Predict the axes of a non-square rectangle and a square before folding. Test the diagonals as well as lines through side midpoints.
Organise the key idea
A figure has reflection symmetry when folding along a full line makes the whole figure coincide. Congruence alone does not specify a reflection relationship between two figures.
Use the lesson question here for paper discussion; the original activity opens separately.
Worked example1
A rectangle has unequal adjacent side lengths. Find all of its symmetry axes and explain why its diagonals are excluded.
Geometry diagram; not drawn to scale except for construction tasks.
Imagine folding along a candidate line and check every vertex, not just the outline near the line.
A symmetry axis must map the entire figure onto itself. Count different full lines only once.
The two lines through the centre joining opposite side midpoints. The diagonals fail for a non-square rectangle.
Check candidate lines by whether corresponding vertices and edges coincide after folding.
The two lines through the centre joining opposite side midpoints. The diagonals fail for a non-square rectangle.
The complete count is 2.
Reference answer
2 axes. The two lines through the centre joining opposite side midpoints. The diagonals fail for a non-square rectangle.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Guided attempt2
Draw every symmetry axis of the square. Use the scaffold to ensure that no type of line is missing.
First draw the two lines through opposite side midpoints.
Then test lines through opposite vertices.
Extend each axis beyond the square and check the total.
Geometry diagram; not drawn to scale except for construction tasks.
Imagine folding along a candidate line and check every vertex, not just the outline near the line.
A symmetry axis must map the entire figure onto itself. Count different full lines only once.
Two lines through opposite side midpoints and the two diagonals.
The two midpoint lines exchange opposite pairs of vertices.
Each diagonal fixes two vertices and exchanges the other two.
All four pass the whole-figure fold test; count each line once.
Reference answer
Four full lines: two joining opposite side midpoints and the two diagonals.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Prerequisite3
Which test defines reflection symmetry for a single figure?
The figure looks balanced from a distance.
Any two parts have equal areas.
Folding along a line makes the whole figure coincide.
The figure contains two equal sides somewhere.
A definition must work for the complete figure.
Check whether vertices and edges coincide, not just areas.
Choose the statement that explicitly uses folding and complete coincidence.
Reflection must match all corresponding points and edges.
Equal area or a visual impression alone is insufficient.
Reference answer
The whole-figure fold test.
Check your work
Identify the fold-and-coincide definition.
Prerequisite4
A pupil draws a short segment inside a symmetric figure and calls it the whole symmetry axis. Which correction is appropriate?
An axis must be a curved line.
Every internal segment is an axis.
An axis cannot pass through the figure.
Extend it in both directions: the axis is a full line.
Distinguish a segment from a line.
A line has no endpoints.
Extend the candidate in both directions.
The line determines a reflection in the whole plane.
Draw it beyond the figure to make the intended full axis clear.
Reference answer
A symmetry axis is a line extending in both directions.
Check your work
Use the term line and show both extensions.
Independent practice5
AB=AC, but △ABC is not equilateral. How many symmetry axes does it have?
Geometry diagram; not drawn to scale except for construction tasks.
Imagine folding along a candidate line and check every vertex, not just the outline near the line.
A symmetry axis must map the entire figure onto itself. Count different full lines only once.
The line through the apex and the base midpoint; the triangle is not equilateral.
Check candidate lines by whether corresponding vertices and edges coincide after folding.
The line through the apex and the base midpoint; the triangle is not equilateral.
The complete count is 1.
Reference answer
1 axes. The line through the apex and the base midpoint; the triangle is not equilateral.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Independent practice6
All three side lengths of a triangle differ. How many symmetry axes does it have?
Geometry diagram; not drawn to scale except for construction tasks.
Imagine folding along a candidate line and check every vertex, not just the outline near the line.
A symmetry axis must map the entire figure onto itself. Count different full lines only once.
No line can fold a scalene triangle onto itself: no pair of its sides is equal.
Check candidate lines by whether corresponding vertices and edges coincide after folding.
No line can fold a scalene triangle onto itself: no pair of its sides is equal.
The complete count is 0.
Reference answer
0 axes. No line can fold a scalene triangle onto itself: no pair of its sides is equal.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Independent practice7
Find the number of symmetry axes of an equilateral triangle and state where each passes.
Geometry diagram; not drawn to scale except for construction tasks.
Imagine folding along a candidate line and check every vertex, not just the outline near the line.
