Day-52-Presentation-Identifying congruent angles of a triangle · corresponding angles in congruent...
Transcript of 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