A \cong \angle B, with the reason “Right Angle Congruence Theorem”. What is wrong?
42. Every right angle measures 90^\circ. Explain what a proof gains by citing the Right Angle Congruence Theorem.
43. Explain why Euclid could not prove that all right angles are equal the way this lesson does.
44. Explain why “supplements of the same angle are congruent” needs no proof of its own once “supplements of congruent angles are congruent” is proved.
45. A proof reads: “\angle 1 and \angle 2 form a linear pair, and \angle 3 and \angle 4 form a linear pair, so \angle 1 \cong \angle 3 by the Congruent Supplements Theorem.” What is missing? Give measures that show the conclusion can fail without it.
46. \angle 1 \cong \angle 3; \angle 1 and \angle 2 are supplementary; \angle 3 and \angle 4 are supplementary. Must \angle 2 \cong \angle 4? Explain.
47. Apart from the lines for the link, the proofs of the Congruent Supplements and Congruent Complements Theorems differ only in 180^\circ and 90^\circ. Explain why the argument works for any fixed total, and name the step that finishes it.
48. Explain why the Congruent Complements Theorem can never show that two obtuse angles are congruent.
49. Explain why Proof 1 of the Vertical Angles Theorem would prove nothing if the proof of the Congruent Supplements Theorem had cited the Vertical Angles Theorem.
50. Show by an example that “if two angles are congruent and share a vertex, they are vertical angles” is false.
51. Euclid had no degree measures. Rewrite Proof 2 of the Vertical Angles Theorem with “two right angles” in place of 180^\circ. Which line then needs the fact that all right angles are equal, and why?
52. Describe the half-turn proof of the Vertical Angles Theorem in a few sentences, and say what it assumes.