A symmetry axis must map the entire figure onto itself. Count different full lines only once.
Each line through a vertex and the opposite side midpoint is an axis.
Check candidate lines by whether corresponding vertices and edges coincide after folding.
Each line through a vertex and the opposite side midpoint is an axis.
The complete count is 3.
Reference answer
3 axes. Each line through a vertex and the opposite side midpoint is an axis.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Independent practice8
For a non-square rectangle, a pupil counts both diagonals as axes. Which reason shows the error?
Diagonals never occur in any symmetry axis.
A rectangle has no symmetry axes.
Every diagonal has to be horizontal.
A diagonal fold would match unequal adjacent sides, so the whole figure does not coincide.
Geometry diagram; not drawn to scale except for construction tasks.
Imagine folding along a candidate line and check every vertex, not just the outline near the line.
A symmetry axis must map the entire figure onto itself. Count different full lines only once.
The two lines through the centre joining opposite side midpoints. The diagonals fail for a non-square rectangle.
A diagonal fixes its endpoints and would exchange the other vertices.
For unequal adjacent sides, those vertices and edges do not coincide.
Retain only the two midpoint axes.
Reference answer
The diagonal fold fails because adjacent sides are unequal; the two midpoint lines are the axes.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Exit question9
Without looking at the worked example, count all symmetry axes of a square.
Geometry diagram; not drawn to scale except for construction tasks.
Imagine folding along a candidate line and check every vertex, not just the outline near the line.
A symmetry axis must map the entire figure onto itself. Count different full lines only once.
Two lines through opposite side midpoints and the two diagonals.
Check candidate lines by whether corresponding vertices and edges coincide after folding.
Two lines through opposite side midpoints and the two diagonals.
The complete count is 4.
Reference answer
4 axes. Two lines through opposite side midpoints and the two diagonals.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Exit question10
Two figures are congruent. Does this fact alone prove that they are reflections of one another in a single line? Explain what still must be checked.
Separate “same shape and size” from “mirror positions”.
One candidate line must serve all corresponding points.
A single matched pair cannot by itself establish the whole relationship.
Congruence specifies equal shape and size.
The two figures may instead be related by a translation or rotation without the required reflection relationship.
Check that one line works for every corresponding pair.
Reference answer
No. One common reflection line must exchange all corresponding points; equal size and shape alone do not supply it.
Check your work
State that congruence alone is insufficient and identify the common-axis requirement.
Further practice11
Which description gives all symmetry axes of a circle?
Only one chosen diameter.
Every tangent line.
Every line through its centre; there are infinitely many.
Only the horizontal and vertical diameters.
Geometry diagram; not drawn to scale except for construction tasks.
Try a slanted diameter as well as horizontal and vertical ones.
There is no special preferred direction through the centre.
All lines through the centre work.
Reflection in a line through the centre preserves the distance to the centre.
Thus every circle point maps to another circle point for any such line.
Reference answer
Every line through the centre is an axis, so there are infinitely many.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
State infinitely many axes and require the centre to lie on each.
Further practice12
Draw the complete symmetry axes of a non-square rectangle, using a dashed line for each.
Geometry diagram; not drawn to scale except for construction tasks.
Imagine folding along a candidate line and check every vertex, not just the outline near the line.
A symmetry axis must map the entire figure onto itself. Count different full lines only once.
The two lines through the centre joining opposite side midpoints. The diagonals fail for a non-square rectangle.
Mark both pairs of opposite-side midpoints.
Join the matching midpoint pairs and extend the lines.
Check that each fold exchanges corresponding corners and edges.
Reference answer
Two axes: the lines through the midpoints of opposite sides.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Further practice13
AB=AC and the triangle is not equilateral. Draw its symmetry axis as a full line and mark the base midpoint.
Geometry diagram; not drawn to scale except for construction tasks.
Imagine folding along a candidate line and check every vertex, not just the outline near the line.
A symmetry axis must map the entire figure onto itself. Count different full lines only once.
The line through the apex and the base midpoint; the triangle is not equilateral.
Mark the midpoint of BC.
Draw the line through A and that midpoint.
It fixes A and exchanges B with C, giving the only axis.
Reference answer
The line joining A to the midpoint of BC, extended in both directions.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Further practice14
Draw every symmetry axis of an equilateral triangle. Mark each opposite-side midpoint to organise your work.
Geometry diagram; not drawn to scale except for construction tasks.
Imagine folding along a candidate line and check every vertex, not just the outline near the line.
