## Key Concepts

- Understand two reflections
- Understand reflections in parallel lines theorem
- Understand reflections in intersecting lines theorem
- Solve reflections in parallel lines
- Solve reflections in intersecting lines

### Two reflections

Compositions of two reflections result in either a translation or a rotation.

Let us understand more about the two reflections with the help of the following theorems.

- Reflections in parallel lines theorem
- Reflections in intersecting lines theorem

### Reflections in parallel lines theorem

#### Theorem 9.5 Reflections in Parallel Lines Theorem

If lines k and m are parallel, then a reflection in line k followed by a reflection in line m is the same as a translation.

**If P” is the image of P, then**:

- PP” -is perpendicular to k and m, and
- PP”= 2d, where d is the distance between k and m.

### Reflections in intersecting lines theorem

#### Theorem 9.6 Reflections in Intersecting Lines Theorem

If lines k and m intersect at point P, then a reflection in k

Followed by a reflection in m is the same as a rotation about point p.

The angle of rotation is 2x^{o}, where x^{o} is the measure of the acute or right angle formed by k and m.

**Note:**

- A reflection followed by a reflection in
**parallel lines**results in**translation.** - A reflection followed by a reflection in
**intersecting lines**results in**rotation.**

**Examples:**

1. In the diagram, a reflection in line *k* maps GH−to G′H’−. A reflection in line *m* maps G′H’− to G”H’−’. Also, HB = 9 and DH” = 4

Name any segments congruent to each segment: HG HB , and GA.

Does AC = BD? Explain.

What is the length of GG” ?

**Solution:**

- HG- ≅≅ H’G’, and HG ≅≅ H”G”. HB ≅≅ H′B. GA≅≅ G’A
- Yes, C = BD because GG”and HH” are perpendicular to both k and m, so BD and AC opposite sides of a rectangle.
- By the properties of reflections, H’B = 9 and H’D = 4. Theorem 9.5 implies that GG” = HH” = 2. BD, so the length of GG” is 2(9 + 4), or 26 units.

2. In the diagram, the figure is reflected in line *k*. The image is then reflected in line *m*. Describe a single transformation that maps F to F”.

**Solution:**

The measure of the acute angle formed between lines* k *and *m* is 70 °.

So, by Theorem 9.6, a single transformation that maps F to F” is a 140° rotation about point P.

You can check that this is correct by tracing lines *k* and *m* and point F, then rotating the point 140°

## Exercise

- In the diagram, the preimage is reflected in line
*k*, then in line*m*. Describe a single transformation that maps the blue figure onto the green figure.

- Find the angle of rotation that maps A onto A”.

- Find the angle of rotation that maps A onto A”.

- Find the angle of rotation which maps A onto A”. Given x = 66 .

- Find the angle between A and A”.

- Describe the type of composition of transformation.

- Find the distance between the parallel lines if the distance ABC between A”B”C” is 1200 mm.

- Find the distance between ABC to A”B”C” , the distance between the parallel lines
*k*and*m*is 500 m.

- The vertices of PQR are P(1, 4), Q(3, -2), and R(7, 1). Use matrix operations to find the image matrix that represents the composition of the given transformations. Then graph PQR and its image.

Translation: (x, y) → (x, y + 5)

Reflection: In the y-axis

- The vertices of PQR are P(1, 4), Q(3, -2), and R(7, 1). Use matrix operations to find the image matrix that represents the composition of the given transformations. Then graph PQR and its image.

Reflection: In the x-axis

Translation: (x, y) → (x-9, y – 4)

### Summary

- Compositions of two reflections result in either a translation or a rotation.
- A reflection followed by a reflection in parallel lines results in translation.
- A reflection followed by a reflection in intersecting lines results in rotation.

### Concept Map

### What we have learned

- Understand two reflections
- Understand reflections in parallel lines theorem
- Understand reflections in intersecting lines theorem
- Solve reflections in parallel lines
- Solve reflections in intersecting lines

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