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5.5 — Claims and Conjectures
In Section 5.4 you investigated how rigid motions preserve geometric relationships. Section 5.5 uses a tessellation—a repeating arrangement of figures—to recognize angle patterns and make conjectures. We will state the relationships clearly, explain why they make sense, and practice identifying them. Section 5.6 will develop more formal proofs.
1. In class: A Solidify Understanding Task
What is a mathematical conjecture?
A conjecture is a statement you think is true based on patterns and known properties. Looking at several examples can suggest a conjecture, but appearance alone does not prove it for every possible figure.
The packet's tessellation is built by repeatedly rotating congruent triangles 180° around side midpoints. These rigid motions preserve lengths and angles, producing repeated straight lines and angle relationships. Instead of copying the busy tessellation, our small diagrams isolate the exact relationships you are supposed to notice.
Observation: Several pairs of opposite angles at line intersections look equal.
Conjecture: When two lines intersect, each pair of vertical angles has equal measure.
Justification to prepare: Each angle in the pair forms a linear pair with the same neighboring angle. Because both sums are 180°, the vertical angles must be equal.
Vertical angles
Vertical angles are the nonadjacent, opposite angles made when two lines intersect. They are congruent, meaning their measures are equal. Adjacent angles at the same intersection form linear pairs and add to 180°.
If angle 1 measures 64°, its vertical angle 3 also measures 64°. An adjacent angle 2 forms a straight angle with 1, so it is 180° − 64° = 116°. Angle 4 is vertical to angle 2 and also measures 116°.
\[m\angle1=m\angle3=64^\circ,\qquad m\angle2=m\angle4=116^\circ.\]Why it works: if angles 1 and 2 total 180°, and angles 2 and 3 total 180°, subtracting the shared angle 2 gives angle 1 = angle 3.
Exterior angles of a triangle
Extend one side of a triangle past a vertex. The angle outside the triangle, between the extension and the other side, is an exterior angle. The two interior angles not touching that exterior angle are its remote interior angles.
Why? The triangle's three interior angles total 180°. The exterior angle and its adjacent interior angle also total 180°. Subtract the adjacent interior angle from both totals.
A triangle has remote interior angles of 43° and 68°. The exterior angle equals their sum:
\[43^\circ+68^\circ=\boxed{111^\circ}.\]The adjacent interior angle is 180° − 111° = 69°. Check: 43° + 68° + 69° = 180°.
An exterior angle is 128°, and one remote interior angle is 52°. The other remote interior angle must be
\[128^\circ-52^\circ=\boxed{76^\circ}.\]The adjacent interior angle is 52° because it is supplementary to 128°. Do not confuse this adjacent angle with a remote one.
Parallel lines cut by a transversal
A transversal is a line crossing two or more other lines at different points. When it crosses parallel lines, the eight angles occur in predictable pairs. Corresponding angles occupy matching positions at the two intersections; alternate interior angles lie between the parallel lines on opposite sides of the transversal; same-side interior angles lie between the parallels on the same side of the transversal.
| Angle relationship | Example from figure | Rule when p ∥ q |
|---|---|---|
| Corresponding | 1 and 5; 2 and 6; 3 and 7; 4 and 8 | Congruent (equal measure) |
| Alternate interior | 3 and 5; 4 and 6 | Congruent |
| Same-side interior | 3 and 6; 4 and 5 | Supplementary (sum 180°) |
| Vertical (at either intersection) | 1 and 3; 5 and 7 | Congruent regardless of parallelism |
Suppose angle 3 measures 115°. Alternate interior angle 5 is 115°. Same-side interior angle 6 is 180° − 115° = 65°. Corresponding angle 7 is 115°, and its vertical angle 5 agrees.
Always identify which relationship you are using before calculating.
A translation taking the upper parallel line to the lower parallel line preserves directions and angle measures. The angle made by the transversal and one parallel line therefore matches the angle in the corresponding position at the other crossing. This supports the conjecture; a full formal proof will be developed in the next lesson.
2. Ready — Quadrilaterals and coordinate-line graphs (1–7)
Study the worked examples first, then try the problems yourself. An answer key at the bottom of this section is provided to check your work.
The interior angles of a quadrilateral add to 360° (problems 1–2)
Draw one diagonal from a vertex of any simple quadrilateral to the opposite vertex, provided it lies inside the figure (for a concave quadrilateral, choose the interior diagonal). It divides the quadrilateral into two triangles. Because each triangle has 180° of interior angles, the quadrilateral has a total of 180° + 180° = 360°.