A symmetry axis must map the entire figure onto itself. Count different full lines only once.
Each line through a vertex and the opposite side midpoint is an axis.
Construct or identify the midpoint of each side.
Join each midpoint to the opposite vertex and extend the line.
The three different lines all pass the fold test.
Reference answer
Three full lines, each through one vertex and the opposite side midpoint.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Extension15
A regular pentagon has five equal sides and five equal angles. Find all its symmetry axes.
Geometry diagram; not drawn to scale except for construction tasks.
Imagine folding along a candidate line and check every vertex, not just the outline near the line.
A symmetry axis must map the entire figure onto itself. Count different full lines only once.
Each axis joins a vertex to the opposite side midpoint; the five vertices give five axes.
Check candidate lines by whether corresponding vertices and edges coincide after folding.
Each axis joins a vertex to the opposite side midpoint; the five vertices give five axes.
The complete count is 5.
Reference answer
5 axes. Each axis joins a vertex to the opposite side midpoint; the five vertices give five axes.
Geometry diagram; not drawn to scale except for construction tasks.
Check your work
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Extension16
The shown scalene triangle and its mirror image are exchanged by the given vertical line. Does either triangle alone need its own symmetry axis? Explain the distinction.
Geometry diagram; not drawn to scale except for construction tasks.
Ask whether the reflection must stay within one triangle or exchange two triangles.
The given line is not an axis of either separate scalene triangle.
It is the common line relating the two different figures.
A figure’s own symmetry maps that figure onto itself.
A pairwise reflection instead maps the first figure to the other.
These triangles are scalene, yet all three corresponding pairs match under the given line.
Reference answer
No. Each scalene triangle has no axis of its own, but the given line exchanges the two whole triangles.
Check your work
Distinguish self-symmetry from two figures being mirror images.
Preparation and assessment notes
Prerequisites
Names and defining properties of rectangles, squares and triangles; matching vertices and edges. A rectangle here is not a square unless stated.
Look for in students’ work
Project one fold test and require a reason for rejected lines. Keep a figure’s own axes separate from a given axis exchanging two figures. A symmetry axis is a line, not a short segment.
Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.
All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.
Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.
Textbook reference and authorship
Original practice aligned to the referenced skills. Use your own textbook for the cited pages.
大湾区初中数学三维课堂（八年级上册）正文.pdf
Year 8, upper volume · text pp. 29–30 · foundation lesson 15
Original questions and explanations; the source textbook and answer PDFs are not reproduced here.
Recognising reflection symmetry and counting axes
Year 8, upper volume · text pp. 29–30 · foundation lesson 15 · 1/2
Name: __________ Class: ______ Date: __________
Worked example1
A rectangle has unequal adjacent side lengths. Find all of its symmetry axes and explain why its diagonals are excluded.
Geometry diagram; not drawn to scale except for construction tasks.
Guided attempt2
Draw every symmetry axis of the square. Use the scaffold to ensure that no type of line is missing.
First draw the two lines through opposite side midpoints.
Then test lines through opposite vertices.
Extend each axis beyond the square and check the total.
Geometry diagram; not drawn to scale except for construction tasks.
Prerequisite3
Which test defines reflection symmetry for a single figure?
The figure looks balanced from a distance.
Any two parts have equal areas.
Folding along a line makes the whole figure coincide.
The figure contains two equal sides somewhere.
Prerequisite4
A pupil draws a short segment inside a symmetric figure and calls it the whole symmetry axis. Which correction is appropriate?
An axis must be a curved line.
Every internal segment is an axis.
An axis cannot pass through the figure.
Extend it in both directions: the axis is a full line.
Recognising reflection symmetry and counting axes
Year 8, upper volume · text pp. 29–30 · foundation lesson 15 · 2/2
Name: __________ Class: ______ Date: __________
Independent practice5
AB=AC, but △ABC is not equilateral. How many symmetry axes does it have?
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice6
All three side lengths of a triangle differ. How many symmetry axes does it have?
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice7
Find the number of symmetry axes of an equilateral triangle and state where each passes.
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice8
For a non-square rectangle, a pupil counts both diagonals as axes. Which reason shows the error?
Diagonals never occur in any symmetry axis.
A rectangle has no symmetry axes.
Every diagonal has to be horizontal.
A diagonal fold would match unequal adjacent sides, so the whole figure does not coincide.
