day-52-presentation-identifying congruent angles of a triangle · corresponding angles in congruent...

DAY 52 – IDENTIFYING

CONGRUENT ANGLES OF A

TRIANGLE

INTRODUCTION

The fundamental property of rigid transformations

of the plane is that they preserve both the size and

shape of a given plane figure. If two plane figures

are congruent, they will have corresponding parts,

and these parts will be congruent. In our previous

lessons on rigid motion, we should have discovered

that when two figures are congruent, there is

always a rigid motion that maps one figure to the

other.

In this lesson, we are going to learn how to

identify congruent angles in triangles based on

rigid motion.

VOCABULARY

1. Congruent Angles

Angles that have the same measure.

2. Orientation

The arrangement of vertices of a plane figure in

relation to one another after a transformation.

3. Corresponding angles in congruent

triangles

Angles in the same position relative to the angles

in the other congruent triangle.

Our main aim is to identify congruent angles

between a triangle and its image after undergoing

all the rigid motions, based on the fact that

corresponding angles in congruent triangles are

equal.

When identifying congruent angles in congruent

triangles, we match angles of one triangle to the

angles of the other triangle, that is, we identify

pairs of corresponding angles.

IDENTIFYING CONGRUENT ANGLES IN

TRIANGLES AFTER A TRANSLATION

In the figure below ΔABC has been mapped onto

ΔPQR after a translation.

A

B

C P

Q

R

Since a translation is a rigid motion, it means that

ΔABC ≅ ΔPQR. The corresponding angles will be

congruent.

Orientation is preserved in a translation. Therefore,

it is easy to identify corresponding parts.

Let us first identify the mapping of corresponding

vertices from the triangles as indicated below.

𝐀 → 𝐏𝐁 → 𝐐𝐂 → 𝐑

Then the mapping of corresponding sides becomes:

AB → PQ; BC → QR; AC → PR

We can now easily identify how the corresponding

angles are mapped as shown below.

∠𝐀 → ∠𝐏∠𝐁 → ∠𝐐∠𝐂 → ∠𝐑

This corresponding angles are congruent. At this

stage we can easily identify pairs of congruent

angles. These pairs are:

∠𝐀 ≅ ∠𝐏∠𝐁 ≅ ∠𝐐∠𝐂 ≅ ∠𝐑

IDENTIFYING CONGRUENT ANGLES IN

TRIANGLES AFTER A REFLECTION

In the figure below ΔABC has been mapped onto

ΔPQR after a reflection along the mirror line 𝑙.

A

B C

P

Q R

𝑙

A reflection is a rigid motion. Consequently, it is

clear that ΔABC ≅ ΔPQR . The corresponding angles

will therefore be congruent.

Orientation is not preserved in a reflection.

Therefore, we should note that the vertices are

reversed in the image.

The distance of vertices from the mirror line

remains the same after reflection. This implies that

vertices near the mirror line will also be near the

mirror line after reflection and vertices further

away from the mirror line will also be further away

after reflection. Basing on these ideas, the mapping

of corresponding vertices becomes:

𝐀 → 𝐏,𝐁 → 𝐑, 𝐂 →Q

The mapping of corresponding sides becomes:

AB → PR; BC → RQ; AC → PQ

We can now easily identify the mapping of

corresponding angles as:

∠𝐀 → ∠𝐏,∠𝐁 → ∠𝐑,∠𝐂 → ∠𝐐

This corresponding angles are congruent. The pairs

of congruent angles in the triangles after the

reflection become:

∠𝐀 ≅ ∠𝐏,∠𝐁 ≅ ∠𝐑, ∠𝐂 ≅ ∠𝐐

IDENTIFYING CONGRUENT ANGLES IN

TRIANGLES AFTER A ROTATION

In the grid below ΔABC has been mapped onto ΔPQRafter a rotation of −90° about point.

A

B C

P

Q R

A rotation is also rigid motion. Consequently, it is

clear that ΔABC ≅ ΔPQR . The corresponding angles

will be congruent.

Orientation is preserved under a rotation; vertices

remain in the same order.

The mapping of corresponding vertices becomes:

𝐀 → 𝐑,𝐁 → 𝐏, 𝐂 →Q

The mapping of corresponding sides becomes:

AB → RP; BC → PQ; AC → RQ

We can now easily identify the mapping of

corresponding angles as:

∠𝐀 → ∠𝐑,∠𝐁 → ∠𝐏,∠𝐂 → ∠𝐐

This corresponding angles are congruent. The pairs

of congruent angles in triangles after the rotation

become.

∠𝐀 ≅ ∠𝐑,∠𝐁 ≅ ∠𝐏,∠𝐂 ≅ ∠𝐐

IDENTIFYING CONGRUENT ANGLES IN

TRIANGLES AFTER A GLIDE REFLECTION

In the figure below ΔABC has been mapped onto

ΔPQR after a glide reflection.

A

B

C

P Q

R

𝑚

A glide reflection is also rigid motion.

Consequently, it is clear that ΔABC ≅ ΔPQR . The

corresponding angles will be congruent.

Orientation is reversed under a reflection but

preserved under translation.

The mapping of corresponding vertices becomes:

𝐀 → 𝐏,𝐁 → 𝐑, 𝐂 →Q

The mapping of corresponding sides becomes:

AB → PR; BC → RQ; AC → PQ

We can now easily identify the mapping of

corresponding angles as:

∠𝐀 → ∠𝐏,∠𝐁 → ∠𝐑,∠𝐂 → ∠𝐐

This corresponding angles are congruent to each

other. The pairs of congruent angles become:

∠𝐀 ≅ ∠𝐏,∠𝐁 ≅ ∠𝐑, ∠𝐂 ≅ ∠𝐐

Thus, we have identified congruent angles in

triangles after a glide reflection.

Example

In the figure below ΔPQR has been mapped onto

ΔXYZ after a translation. Identify all the pairs of

congruent angles.

A B

C

P Q

R

Solution

A translation preserves orientation, therefore the

order of the vertices remain the same. The

mapping of corresponding angles becomes:

∠𝐀 → ∠𝐏, ∠𝐁 → ∠𝐐,∠𝐂 → ∠R

These angles are congruent forming the following

pairs of congruent angles.

∠𝐀 ≅ ∠𝐏,∠𝐁 ≅ ∠𝐐,∠𝐂 ≅ ∠𝐑

HOMEWORK

In the figure below ΔPQR has been mapped onto

ΔXYZ after a reflection along the line 𝑚. Identify all

the pairs of congruent angles.

P

Q R

Z

X Y𝑚

ANSWERS TO HOMEWORK

∠𝑃 ≅ ∠Z∠𝑄 ≅ ∠Y∠𝑅 ≅ ∠X

THE END