A quadrilateral has three interior angles of 82°, 107° and 94°. Let its fourth angle be x.
\[x+82^\circ+107^\circ+94^\circ=360^\circ\]\[x=360^\circ-283^\circ=\boxed{77^\circ}.\]In packet Ready 2, the four angles are labeled \(4x+23\), \(7x-3\), \(16x+15\), and \(6x-5\). Their sum is 360°, so \[(4x+23)+(7x-3)+(16x+15)+(6x-5)=360\] \[33x+30=360\implies\boxed{x=10}.\] The angle measures are 63°, 67°, 175° and 55°, and they total 360°.
Recognize lines from slope and intercept (problems 3–7)
In slope-intercept form \(y=mx+b\), \(m\) is the slope (rise divided by run) and \(b\) is the y-intercept. A positive slope rises as x increases; a negative slope falls. Two nonvertical lines with equal slopes are parallel. Changing only b shifts a line up or down.
Imagine a graph with two rising lines of slope \(\tfrac34\) and two falling lines of slope \(-\tfrac34\). The rising line passing through (0,4) is \(y=\tfrac34 x+4\), while the one through (0,2) is \(y=\tfrac34 x+2\). The falling lines with those intercepts are \(y=-\tfrac34x+4\) and \(y=-\tfrac34x+2\).
For each packet equation, read its printed fraction carefully: choose a line by slope first, then by y-intercept. Pairwise intersections of two slopes and two intercepts may form a parallelogram. Look for parallel opposite edges.
Check your work — Ready 1–7 answer key
- Draw an interior diagonal to divide the quadrilateral into two triangles. Their interior angles together are exactly the quadrilateral’s angles, so the total is \(180^\circ+180^\circ=360^\circ\).
- \(x=10\), since \(33x+30=360\).
- Line s.
- Line q.
- Line r.
- Line p.
- A rhombus (also a parallelogram), with vertices \((0,2),(\frac43,3),(0,4),(-\frac43,3)\). All four sides have length \(\frac53\); opposite sides are parallel, opposite angles are congruent, and consecutive angles are supplementary. The diagonals have lengths 2 and \(\frac83\), are perpendicular, and bisect each other at \((0,3)\); this rhombus is not a square.
3. Set — Recognize and calculate angle relationships (8–25)
Study the worked examples first, then try the problems yourself. An answer key at the bottom of this section is provided to check your work.
For the shared ideas used in these exercises, revisit the In class explanation.
Problems 8–16: search each drawing for three possible features: (a) an intersection of two straight lines producing vertical angles, (b) a triangle with one side extended creating an exterior angle, and (c) two explicitly parallel lines crossed by another line. A single complicated picture can contain all three features, but do not assume an unlabeled pair of lines is parallel just because they look parallel.
A picture shows two crossing lines. Mark the opposite pair of angles to identify vertical angles. If a triangle is also present and one side is extended, mark the outside angle at that vertex. To label a transversal angle relationship, identify two confirmed parallel lines and the line crossing both of them.
Problems 17–19: when an exterior angle and one remote interior angle are given, find the second remote interior angle by subtraction. If instead the adjacent interior angle is given, subtract it from 180° first to obtain the exterior angle.
The interior angle adjacent to an exterior angle is 57°, and one remote interior angle is 49°. The exterior angle is 180° − 57° = 123°. The remaining remote angle is 123° − 49° = \(\boxed{74^\circ}\).
Problems 20–25: given p ∥ q, decide whether each angle pair is congruent or supplementary. The appropriate relationship—not how the angle appears—is the justification. Start by identifying corresponding, alternate interior, same-side interior, or vertical positions; if necessary, combine relationships (such as a linear pair followed by corresponding angles).
The worksheet tells students to classify pairs “by trusting how they look,” even though the mathematical reason must use the given parallel lines and the relevant angle relationships. A more reliable instruction is: “Use the given fact that p and q are parallel and known angle relationships to decide whether each pair is congruent or supplementary.”
Check your work — Set 8–25 answer key
- Vertical angles at the scissors’ pivot: opposite angles between the two blade directions. Highlighting not shown.
- Parallel fence rails cut by the diagonal brace (or vertical pickets). Highlighting not shown.
- Vertical angles where two star segments cross, and exterior angles of the small triangles along continued star segments. No pair of parallel straight segments is established by this picture. Highlighting not shown.
- Parallel rails crossed by ties (transversals). Highlighting not shown.
- Vertical angles where the extended directions of two crossing sticks meet; small triangles can also have exterior angles along continuing sticks. Highlighting not shown.
- Vertical angles at the center between opposite straight radial lines. Highlighting not shown.
- Parallel crop rows cut by a field boundary or another row direction. Highlighting not shown.
- Exterior angles of the small triangles along continued diagonal edges; vertical angles at the center, and parallel top and bottom horizontal edges cut by a diagonal. Highlighting not shown.