Geometry diagram; not drawn to scale except for construction tasks.
Exit question9
Without looking at the worked example, count all symmetry axes of a square.
Geometry diagram; not drawn to scale except for construction tasks.
Exit question10
Two figures are congruent. Does this fact alone prove that they are reflections of one another in a single line? Explain what still must be checked.
Recognising reflection symmetry and counting axes — Further practice
Year 8, upper volume · text pp. 29–30 · foundation lesson 15
Further practice11
Which description gives all symmetry axes of a circle?
Only one chosen diameter.
Every tangent line.
Every line through its centre; there are infinitely many.
Only the horizontal and vertical diameters.
Geometry diagram; not drawn to scale except for construction tasks.
Further practice12
Draw the complete symmetry axes of a non-square rectangle, using a dashed line for each.
Geometry diagram; not drawn to scale except for construction tasks.
Further practice13
AB=AC and the triangle is not equilateral. Draw its symmetry axis as a full line and mark the base midpoint.
Geometry diagram; not drawn to scale except for construction tasks.
Further practice14
Draw every symmetry axis of an equilateral triangle. Mark each opposite-side midpoint to organise your work.
Geometry diagram; not drawn to scale except for construction tasks.
Extension15
A regular pentagon has five equal sides and five equal angles. Find all its symmetry axes.
Geometry diagram; not drawn to scale except for construction tasks.
Extension16
The shown scalene triangle and its mirror image are exchanged by the given vertical line. Does either triangle alone need its own symmetry axis? Explain the distinction.
Geometry diagram; not drawn to scale except for construction tasks.
Recognising reflection symmetry and counting axes — Teacher answers
Year 8, upper volume · text pp. 29–30 · foundation lesson 15
Project one fold test and require a reason for rejected lines. Keep a figure’s own axes separate from a given axis exchanging two figures. A symmetry axis is a line, not a short segment.
Worked example1
A rectangle has unequal adjacent side lengths. Find all of its symmetry axes and explain why its diagonals are excluded.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
2 axes. The two lines through the centre joining opposite side midpoints. The diagonals fail for a non-square rectangle.
Check candidate lines by whether corresponding vertices and edges coincide after folding.
The two lines through the centre joining opposite side midpoints. The diagonals fail for a non-square rectangle.
The complete count is 2.
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Geometry diagram; not drawn to scale except for construction tasks.
Guided attempt2
Draw every symmetry axis of the square. Use the scaffold to ensure that no type of line is missing.
First draw the two lines through opposite side midpoints.
Then test lines through opposite vertices.
Extend each axis beyond the square and check the total.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
Four full lines: two joining opposite side midpoints and the two diagonals.
The two midpoint lines exchange opposite pairs of vertices.
Each diagonal fixes two vertices and exchanges the other two.
All four pass the whole-figure fold test; count each line once.
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Geometry diagram; not drawn to scale except for construction tasks.
Prerequisite3
Which test defines reflection symmetry for a single figure?
The figure looks balanced from a distance.
Any two parts have equal areas.
Folding along a line makes the whole figure coincide.
The figure contains two equal sides somewhere.
Reference answer
The whole-figure fold test.
Reflection must match all corresponding points and edges.
Equal area or a visual impression alone is insufficient.
Identify the fold-and-coincide definition.
Prerequisite4
A pupil draws a short segment inside a symmetric figure and calls it the whole symmetry axis. Which correction is appropriate?
An axis must be a curved line.
Every internal segment is an axis.
An axis cannot pass through the figure.
Extend it in both directions: the axis is a full line.
Reference answer
A symmetry axis is a line extending in both directions.
The line determines a reflection in the whole plane.
Draw it beyond the figure to make the intended full axis clear.
Use the term line and show both extensions.
Independent practice5
AB=AC, but △ABC is not equilateral. How many symmetry axes does it have?
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
1 axes. The line through the apex and the base midpoint; the triangle is not equilateral.
Check candidate lines by whether corresponding vertices and edges coincide after folding.
The line through the apex and the base midpoint; the triangle is not equilateral.
The complete count is 1.
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice6
All three side lengths of a triangle differ. How many symmetry axes does it have?
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
0 axes. No line can fold a scalene triangle onto itself: no pair of its sides is equal.
Check candidate lines by whether corresponding vertices and edges coincide after folding.
No line can fold a scalene triangle onto itself: no pair of its sides is equal.
The complete count is 0.
