Constructions and Tilings · Lesson 4 of 12
Repeating Units and Repeating Angles
“Copy an angle exactly, then use equal corresponding angles to construct parallel lines and repeated patterns.”
• Identify the equal lengths and angles in a repeating unit. • Copy a given angle using arcs and a transferred chord. • Explain angle copying through SSS congruence. • Construct a line parallel to a given line through a chosen point. • Use parallel and repeated directions in a design.
Repeating Units and Repeating Angles
A repeating pattern is built from units that match, even when they point in different directions. The unit in the source design has two arms with a fixed angle between them and an arc joining the ends. Equal arm lengths alone do not keep the unit unchanged: a wider angle would produce a wider shape.
A compass can transfer a length by carrying the same opening to a new position. To copy the whole unit, we must also transfer its angle. Measuring with a protractor could approximate the opening, but a ruler-and-compass construction can copy it using equal distances.
The key is to choose two points on the original angle at the same distance from its vertex. They form an isosceles triangle, which is a triangle with two equal sides. If we reproduce all three sides of that triangle at a new vertex, SSS congruence reproduces the angle too.
Steps of Construction to Copy an Angle
Let A be the original vertex. We want the same opening at a new point X, where one starting ray is already drawn. A copy may face a different direction; the construction must preserve the opening rather than the position on the page.
- From A, draw an arc meeting the original rays at B and C. Keep its radius available.
- Draw an arc of the same radius from X, meeting the new starting ray at Z.
- Set the compass opening to BC, the straight distance between the two original arc intersections.
- With centre Z, transfer this opening onto the new arc to locate Y on the intended side of the starting ray.
- Draw ray XY. The angle YXZ is a copy of the angle BAC.
Why the Copied Angle Is Equal
Every step supplies one equality needed for congruence. Keeping track of these equalities also helps you find an error if the copied angle looks different from the original. Look for a changed compass opening or the wrong transferred distance.
AB = XY and AC = XZ because all four lengths use the original arc radius. BC = YZ because the compass transfers that chord length. Thus triangles ABC and XYZ are congruent by SSS, with A matching X, B matching Y, and C matching Z. Corresponding angles give ∠BAC = ∠YXZ.
The chord is the straight segment joining two points on a circle. It is not the curved part of the circumference. Trying to transfer the curved arc length would not reproduce the third side of the triangle. For an ordinary angle less than 180°, the suitable intersection on the chosen side provides the required copy.
Problem
A narrow angle is drawn without a stated numerical measure. Copy it at X.
- 1.Choose any convenient radius at the original vertex and mark its two ray intersections.
- 2.Use the same radius at X to mark Z on the new ray.
- 3.Measure the original chord with the compass and transfer it from Z to the new arc, giving Y.
- 4.Draw XY. SSS establishes equal openings even though the original angle was never measured.
Problem
Copy a given angle wider than 90° at a different orientation.
- 1.Draw the same-radius arcs at the old and new vertices.
- 2.Transfer the chord between the original arc intersections. This chord is longer than that of a smaller angle with the same radius.
- 3.Choose the new crossing that places the angle on the intended side of the starting ray and draw the second ray.
- 4.The three side equalities still give SSS. The method does not require the angle to be acute.
Copy the straight chord distance between the two marked points, not a length measured along the arc. Also keep the first arc radius unchanged between the source and destination; otherwise the SSS equalities are lost.
Construction of a Line Parallel to the Given Line
Angle copying has an important application beyond designs. When a transversal crosses two lines, equal corresponding angles establish that those lines are parallel. A transversal is simply a line that crosses both lines.
Let m be the given line and B a point outside it. Draw a transversal l through B that crosses m at A. At B, copy the angle that m makes with l at A, placing it in the corresponding position. The new ray through B supplies a line n. Because the corresponding angles are equal, m and n are parallel.
Corresponding position matters. In the illustration, each highlighted angle is on the same side of the transversal and in the same relative position at its crossing. Copying an angle into an arbitrary quadrant may produce another line, so name the starting ray and intended side before transferring the chord.
Problem
Construct a line through B parallel to m using only a straightedge and compass.
- 1.Choose A on m and draw AB as the transversal, extending it where needed.
- 2.Draw equal-radius arcs centred at A and B to locate matching distances on the relevant rays of the transversal.
- 3.Transfer the chord of the angle at A to the corresponding side at B, then draw the new ray.
- 4.Extend that ray to line n. Equal corresponding angles prove m ∥ n; B belongs to n by construction.
Designs Using Parallel Lines
Once an angle can be copied, a unit can be repeated without slowly changing its shape. Use the compass for the matching lengths and the copied angle for the opening. Both conditions are needed for an exact copy.
Recreate the repeating-unit design by building one unit first, then copying it in the second orientation and alternating the two. For the star-like design below, draw the support directions and use parallel edges to reproduce matching arms. Check corresponding lengths and angles before darkening the final outline. Try four pairs of parallel lines in different orientations; none has to be horizontal.
Check Your Understanding
Use these questions to check the conditions behind each method, as well as the result. For a diagram-based question, follow the labels and given information rather than judging by appearance.
Quiz
Which distance is transferred when copying an angle?
Which congruence condition justifies the copy?
What must match in two copies of the repeating unit?
What proves that the newly constructed line is parallel to the given line?
Can the angle-copying method be used for an obtuse angle?
Practice Problems
- Draw four angles in different orientations and copy each without measuring its angle.
- In triangles ABC and XYZ, state the three equalities needed for the copying proof.
- Explain why copying an arc length is not the same as transferring BC.
- Construct a parallel to a sloping line through an outside point. Label the corresponding angles.
- Recreate the repeating-unit pattern shown in the lesson.
- Recreate the star-like design using support directions and parallel edges. Explain one pair of parallel edges.
For each, use equal-radius arcs at the two vertices, transfer the chord, and draw the new second ray on the chosen side.
Key Takeaways
• An exact repeated unit preserves both lengths and angles. • Equal-radius arcs and a transferred chord reproduce three triangle sides. • SSS explains why the copied angle matches the original. • A chord is straight; it is different from a curved arc. • Equal corresponding angles can be used to construct parallel lines. • The copied angle must be placed in the intended corresponding position.