- Vertical angles where the two ski directions intersect; the bicycle frame also contains triangles with exterior angles along continued frame members. Highlighting not shown.
- \(x=15\); the remote angles are \(40^\circ\) at G and \(80^\circ\) at H, since \((2x+10)+(6x-10)=120\).
- \(x=10\); the remote angles are \(38^\circ\) at G and \(34^\circ\) at J, since \((4x-2)+(3x+4)=72\).
- \(x=8\); the remote angles are \(50^\circ\) at B and \(61^\circ\) at C, since \((8x-14)+(7x+5)=111\).
- Congruent — vertical angles.
- Congruent — corresponding angles.
- Supplementary — angle 2 corresponds to 6, and 6 and 8 are a linear pair.
- Supplementary — same-side interior angles.
- Supplementary — same-side interior angles.
- Supplementary — a linear pair.
4. Go — Complementary and supplementary angles (26–31)
Study the worked examples first, then try the problems yourself. An answer key at the bottom of this section is provided to check your work.
Complementary angles have measures adding to 90°. Supplementary angles have measures adding to 180°. For an angle of size \(\theta\):
A nonnegative complement exists only for \(0^\circ\leq\theta\leq90^\circ\); a nonnegative supplement exists only for \(0^\circ\leq\theta\leq180^\circ\). At 90°, the complement is a zero-degree angle by the nonnegative-measure convention; some classrooms require positive angles and say “none.” Follow your class convention.
| Packet given angle | Complement | Supplement |
|---|---|---|
| 26. 37° | 53° | 143° |
| 27. 59° | 31° | 121° |
| 28. 89° | 1° | 91° |
| 29. 111° | No nonnegative complement | 69° |
| 30. 3° | 87° | 177° |
| 31. 90° | 0° (if allowed) | 90° |
Additional fully worked Go problems
Find the complement and supplement of 26°.
\[90^\circ-26^\circ=\boxed{64^\circ},\qquad180^\circ-26^\circ=\boxed{154^\circ}.\]Check: 26°+64°=90°, and 26°+154°=180°.
An angle is 38° larger than its complement. Find both angles and the supplement of the original angle.
Let the original angle be \(x\). Its complement is \(90-x\). Then
\[x=(90-x)+38\implies 2x=128\implies x=64^\circ.\]The complement is 26° and its supplement is \(180^\circ-64^\circ=\boxed{116^\circ}\).
Find the complement and supplement of a 138° angle.
Because 138° is greater than 90°, it has no nonnegative complement. Its supplement is \(180^\circ-138^\circ=\boxed{42^\circ}\). A negative result from subtracting 138° from 90° is not a physical angle measure for this task.
Check your work — Go 26–31 answer key
- Complement \(53^\circ\); supplement \(143^\circ\).
- Complement \(31^\circ\); supplement \(121^\circ\).
- Complement \(1^\circ\); supplement \(91^\circ\).
- No nonnegative complement; supplement \(69^\circ\).
- Complement \(87^\circ\); supplement \(177^\circ\).
- Complement \(0^\circ\) if zero angles are allowed (otherwise none); supplement \(90^\circ\).
5. Resources
For additional practice, see Khan Academy Geometry. All essential concepts are explained in this lesson without needing the external site.
6. Teacher's review (teachers and parents)
Set 20–25, “by trusting how they look.” The wording explicitly encourages visual guessing instead of deduction from the given parallelism. It risks reinforcing a non-rigorous approach to angle relationships. Revise it to: “Using p ∥ q, identify the relationships that show each angle pair is congruent or supplementary.”
Go 31, complement of 90°. The answer is convention-dependent if the word “angle” is intended to require positive measure. A zero-degree angle is mathematically meaningful in common geometric conventions; if zero angles are excluded, state that convention explicitly.
7. Coverage of the assigned packet
| 5.5 component | Instruction |
|---|---|
| Main task: vertical angles | In class with example and diagram |
| Main task: exterior angles | In class with theorem and diagram |
| Main task: transversal angle pairs | In class with labeled diagram and comparison table |
| Ready 1–2: quadrilateral angles | Section 2 |
| Ready 3–7: slope, intercept and quadrilateral formed by lines | Section 2 |
| Set 8–16: identify geometric features | Section 3 |
| Set 17–19: remote interior angles | Section 3 |
| Set 20–25: congruent/supplementary angle pairs | In class and Section 3 |
| Go 26–31: complements/supplements | Section 4 plus three original worked problems |
Independent instructional companion to Mathematics Vision Project, Secondary Mathematics II, Module 5: Geometric Figures (2017), Section 5.5. Diagrams are mathematically constrained SVG. Internet access is needed for MathJax equation formatting.
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