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice7
Find the number of symmetry axes of an equilateral triangle and state where each passes.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
3 axes. Each line through a vertex and the opposite side midpoint is an axis.
Check candidate lines by whether corresponding vertices and edges coincide after folding.
Each line through a vertex and the opposite side midpoint is an axis.
The complete count is 3.
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Geometry diagram; not drawn to scale except for construction tasks.
Independent practice8
For a non-square rectangle, a pupil counts both diagonals as axes. Which reason shows the error?
Diagonals never occur in any symmetry axis.
A rectangle has no symmetry axes.
Every diagonal has to be horizontal.
A diagonal fold would match unequal adjacent sides, so the whole figure does not coincide.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The diagonal fold fails because adjacent sides are unequal; the two midpoint lines are the axes.
A diagonal fixes its endpoints and would exchange the other vertices.
For unequal adjacent sides, those vertices and edges do not coincide.
Retain only the two midpoint axes.
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Geometry diagram; not drawn to scale except for construction tasks.
Exit question9
Without looking at the worked example, count all symmetry axes of a square.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
4 axes. Two lines through opposite side midpoints and the two diagonals.
Check candidate lines by whether corresponding vertices and edges coincide after folding.
Two lines through opposite side midpoints and the two diagonals.
The complete count is 4.
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Geometry diagram; not drawn to scale except for construction tasks.
Exit question10
Two figures are congruent. Does this fact alone prove that they are reflections of one another in a single line? Explain what still must be checked.
Reference answer
No. One common reflection line must exchange all corresponding points; equal size and shape alone do not supply it.
Congruence specifies equal shape and size.
The two figures may instead be related by a translation or rotation without the required reflection relationship.
Check that one line works for every corresponding pair.
State that congruence alone is insufficient and identify the common-axis requirement.
Further practice11
Which description gives all symmetry axes of a circle?
Only one chosen diameter.
Every tangent line.
Every line through its centre; there are infinitely many.
Only the horizontal and vertical diameters.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
Every line through the centre is an axis, so there are infinitely many.
Reflection in a line through the centre preserves the distance to the centre.
Thus every circle point maps to another circle point for any such line.
State infinitely many axes and require the centre to lie on each.
Geometry diagram; not drawn to scale except for construction tasks.
Further practice12
Draw the complete symmetry axes of a non-square rectangle, using a dashed line for each.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
Two axes: the lines through the midpoints of opposite sides.
Mark both pairs of opposite-side midpoints.
Join the matching midpoint pairs and extend the lines.
Check that each fold exchanges corresponding corners and edges.
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Geometry diagram; not drawn to scale except for construction tasks.
Further practice13
AB=AC and the triangle is not equilateral. Draw its symmetry axis as a full line and mark the base midpoint.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
The line joining A to the midpoint of BC, extended in both directions.
Mark the midpoint of BC.
Draw the line through A and that midpoint.
It fixes A and exchanges B with C, giving the only axis.
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Geometry diagram; not drawn to scale except for construction tasks.
Further practice14
Draw every symmetry axis of an equilateral triangle. Mark each opposite-side midpoint to organise your work.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
Three full lines, each through one vertex and the opposite side midpoint.
Construct or identify the midpoint of each side.
Join each midpoint to the opposite vertex and extend the line.
The three different lines all pass the fold test.
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Geometry diagram; not drawn to scale except for construction tasks.
Extension15
A regular pentagon has five equal sides and five equal angles. Find all its symmetry axes.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
5 axes. Each axis joins a vertex to the opposite side midpoint; the five vertices give five axes.
Check candidate lines by whether corresponding vertices and edges coincide after folding.
Each axis joins a vertex to the opposite side midpoint; the five vertices give five axes.
The complete count is 5.
List or draw all valid full lines without counting one line twice.
Explain the fold test and exclude invalid candidates.
Geometry diagram; not drawn to scale except for construction tasks.
Extension16
The shown scalene triangle and its mirror image are exchanged by the given vertical line. Does either triangle alone need its own symmetry axis? Explain the distinction.
Geometry diagram; not drawn to scale except for construction tasks.
Reference answer
No. Each scalene triangle has no axis of its own, but the given line exchanges the two whole triangles.
A figure’s own symmetry maps that figure onto itself.
A pairwise reflection instead maps the first figure to the other.
These triangles are scalene, yet all three corresponding pairs match under the given line.
Distinguish self-symmetry from two figures being mirror images.
